Business Statistics A Decision Making Approach 9th Groebner Shannon Fr

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9 781292 023359 ISBN 978-1-29202-335-9 Business Statistics A Decision-Making Approach Groebner Shannon Fry Ninth Edition Business Statistics Groebner Shannon Fry 9e

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Business Statistics A Decision-Making Approach Groebner Shannon Fry Ninth Edition

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Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd Saffron House 6–10 Kirby Street London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America ISBN 10: 1-292-02335-X ISBN 13: 978-1-292-02335-9 ISBN 10: 1-292-02335-X ISBN 13: 978-1-292-02335-9

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Table of Contents PEARSON C U S T OM LIBRA R Y I 1. The Where Why and How of Data Collection 1 1 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 2. Graphs Charts and Tables - Describing Y our Data 33 33 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 3. Describing Data Using Numerical Measures 87 87 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 4. Special Review Section I 143 143 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 5. Introduction to Probability 151 151 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 6. Discrete Probability Distributions 197 197 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 7. Introduction to Continuous Probability Distributions 243 243 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 8. Introduction to Sampling Distributions 277 277 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 9. Estimating Single Population Parameters 319 319 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 10. Introduction to Hypothesis Testing 363 363 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 11. Estimation and Hypothesis Testing for Two Population Parameters 417 417 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 12. Hypothesis Tests and Estimation for Population Variances 469 469 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 13. Analysis of Variance 497 497 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith

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II 14. Special Review Section II 551 551 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 15. Goodness-of-Fit Tests and Contingency Analysis 569 569 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 16. Introduction to Linear Regression and Correlation Analysis 601 601 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 17. Multiple Regression Analysis and Model Building 657 657 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 18. Analyzing and Forecasting Time-Series Data 733 733 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 19. Introduction to Nonparametric Statistics 797 797 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 20. Introduction to Quality and Statistical Process Control 831 831 David F. Groebner/Patrick W. Shannon/Phillip C. Fry/Kent D. Smith 861 861 Index

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The Where Why and How of Data Collection Quick Prep Links Locate a recent copy of a business periodical such as Fortune or Business Week and take note of the graphs charts and tables that are used in the articles and advertisements. Recall any recent experiences you have had in which you were asked to complete a written survey or respond to a telephone survey. Make sure that you have access to Excel software. Open Excel and familiarize yourself with the software. What Is Business Statistics Procedures for Collecting Data  Populations Samples and Sampling Techniques  Data Types and Data Measurement Levels  A Brief Introduction to Data Mining  Outcome 1. Know the key data collection methods. Why you need to know A transformation is taking place in many organizations involving how managers are using data to help improve their decision making. Because of the recent advances in software and database systems managers are able to analyze data in more depth than ever before. A new discipline called data mining is growing and one of the fastest-growing career areas is referred to as business intelligence. Data mining or knowledge discovery is an interdisciplinary field involving primarily computer science and statistics. People working in this field are referred to as “data scientists.” Doing an Internet search on data mining will yield a large number of sites talking about the field. In today’s workplace you can have an immediate competitive edge over other new employees and even those with more experience by applying statistical analysis skills to real-world decision making. The purpose of this text is to assist in your learning process and to complement your instructor’s efforts in conveying how to apply a variety of important statistical procedures. The major automakers such as GM Ford and Toyota maintain databases with information on production quality customer satisfaction safety records and much more. Walmart the world’s largest retail chain collects and manages mas- sive amounts of data related to the operation of its stores throughout the world. Its highly sophisticated database systems contain sales data detailed customer data employee satisfaction data and much more. Governmental agencies amass extensive data on such things as unemployment interest rates incomes and education. However access to data is not limited to large companies. The rela- tively low cost of computer hard drives with 100-gigabyte or larger capacities makes it possible for small firms and even individuals to store vast amounts of Outcome 2. Know the difference between a population and a sample. Outcome 3. Understand the similarities and differences between different sampling methods. Outcome 4. Understand how to categorize data by type and level of measurement. Data Mining The application of statistical techniques and algorithms to the analysis of large data sets. Business Intelligence The application of tools and technologies for gathering storing retrieving and analyzing data that businesses collect and use. Outcome 5. Become familiar with the concept of data mining and some of its applications. Anton Foltin/Shutterstock From Chapter 1 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved.

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The Where Why and How of Data Collection data on desktop computers. But without some way to transform the data into useful information the data these compa- nies have gathered are of little value. Transforming data into information is where business statistics comes in—the statistical procedures introduced in this text are those that are used to help transform data into information. This text focuses on the practical applica- tion of statistics we do not develop the theory you would find in a mathematical statistics course. Will you need to use math in this course Yes but mainly the concepts covered in your college algebra course. Statistics does have its own terminology. You will need to learn various terms that have special statistical mean- ing. You will also learn certain dos and don’ts related to statistics. But most importantly you will learn specific meth- ods to effectively convert data into information. Don’t try to memorize the concepts rather go to the next level of learning called understanding. Once you understand the underlying concepts you will be able to think statistically. Because data are the starting point for any statistical analysis this text is devoted to discussing various aspects of data from how to collect data to the different types of data that you will be analyzing. You need to gain an under- standing of the where why and how of data and data collection. What Is Business Statistics Articles in your local newspaper news stories on television and national publications such as the Wall Street Journal and Fortune discuss stock prices crime rates government-agency budgets and company sales and profit figures. These values are statistics but they are just a small part of the discipline called business statistics which provides a wide variety of methods to assist in data analysis and decision making. Descriptive Statistics Business statistics can be segmented into two general categories. The first category involves the procedures and techniques designed to describe data such as charts graphs and numeri- cal measures. The second category includes tools and techniques that help decision makers draw inferences from a set of data. Inferential procedures include estimation and hypothesis testing. A brief discussion of these techniques follows. BUSINESS APPLICATION DESCRIBING DATA INDEPENDENT TEXTBOOK PUBLISHING INC. Independent Textbook Publishing Inc. publishes 15 college-level texts in the business and social sciences areas. Figure 1 shows an Excel spreadsheet containing data for each of these 15 textbooks. Each column Business Statistics A collection of procedures and techniques that are used to convert data into meaningful information in a business environment. Excel 2010 Instructions: 1. Open File: Independent Textbook.xlsx. FIGURE 1 | Excel 2010 Spreadsheet of Independent Textbook Publishing Inc.

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The Where Why and How of Data Collection in the spreadsheet corresponds to a different factor for which data were collected. Each row corresponds to a different textbook. Many statistical procedures might help the owners describe these textbook data including descriptive techniques such as charts graphs and numerical measures. Charts and Graphs Other text will discuss many different charts and graphs—such as the one shown in Figure 2 called a histogram. This graph displays the shape and spread of the distribution of number of copies sold. The bar chart shown in Figure 3 shows the total num- ber of textbooks sold broken down by the two markets business and social sciences. Bar charts and histograms are only two of the techniques that could be used to graphi- cally analyze the data for the textbook publisher. BUSINESS APPLICATION DESCRIBING DATA CROWN INVESTMENTS At Crown Investments a senior analyst is preparing to present data to upper management on the 100 fastest-growing companies on the Hong Kong Stock Exchange. Figure 4 shows an Excel worksheet containing a subset of the data. The columns correspond to the different items of interest growth percentage sales and so on. The data for each company are in a single row. The entire data are in a file called Fast100. Number of Books Under 50000 50000 100000 100000 150000 150000 200000 Number of Copies Sold Independent Textbook Publishing Inc. Distribution of Copies Sold 0 1 2 3 4 5 6 7 8 FIGURE 2 | Histogram Showing the Copies Sold Distribution 0 100000 200000 300000 400000 500000 600000 700000 800000 Market Classifcation Total Copies Sold Total Copies Sold by Market Class Social Sciences Business FIGURE 3 | Bar Chart Showing Copies Sold by Sales Category

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The Where Why and How of Data Collection In addition to preparing appropriate graphs the analyst will compute important numeri- cal measures. One of the most basic and most useful measures in business statistics is one with which you are already familiar: the arithmetic mean or average. Arithmetic Mean or Average The sum of all values divided by the number of values. –99 indicates missing data Excel 2010 Instructions: 1. Open file: Fast100.xlsx. FIGURE 4 | Crown Investment Example Average The sum of all the values divided by the number of values. In equation form: Average a N i 1 x i N Sum of all data values Number of data values 1 where: N Number of data values x i ith data value The analyst may be interested in the average profit that is the average of the col- umn labeled “Profits” for the 100 companies. The total profit for the 100 companies is 3193.60 but profits are given in millions of dollars so the total profit amount is actually 3193600000. The average is found by dividing this total by the number of companies: Average +3193600000 100 +31936000 or +31.936 million The average or mean is a measure of the center of the data. In this case the ana- lyst may use the average profit as an indicator—firms with above-average profits are rated higher than firms with below-average profits. The graphical and numerical measures illustrated here are only some of the many descriptive procedures that will be introduced elsewhere. The key to remember is that the purpose of any descriptive procedure is to describe data. Your task will be to select the proce- dure that best accomplishes this. As Figure 5 reminds you the role of statistics is to convert data into meaningful information.

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The Where Why and How of Data Collection Inferential Procedures Advertisers pay for television ads based on the audience level so knowing how many viewers watch a particular program is important millions of dollars are at stake. Clearly the networks don’t check with everyone in the country to see if they watch a particular program. Instead they pay a fee to the Nielsen company http://www.nielsen.com/ which uses statistical inference procedures to estimate the number of viewers who watch a particular television program. There are two primary categories of statistical inference procedures: estimation and hypothesis testing. These procedures are closely related but serve very different purposes. Estimation In situations in which we would like to know about all the data in a large data set but it is impractical to work with all the data decision makers can use techniques to esti- mate what the larger data set looks like. The estimates are formed by looking closely at a subset of the larger data set. BUSINESS APPLICATION STATISTICAL INFERENCE NEW PRODUCT INTRODUCTION Energy-boosting drinks such as Red Bull Go Girl Monster and Full Throttle have become very popular among college students and young professionals. But how do the companies that make these products determine whether they will sell enough to warrant the product introduction A typical approach is to do market research by introducing the product into one or more test markets. People in the targeted age income and educational categories target market are asked to sample the product and indicate the likelihood that they would purchase the product. The percentage of people who say that they will buy forms the basis for an estimate of the true percentage of all people in the target market who will buy. If that estimate is high enough the company will introduce the product. Hypothesis Testing Television advertising is full of product claims. For example we might hear that “Goodyear tires will last at least 60000 miles” or that “more doctors recommend Bayer Aspirin than any other brand.” Other claims might include statements like “General Electric light bulbs last longer than any other brand” or “customers prefer McDonald’s over Burger King.” Are these just idle boasts or are they based on actual data Probably some of both However consumer research organizations such as Consumers Union publisher of Consumer Reports regularly test these types of claims. For example in the hamburger case Consumer Reports might select a sample of customers who would be asked to blind taste test Burger King’s and McDonald’s hamburgers under the hypoth- esis that there is no difference in customer preferences between the two restaurants. If the sample data show a substantial difference in preferences then the hypothesis of no differ- ence would be rejected. If only a slight difference in preferences was detected then Con- sumer Reports could not reject the hypothesis. Statistical Inference Procedures Procedures that allow a decision maker to reach a conclusion about a set of data based on a subset of that data. FIGURE 5 | The Role of Business Statistics Information Data Descriptive Inferential Statistical Procedures

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The Where Why and How of Data Collection MyStatLab Journal. Find three examples of the use of a graph to display data. For each graph a. Give the name date and page number of the periodical in which the graph appeared. b. Describe the main point made by the graph. c. Analyze the effectiveness of the graphs. 1-12. The human resources manager of an automotive supply store has collected the following data showing the number of employees in each of five categories by the number of days missed due to illness or injury during the past year. Missed Days 0–2 days 3–5 days 6–8 days 8–10 days Employees 159 67 32 10 Construct the appropriate chart for these data. Be sure to use labels and to add a title to your chart. 1-13. Suppose Fortune would like to determine the average age and income of its subscribers. How could statistics be of use in determining these values 1-14. Locate an example from a business periodical or newspaper in which estimation has been used. a. What specifically was estimated b. What conclusion was reached using the estimation c. Describe how the data were extracted and how they were used to produce the estimation. d. Keeping in mind the goal of the estimation discuss whether you believe that the estimation was successful and why. e. Describe what inferences were drawn as a result of the estimation. 1-15. Locate one of the online job Web sites and pick several job listings. For each job type discuss one or more situations in which statistical analyses would be used. Base your answer on research Internet business periodicals personal interviews etc.. Indicate whether the situations you are describing involve descriptive statistics or inferential statistics or a combination of both. 1-16. Suppose Super-Value a major retail food company is thinking of introducing a new product line into a market area. It is important to know the age characteristics of the people in the market area. a. If the executives wish to calculate a number that would characterize the “center” of the age data what statistical technique would you suggest Explain your answer. b. The executives need to know the percentage of people in the market area that are senior citizens. Name the basic category of statistical procedure they would use to determine this information. c. Describe a hypothesis the executives might wish to test concerning the percentage of senior citizens in the market area. Skill Development 1-1. For the following situation indicate whether the statistical application is primarily descriptive or inferential. “The manager of Anna’s Fabric Shop has collected data for 10 years on the quantity of each type of dress fabric that has been sold at the store. She is interested in making a presentation that will illustrate these data effectively.” 1-2. Consider the following graph that appeared in a company annual report. What type of graph is this Explain. 45000 40000 35000 30000 25000 20000 15000 10000 5000 Fruit Vegetables Meat and Poultry FOOD STORE SALES Monthly Sales Canned Goods Department Cereal and Dry Goods Other 0 1-3. Review Figures 2 and 3 and discuss any differences you see between the histogram and the bar chart. 1-4. Think of yourself as working for an advertising firm. Provide an example of how hypothesis testing can be used to evaluate a product claim. 1-5. Define what is meant by hypothesis testing. Provide an example in which you personally have tested a hypothesis even if you didn’t use formal statistical techniques to do so. 1-6. In what situations might a decision maker need to use statistical inference procedures 1-7. Explain under what circumstances you would use hypothesis testing as opposed to an estimation procedure. 1-8. Discuss any advantages a graph showing a whole set of data has over a single measure such as an average. 1-9. Discuss any advantages a single measure such as an average has over a table showing a whole set of data. Business Applications 1-10. Describe how statistics could be used by a business to determine if the dishwasher parts it produces last longer than a competitor’s brand. 1-11. Locate a business periodical such as Fortune or Forbes or a business newspaper such as The Wall Street 1- Exercises END EXERCISES 1-1

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The Where Why and How of Data Collection Procedures for Collecting Data We have defined business statistics as a set of procedures that are used to transform data into information. Before you learn how to use statistical procedures it is important that you become familiar with different types of data collection methods. Data Collection Methods There are many methods and procedures available for collecting data. The following are con- sidered some of the most useful and frequently used data collection methods: ● Experiments ● Telephone surveys ● Written questionnaires and surveys ● Direct observation and personal interviews BUSINESS APPLICATION EXPERIMENTS FOOD PROCESSING A company often must conduct a specific experiment or set of experiments to get the data managers need to make informed decisions. For example Lamb Weston McCain and the J. R. Simplot Company are the primary suppliers of french fries to McDonald’s in North America. At its Caldwell Idaho factory the J. R. Simplot Company has a test center that among other things houses a mini french fry plant used to conduct experiments on its potato manufacturing process. McDonald’s has strict standards on the quality of the french fries it buys. One important attribute is the color of the fries after cooking. They should be uniformly “golden brown”—not too light or too dark. French fries are made from potatoes that are peeled sliced into strips blanched partially cooked and then freeze-dried—not a simple process. Because potatoes differ in many ways such as sugar content and moisture blanching time cooking temperature and other factors vary from batch to batch. Simplot employees start their experiments by grouping the raw potatoes into batches with similar characteristics. They run some of the potatoes through the line with blanch time and temperature settings set at specific levels defined by an experimental design. After measuring one or more output variables for that run employees change the settings and run another batch again measuring the output variables. Figure 6 shows a typical data collection form. The output variable for example percent- age of fries without dark spots for each combination of potato category blanch time and temperature is recorded in the appropriate cell in the table. Experiment A process that produces a single outcome whose result cannot be predicted with certainty. Experimental Design A plan for performing an experiment in which the variable of interest is defined. One or more factors are identified to be manipulated changed or observed so that the impact or influence on the variable of interest can be measured or observed. Chapter Outcome 1. FIGURE 6 | Data Layout for the French Fry Experiment Potato Category 1 234 Blanch Temperature Blanch Time 100 110 120 10 minutes 100 110 120 15 minutes 100 110 120 20 minutes 100 110 120 25 minutes

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The Where Why and How of Data Collection BUSINESS APPLICATION TELEPHONE SURVEYS PUBLIC ISSUES Chances are that you have been on the receiving end of a telephone call that begins something like: “Hello. My name is Mary Jane and I represent the XYZ organization. I am conducting a survey on …” Political groups use telephone surveys to poll people about candidates and issues. Marketing research companies use phone surveys to learn likes and dislikes of potential customers. Telephone surveys are a relatively inexpensive and efficient data collection procedure. Of course some people will refuse to respond to a survey others are not home when the calls come and some people do not have home phones—only have a cell phone—or cannot be reached by phone for one reason or another. Figure 7 shows the major steps in conducting a telephone survey. This example survey was run a few years ago by a Seattle television sta- tion to determine public support for using tax dollars to build a new football stadium for the National Football League’s Seattle Seahawks. The survey was aimed at property tax payers only. Because most people will not stay on the line very long the phone survey must be short—usually one to three minutes. The questions are generally what are called closed-end questions. For example a closed-end question might be “To which political party do you belong Republican Democrat Or other” The survey instrument should have a short statement at the beginning explaining the purpose of the survey and reassuring the respondent that his or her responses will remain confidential. The initial section of the survey should contain questions relating to the central issue of the survey. The last part of the survey should contain demographic questions such as gender income level education level that will allow you to break down the responses and look deeper into the survey results. Closed-End Questions Questions that require the respondent to select from a short list of defined choices. Demographic Questions Questions relating to the respondents’ characteristics backgrounds and attributes. FIGURE 7 | Major Steps for a Telephone Survey Determine Sample Size and Sampling Method Pretest the Survey Defne the Population of Interest Select Sample and Make Calls Develop Survey Questions Defne the Issue Do taxpayers favor a special bond to build a new football stadium for the Seahawks If so should the Seahawks’ owners share the cost Population is all residential property tax payers in King County Washington. Te survey will be conducted among this group only. Limit the number of questions to keep survey short. Ask important questions frst. Provide specifc response options when possible. Establish eligibility. “Do you own a residence in King County” Add demographic questions at the end: age income etc. Introduction should explain purpose of survey and who is conducting it—stress that answers are anonymous. Try the survey out on a small group from the population. Check for length clarity and ease of conducting. Have we forgotten anything Make changes if needed. Get phone numbers from a computer-generated or “current” list. Develop “callback” rule for no answers. Callers should be trained to ask questions fairly. Do not lead the respondent. Record responses on data sheet. Sample size is dependent on how confdent we want to be of our results how precise we want the results to be and how much opinions difer among the population members. Various sampling methods are available.

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The Where Why and How of Data Collection A survey budget must be considered. For example if you have 3000 to spend on calls and each call costs 10 to make you obviously are limited to making 300 calls. However keep in mind that 300 calls may not result in 300 usable responses. The phone survey should be conducted in a short time period. Typically the prime calling time for a voter survey is between 7:00 p.m. and 9:00 p.m. However some people are not home in the evening and will be excluded from the survey unless there is a plan for conducting callbacks. Written Questionnaires and Surveys The most frequently used method to collect opinions and factual data from people is a written questionnaire. In some instances the questionnaires are mailed to the respondent. In others they are administered directly to the potential respondents. Written questionnaires are generally the least expensive means of collecting survey data. If they are mailed the major costs include postage to and from the respondents questionnaire development and printing costs and data anal- ysis. Figure 8 shows the major steps in conducting a written survey. Note how written surveys are similar to telephone surveys however written surveys can be slightly more involved and therefore take more time to complete than those used for a telephone survey. However you must be careful to construct a questionnaire that can be easily completed without requiring too much time. A written survey can contain both closed-end and open-end questions. Open-end ques- tions provide the respondent with greater flexibility in answering a question however the responses can be difficult to analyze. Note that telephone surveys can use open-end ques- tions too. However the caller may have to transcribe a potentially long response and there is risk that the interviewees’ comments may be misinterpreted. Written surveys also should be formatted to make it easy for the respondent to provide accurate and reliable data. This means that proper space must be provided for the responses Open-End Questions Questions that allow respondents the freedom to respond with any value words or statements of their own choosing. FIGURE 8 | Written Survey Steps Determine Sample Size and Sampling Method Pretest the Survey Defne the Population of Interest Select Sample and Send Surveys Design the Survey Instrument Defne the Issue Clearly state the purpose of the survey. Defne the objectives. What do you want to learn from the survey Make sure there is agreement before you proceed. Defne the overall group of people to be potentially included in the survey and obtain a list of names and addresses of those individuals in this group. Limit the number of questions to keep the survey short. Ask important questions frst. Provide specifc response options when possible. Add demographic questions at the end: age income etc. Introduction should explain purpose of survey and who is conducting it—stress that answers are anonymous. Layout of the survey must be clear and attractive. Provide location for responses. Try the survey out on a small group from the population. Check for length clarity and ease of conducting. Have we forgotten anything Make changes if needed. Mail survey to a subset of the larger group. Include a cover letter explaining the purpose of the survey. Include pre-stamped return envelope for returning the survey. Sample size is dependent on how confdent we want to be of our results how precise we want the results to be and how much opinions difer among the population members. Various sampling methods are available.

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The Where Why and How of Data Collection and the directions must be clear about how the survey is to be completed. A written survey needs to be pleasing to the eye. How it looks will affect the response rate so it must look professional. Y ou also must decide whether to manually enter or scan the data gathered from your writ- ten survey. The survey design will be affected by the approach you take. If you are adminis- tering a large number of surveys scanning is preferred. It cuts down on data entry errors and speeds up the data gathering process. However you may be limited in the form of responses that are possible if you use scanning. If the survey is administered directly to the desired respondents you can expect a high response rate. For example you probably have been on the receiving end of a written survey many times in your college career when you were asked to fill out a course evaluation form at the end of the term. Most students will complete the form. On the other hand if a survey is administered through the mail you can expect a low response rate—typically 5 to 20. Therefore if you want 200 responses you should mail out 1000 to 4000 questionnaires. Overall written surveys can be a low-cost effective means of collecting data if you can overcome the problems of low response. Be careful to pretest the survey and spend extra time on the format and look of the survey instrument. Developing a good written questionnaire or telephone survey instrument is a major chal- lenge. Among the potential problems are the following: ● Leading questions Example: “Do you agree with most other reasonably minded people that the city should spend more money on neighborhood parks” Issue: In this case the phrase “Do you agree” may suggest that you should agree. Also by suggesting that “most reasonably minded people” already agree the respondent might be compelled to agree so that he or she can also be consid- ered “reasonably minded.” Improvement: “In your opinion should the city increase spending on neighbor- hood parks” Example: “To what extent would you support paying a small increase in your prop- erty taxes if it would allow poor and disadvantaged children to have food and shelter” Issue: The question is ripe with emotional feeling and may imply that if you don’t support additional taxes you don’t care about poor children. Improvement: “Should property taxes be increased to provide additional funding for social services” ● Poorly worded questions Example: “How much money do you make at your current job” Issue: The responses are likely to be inconsistent. When answering does the respondent state the answer as an hourly figure or as a weekly or monthly total Also many people refuse to answer questions regarding their income. Improvement: “Which of the following categories best reflects your weekly income from your current job Under 500 500–1000 Over 1000” Example: “After trying the new product please provide a rating from 1 to 10 to indicate how you like its taste and freshness.” Issue: First is a low number or a high number on the rating scale considered a positive response Second the respondent is being asked to rate two factors taste and freshness in a single rating. What if the product is fresh but does not taste good Improvement: “After trying the new product please rate its taste on a 1 to 10 scale with 1 being best. Also rate the product’s freshness using the same 1 to 10 scale. Taste Freshness” The way a question is worded can influence the responses. Consider an example that occurred in September 2008 during the financial crisis that resulted from the sub-prime

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The Where Why and How of Data Collection mortgage crisis and bursting of the real estate bubble. Three surveys were conducted on the same basic issue. The following questions were asked: “Do you approve or disapprove of the steps the Federal Reserve and Treasury Depart- ment have taken to try to deal with the current situation involving the stock market and major financial institutions” ABC News/Washington Post 44 Approve — 42 Disappro v e —14 Unsure “Do you think the government should use taxpayers’ dollars to rescue ailing private financial firms whose collapse could have adverse effects on the economy and market or is it not the government’s responsibility to bail out private companies with taxpayer dollars” LA Times/Bloomberg 31 Use Tax Payers’ Dollars — 55 Not Government’s Responsibility— 14 Unsure “As you may know the government is potentially investing billions to try and keep financial institutions and markets secure. Do you think this is the right thing or the wrong thing for the government to be doing” Pew Research Center 57 Right Thing — 30 Wrong Thing—13 Unsure Note the responses to each of these questions. The way the question is worded can affect the responses. Direct Observation and Personal Interviews Direct observation is another procedure that is often used to collect data. As implied by the name this technique requires the pro- cess from which the data are being collected to be physically observed and the data recorded based on what takes place in the process. Possibly the most basic way to gather data on human behavior is to watch people. If you are trying to decide whether a new method of displaying your product at the supermarket will be more pleasing to customers change a few displays and watch customers’ reactions. If as a member of a state’s transportation department you want to determine how well motorists are complying with the state’s seat belt laws place observers at key spots throughout the state to monitor people’s seat belt habits. A movie producer seeking information on whether a new movie will be a success holds a preview showing and observes the reactions and com- ments of the movie patrons as they exit the screening. The major constraints when collecting observations are the amount of time and money required. For observations to be effective trained observers must be used which increases the cost. Personal observation is also time- consuming. Finally personal perception is subjective. There is no guarantee that different observers will see a situation in the same way much less report it the same way. Personal interviews are often used to gather data from people. Interviews can be either structured or unstructured depending on the objectives and they can utilize either open-end or closed-end questions. Regardless of the procedure used for data collection care must be taken that the data collected are accurate and reliable and that they are the right data for the purpose at hand. Other Data Collection Methods Data collection methods that take advantage of new technologies are becoming more prev- alent all the time. For example many people believe that Walmart is one of the best com- panies in the world at collecting and using data about the buying habits of its customers. Most of the data are collected automatically as checkout clerks scan the UPC bar codes on the products customers purchase. Not only are Walmart’s inventory records automatically updated but information about the buying habits of customers is also recorded. This allows Walmart to use analytics and data mining to drill deep into the data to help with its deci- sion making about many things including how to organize its stores to increase sales. For instance Walmart apparently decided to locate beer and disposable diapers close together when it discovered that many male customers also purchase beer when they are sent to the store for diapers. Bar code scanning is used in many different data collection applications. In a DRAM dynamic random-access memory wafer fabrication plant batches of silicon wafers have bar codes. As the batch travels through the plant’s workstations its progress and quality are tracked through the data that are automatically obtained by scanning. Unstructured Interview Interviews that begin with one or more broadly stated questions with further questions being based on the responses. Structured Interview Interviews in which the questions are scripted.

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The Where Why and How of Data Collection Every time you use your credit card data are automatically collected by the retailer and the bank. Computer information systems are developed to store the data and to provide decision makers with procedures to access the data. In many instances your data collection method will require you to use physical measurement. For example the Andersen Window Company has quality analysts physically measure the width and height of its windows to assure that they meet customer specifications and a state Department of Weights and Measures will physically test meat and produce scales to determine that customers are being properly charged for their purchases. Data Collection Issues Data Accuracy When you need data to make a decision we suggest that you first see if appropriate data have already been collected because it is usually faster and less expensive to use existing data than to collect data yourself. However before you rely on data that were collected by someone else for another purpose you need to check out the source to make sure that the data were collected and recorded properly. Such organizations as Bloomberg V alue Line and Fortune have built their reputations on providing quality data. Although data errors are occasionally encountered they are few and far between. You really need to be concerned with data that come from sources with which you are not familiar. This is an issue for many sources on the World Wide Web. Any organiza- tion or any individual can post data to the Web. Just because the data are there doesn’t mean they are accurate. Be careful. Interviewer Bias There are other general issues associated with data collection. One of these is the potential for bias in the data collection. There are many types of bias. For exam- ple in a personal interview the interviewer can interject bias either accidentally or on pur- pose by the way she asks the questions by the tone of her voice or by the way she looks at the subject being interviewed. We recently allowed ourselves to be interviewed at a trade show. The interviewer began by telling us that he would only get credit for the interview if we answered all of the questions. Next he asked us to indicate our satisfaction with a particular display. He wasn’t satisfied with our less-than-enthusiastic rating and kept asking us if we really meant what we said. He even asked us if we would consider upgrading our rating How reliable do you think these data will be Nonresponse Bias Another source of bias that can be interjected into a survey data collection process is called nonresponse bias. We stated earlier that mail surveys suffer from a high percentage of unreturned surveys. Phone calls don’t always get through or people refuse to answer. Subjects of personal interviews may refuse to be interviewed. There is a potential problem with nonresponse. Those who respond may provide data that are quite different from the data that would be supplied by those who choose not to respond. If you aren’t careful the responses may be heavily weighted by people who feel strongly one way or another on an issue. Selection Bias Bias can be interjected through the way subjects are selected for data collection. This is referred to as selection bias. A study on the virtues of increasing the stu- dent athletic fee at your university might not be best served by collecting data from students attending a football game. Sometimes the problem is more subtle. If we do a telephone sur- vey during the evening hours we will miss all of the people who work nights. Do they share the same views income education levels and so on as people who work days If not the data are biased. Written and phone surveys and personal interviews can also yield flawed data if the inter- viewees lie in response to questions. For example people commonly give inaccurate data about such sensitive matters as income. Lying is also an increasing problem with exit polls in which voters are asked who they voted for immediately after casting their vote. Sometimes the data errors are not due to lies. The respondents may not know or have accurate informa- tion to provide the correct answer. Observer Bias Data collection through personal observation is also subject to problems. People tend to view the same event or item differently. This is referred to as observer bias. Bias An effect that alters a statistical result by systematically distorting it different from a random error which may distort on any one occasion but balances out on the average.

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The Where Why and How of Data Collection One area in which this can easily occur is in safety check programs in companies. An impor- tant part of behavioral-based safety programs is the safety observation. Trained data collec- tors periodically conduct a safety observation on a worker to determine what if any unsafe acts might be taking place. We have seen situations in which two observers will conduct an observation on the same worker at the same time yet record different safety data. This is especially true in areas in which judgment is required on the part of the observer such as the distance a worker is from an exposed gear mechanism. People judge distance differently. Measurement Error A few years ago we were working with a wood window manufac- turer. The company was having a quality problem with one of its saws. A study was devel- oped to measure the width of boards that had been cut by the saw. Two people were trained to use digital calipers and record the data. This caliper is a U-shaped tool that measures distance in inches to three decimal places. The caliper was placed around the board and squeezed tightly against the sides. The width was indicated on the display. Each person measured 500 boards during an 8-hour day. When the data were analyzed it looked like the widths were coming from two different saws one set showed considerably narrower widths than the other. Upon investigation we learned that the person with the narrower width measurements was pressing on the calipers much more firmly. The soft wood reacted to the pressure and gave narrower readings. Fortunately we had separated the data from the two data collectors. Had they been merged the measurement error might have gone undetected. Internal Validity When data are collected through experimentation you need to make sure that proper controls have been put in place. For instance suppose a drug company such as Pfizer is conducting tests on a drug that it hopes will reduce cholesterol. One group of test participants is given the new drug while a second group a control group is given a placebo. Suppose that after several months the group using the drug saw significant cholesterol reduc- tion. For the results to have internal validity the drug company would have had to make sure the two groups were controlled for the many other factors that might affect cholesterol such as smoking diet weight gender race and exercise habits. Issues of internal validity are generally addressed by randomly assigning subjects to the test and control groups. However if the extraneous factors are not controlled there could be no assurance that the drug was the factor influencing reduced cholesterol. For data to have internal validity the extraneous factors must be controlled. External Validity Even if experiments are internally valid you will always need to be con- cerned that the results can be generalized beyond the test environment. For example if the cholesterol drug test had been performed in Europe would the same basic results occur for people in North America South America or elsewhere For that matter the drug company would also be interested in knowing whether the results could be replicated if other subjects are used in a similar experiment. If the results of an experiment can be replicated for groups different from the original population then there is evidence the results of the experiment have external validity. An extensive discussion of how to measure the magnitude of bias and how to reduce bias and other data collection problems is beyond the scope of this text. However you should be aware that data may be biased or otherwise flawed. Always pose questions about the potential for bias and determine what steps have been taken to reduce its effect. Internal Validity A characteristic of an experiment in which data are collected in such a way as to eliminate the effects of variables within the experimental environment that are not of interest to the researcher. External Validity A characteristic of an experiment whose results can be generalized beyond the test environment so that the outcomes can be replicated when the experiment is repeated. Skill Development 1-17. If a pet store wishes to determine the level of customer satisfaction with its services would it be appropriate to conduct an experiment Explain. 1-18. Define what is meant by a leading question. Provide an example. 1-19. Briefly explain what is meant by an experiment and an experimental design. 1- Exercises MyStatLab

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The Where Why and How of Data Collection day you receive an e-mail containing a questionnaire asking you to rate the quality of the experience. Discuss both the advantages and disadvantages of using this form of questionnaire delivery. 1-28. In your capacity as assistant sales manager for a large office products retailer you have been assigned the task of interviewing purchasing managers for medium and large companies in the San Francisco Bay area. The objective of the interview is to determine the office product buying plans of the company in the coming year. Develop a personal interview form that asks both issue-related questions as well as demographic questions. 1-29. The regional manager for Macy’s is experimenting with two new end-of-aisle displays of the same product. An end-of-aisle display is a common method retail stores use to promote new products. You have been hired to determine which is more effective. Two measures you have decided to track are which display causes the highest percentage of people to stop and for those who stop which causes people to view the display the longest. Discuss how you would gather such data. 1-30. In your position as general manager for United Fitness Center you have been asked to survey the customers of your location to determine whether they want to convert the racquetball courts to an aerobic exercise space. The plan calls for a written survey to be handed out to customers when they arrive at the fitness center. Your task is to develop a short questionnaire with at least three “issue” questions and at least three demographic questions. You also need to provide the finished layout design for the questionnaire. 1-31. According to a national CNN/USA/Gallup survey of 1025 adults conducted March 14–16 2008 63 say they have experienced a hardship because of rising gasoline prices. How do you believe the survey was conducted and what types of bias could occur in the data collection process 1-20. Refer to the three questions discussed in this section involving the financial crises of 2008 and 2009 and possible government intervention. Note that the questions elicited different responses. Discuss the way the questions were worded and why they might have produced such different results. 1-21. Suppose a survey is conducted using a telephone survey method. The survey is conducted from 9 a.m. to 11 a.m. on Tuesday. Indicate what potential problems the data collectors might encounter. 1-22. For each of the following situations indicate what type of data collection method you would recommend and discuss why you have made that recommendation: a. collecting data on the percentage of bike riders who wear helmets b. collecting data on the price of regular unleaded gasoline at gas stations in your state c. collecting data on customer satisfaction with the service provided by a major U.S. airline 1-23. Assume you have received a class assignment to determine the attitude of students in your school toward the school’s registration process. What are the validity issues you should be concerned with Business Applications 1-24. According to a report issued by the U.S. Department of Agriculture USDA the agency estimates that Southern fire ants spread at a rate of 4 to 5 miles a year. What data collection method do you think was used to collect this data Explain your answer. 1-25. Suppose you are asked to survey students at your university to determine if they are satisfied with the food service choices on campus. What types of biases must you guard against in collecting your data 1-26. Briefly describe how new technologies can assist businesses in their data collection efforts. 1-27. Assume you have used an online service such as Orbitz or Travelocity to make an airline reservation. The following END EXERCISES 1-2 Populations Samples and Sampling Techniques Populations and Samples Two of the most important terms in statistics are population and sample. The list of all objects or individuals in the population is referred to as the frame. Each object or individual in the frame is known as a sampling unit. The choice of the frame depends on what objects or individuals you wish to study and on the availability of the list of these objects or individuals. Once the frame is defined it forms the list of sampling units. The next example illustrates this concept. BUSINESS APPLICATION POPULATIONS AND SAMPLES U.S. BANK We can use U.S. Bank to illustrate the difference between a population and a sample. U.S. Bank is very concerned about the time customers spend waiting in the drive-up teller line. At a particular U.S. Bank on a given day 347 cars arrived at the drive-up. Population The set of all objects or individuals of interest or the measurements obtained from all objects or individuals of interest. Sample A subset of the population. Chapter Outcome 2.

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The Where Why and How of Data Collection A population includes measurements made on all the items of interest to the data gath- erer. In our example the U.S. Bank manager would define the population as the waiting time for all 347 cars. The list of these cars possibly by license number forms the frame. If she examines the entire population she is taking a census. But suppose 347 cars are too many to track. The U.S. Bank manager could instead select a subset of these cars called a sample. The manager could use the sample results to make statements about the population. For example she might calculate the average waiting time for the sample of cars and then use that to con- clude what the average waiting time is for the population. There are trade-offs between taking a census and taking a sample. Usually the main trade-off is whether the information gathered in a census is worth the extra cost. In organiza- tions in which data are stored on computer files the additional time and effort of taking a census may not be substantial. However if there are many accounts that must be manually checked a census may be impractical. Another consideration is that the measurement error in census data may be greater than in sample data. A person obtaining data from fewer sources tends to be more complete and thorough in both gathering and tabulating the data. As a result with a sample there are likely to be fewer human errors. Parameters and Statistics Descriptive numerical measures such as an average or a pro- portion that are computed from an entire population are called parameters. Corresponding measures for a sample are called statistics. Suppose in the previous example the U.S. Bank manager timed every car that arrived at the drive-up teller on a particular day and calculated the average. This population average waiting time would be a parameter. However if she selected a sample of cars from the population the average waiting time for the sampled cars would be a statistic. Sampling Techniques Once a manager decides to gather information by sampling he or she can use a sampling technique that falls into one of two categories: statistical or nonstatistical. Both nonstatistical and statistical sampling techniques are commonly used by decision makers. Regardless of which technique is used the decision maker has the same objective— to obtain a sample that is a close representative of the population. There are some advantages to using a statistical sampling technique as we will discuss many times throughout this text. However in many cases nonstatistical sampling represents the only feasible way to sample as illustrated in the following example. BUSINESS APPLICATION NONSTATISTICAL SAMPLING SUN-CITRUS ORCHARDS Sun-Citrus Orchards owns and operates a large fruit orchard and fruit-packing plant in Florida. During harvest time in the orange grove pickers load 20-pound sacks with oranges which are then transported to the packing plant. At the packing plant the oranges are graded and boxed for shipping nationally and internationally. Because of the volume of oranges involved it is impossible to assign a quality grade to each individual orange. Instead as each sack moves up the conveyor into the packing plant a quality manager selects an orange sack every so often grades the individual oranges in the sack as to size color and so forth and then assigns an overall quality grade to the entire shipment from which the sample was selected. Because of the volume of oranges the quality manager at Sun-Citrus uses a nonstatis- tical sampling method called convenience sampling. In doing so the quality manager is willing to assume that orange quality size color etc. is evenly spread throughout the many sacks of oranges in the shipment. That is the oranges in the sacks selected are of the same quality as those that were not inspected. There are other nonstatistical sampling methods such as judgment sampling and ratio sampling which are not discussed here. Instead the most frequently used statistical sampling techniques will now be discussed. Census An enumeration of the entire set of measurements taken from the whole population. Statistical Sampling Techniques Those sampling methods that use selection techniques based on chance selection. Nonstatistical Sampling Techniques Those methods of selecting samples using convenience judgment or other nonchance processes. Convenience Sampling A sampling technique that selects the items from the population based on accessibility and ease of selection. Elena Elisseeva/Shutterstock

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The Where Why and How of Data Collection Statistical Sampling Statistical sampling methods also called probability sampling allow every item in the population to have a known or calculable chance of being included in the sample. The fundamental statistical sample is called a simple random sample. Other types of statistical sampling discussed in this text include stratified random sampling systematic sampling and cluster sampling. BUSINESS APPLICATION SIMPLE RANDOM SAMPLING CABLE-ONE A salesperson at Cable-One wishes to estimate the percentage of people in a local subdivision who have satellite television service such as Direct TV. The result would indicate the extent to which the satellite industry has made inroads into Cable-One’s market. The population of interest consists of all families living in the subdivision. For this example we simplify the situation by saying that there are only five families in the subdivision: James Sanchez Lui White and Fitzpatrick. We will let N represent the pop- ulation size and n the sample size. From the five families N 5 we select three n 3 for the sample. There are 10 possible samples of size 3 that could be selected. James Sanchez Lui James Sanchez White James Sanchez Fitzpatrick James Lui White James Lui Fitzpatrick James White Fitzpatrick Sanchez Lui White Sanchez Lui Fitzpatrick Sanchez White Fitzpatrick Lui White Fitzpatrick Note that no family is selected more than once in a given sample. This method is called sam- pling without replacement and is the most commonly used method. If the families could be selected more than once the method would be called sampling with replacement. Simple random sampling is the method most people think of when they think of ran- dom sampling. In a correctly performed simple random sample each of these samples would have an equal chance of being selected. For the Cable-One example a simplified way of selecting a simple random sample would be to put each sample of three names on a piece of paper in a bowl and then blindly reach in and select one piece of paper. However this method would be difficult if the number of possible samples were large. For example if N 50 and a sample of size n 10 is to be selected there are more than 10 billion possible samples. Try finding a bowl big enough to hold those Simple random samples can be obtained in a variety of ways. We present two examples to illustrate how simple random samples are selected in practice. BUSINESS APPLICATION RANDOM NUMBERS STATE SOCIAL SERVICES Suppose the state director for a Midwestern state’s social services system is considering changing the timing on food stamp distribution from once a month to once every two weeks. Before making any decisions he wants to survey a sample of 100 citizens who are on food stamps in a particular county from the 300 total food stamp recipients in that county. He first assigns recipients a number 001 to 300. He can then use the random number function in Excel to determine which recipients to include in the sample. Figure 9 shows the results when Excel chooses 10 random numbers. The first recipient sampled is number 115 followed by 31 and so forth. The important thing to remember is that assigning each recipient a number and then randomly selecting a sample from those numbers gives each possible sample an equal chance of being selected. RANDOM NUMBERS TABLE If you don’t have access to computer software such as Excel the items in the population to be sampled can be determined by using the random numbers table. Begin by selecting a starting point in the random numbers table row and digit. Suppose we use row 5 digit 8 as the starting point. Go down 5 rows and over 8 digits. Verify that the digit in this location is 1. Ignoring the blanks between columns that are there only to make the table more readable the first three-digit number is 149. Recipient number 149 is the first one selected in the sample. Each subsequent random number is obtained from the random numbers in the next row down. For instance the second number is 127. The procedure continues selecting numbers from top to bottom in each subsequent column. Numbers exceeding 300 and duplicate numbers are skipped. When enough numbers are Simple Random Sampling A method of selecting items from a population such that every possible sample of a specified size has an equal chance of being selected. Chapter Outcome 3. Excel Tutorial Excel tutorials

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The Where Why and How of Data Collection found for the desired sample size the process is completed. Food-stamp recipients whose numbers are chosen are then surveyed. BUSINESS APPLICATION STRATIFIED RANDOM SAMPLING FEDERAL RESERVE BANK Sometimes the sample size required to obtain a needed level of information from a simple random sampling may be greater than our budget permits. At other times it may take more time to collect than is available. Stratified random sampling is an alternative method that has the potential to provide the desired information with a smaller sample size. The following example illustrates how stratified sampling is performed. Each year the Federal Reserve Board asks its staff to estimate the total cash holdings of U.S. financial institutions as of July 1. The staff must base the estimate on a sample. Note that not all financial institutions banks credit unions and the like are the same size. A majority are small some are medium sized and only a few are large. However the few large institu- tions have a substantial percentage of the total cash on hand. To make sure that a simple random sample includes an appropriate number of small medium and large institutions the sample size might have to be quite large. As an alternative to the simple random sample the Federal Reserve staff could divide the institutions into three groups called strata: small medium and large. Staff members could then select a simple random sample of institutions from each stratum and estimate the total cash on hand for all institutions from this combined sample. Figure 10 shows the stratified random sampling concept. Note that the combined sample size n 1 + n 2 + n 3 is the sum of the simple random samples taken from each stratum. The key behind stratified sampling is to develop a stratum for each characteristic of inter- est such as cash on hand that has items that are quite homogeneous. In this example the size of the financial institution may be a good factor to use in stratifying. Here the combined sample size n 1 + n 2 + n 3 will be less than the sample size that would have been required if no stratification had occurred. Because sample size is directly related to cost in both time and money a stratified sample can be more cost effective than a simple random sample. Multiple layers of stratification can further reduce the overall sample size. For example the Federal Reserve might break the three strata in Figure 10 into substrata based on type of institution: state bank interstate bank credit union and so on. Most large-scale market research studies use stratified random sampling. The well-known political polls such as the Gallup and Harris polls use this technique also. For instance the Gallup poll typically samples between 1800 and 2500 people nationwide to estimate how more than 60 million people will vote in a presidential election. We encourage you to go to the Web site http://www.gallup.com/poll/101872/how-does-gallup-polling-work.aspx to read a very good discussion about how the Gallup polls are conducted. The Web site discusses how samples are selected and many other interesting issues associated with polling. Stratified Random Sampling A statistical sampling method in which the population is divided into subgroups called strata so that each population item belongs to only one stratum. The objective is to form strata such that the population values of interest within each stratum are as much alike as possible. Sample items are selected from each stratum using the simple random sampling method. Excel 2010 Instructions: 1. On the Data tab click Data Analysis. 2. Select Random Number Generation option. 3. Set the Number of Random Numbers to 10. 4. Select Uniform as the distribution. 5. Defne range as between 1 and 300. 6. Indicate that the results are to go in cell A1. 7. Click OK. To convert numbers to integers select the data in column A and on the Home tab in the Number group. Click the Decrease decimal button several times to remove the decimal places. FIGURE 9 | Excel 2010 Output of Random Numbers for State Social Services Example Jonathan Larsen/Shutterstock

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The Where Why and How of Data Collection BUSINESS APPLICATION SYSTEMATIC RANDOM SAMPLING STATE UNIVERSITY ASSOCIATED STUDENTS A few years ago elected student council officers at mid-sized state university in the Northeast decided to survey fellow students on the issue of the legality of carrying firearms on campus. To determine the opinion of its 20000 students a questionnaire was sent to a sample of 500 students. Although simple random sampling could have been used an alternative method called systematic random sampling was chosen. The university’s systematic random sampling plan called for it to send the question- naire to every 40th student 20000500 40 from an alphabetic list of all students. The process could begin by using Excel to generate a single random number in the range 1 to 40. Suppose this value was 25. The 25th student in the alphabetic list would be selected. After that every 40th students would be selected 25 65 105 145 . . . until there were 500 students selected. Systematic sampling is frequently used in business applications. Use it as an alternative to simple random sampling only when you can assume the population is randomly ordered with respect to the measurement being addressed in the survey. In this case students’ views on firearms on campus are likely unrelated to the spelling of their last name. BUSINESS APPLICATION CLUSTER SAMPLING OAKLAND RAIDERS FOOTBALL TEAM The Oakland Raiders of the National Football League plays its home games at O.co formerly Overstock.com Coliseum in Oakland California. Despite its struggles to win in recent years the team has a passionate fan base. Recently an outside marketing group was retained by the Raiders to interview season ticket holders about the potential for changing how season ticket pricing is structured. The Oakland Raiders Web site http://www.raiders.com/tickets/seating-price-map.html shows the layout of the O.co Coliseum. The marketing firm plans to interview season ticket holders just prior to home games during the current season. One sampling technique is to select a simple random sample of size n from the population of all season ticket holders. Unfortunately this technique would likely require that interviewers go to each section in the stadium. This would prove to be an expensive and time-consuming process. A systematic or stratified sampling procedure also would probably require visiting each section in the stadium. The geographical spread of those being interviewed in this case causes problems. A sampling technique that overcomes the geographical spread problem is cluster sampling. The stadium sections would be the clusters. Ideally the clusters would each have the same characteristics as the population as a whole. Systematic Random Sampling A statistical sampling technique that involves selecting every kth item in the population after a randomly selected starting point between 1 and k. The value of k is determined as the ratio of the population size over the desired sample size. Cluster Sampling A method by which the population is divided into groups or clusters that are each intended to be mini-populations. A simple random sample of m clusters is selected. The items chosen from a cluster can be selected using any probability sampling technique. Large Institutions Population: Cash Holdings of All Financial Institutions in the United States Stratifed Population Financial Institutions Medium-Size Institutions Small Institutions Stratum 1 Stratum 2 Stratum 3 Select n 1 Select n 2 Select n 3 FIGURE 10 | Stratified Sampling Example

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The Where Why and How of Data Collection After the clusters have been defined a sample of m clusters is selected at random from the list of possible clusters. The number of clusters to select depends on various factors including our survey budget. Suppose the marketing firm randomly selects eight clusters: 104 - 142 - 147 - 218 - 228 - 235 - 307 - 327 These are the primary clusters. Next the marketing company can either survey all the ticketholders in each cluster or select a simple random sample of ticketholders from each cluster depending on time and budget considerations. Skill Development 1-32. Indicate which sampling method would most likely be used in each of the following situations: a. an interview conducted with mayors of a sample of cities in Florida b. a poll of voters regarding a referendum calling for a national value-added tax c. a survey of customers entering a shopping mall in Minneapolis 1-33. A company has 18000 employees. The file containing the names is ordered by employee number from 1 to 18000. If a sample of 100 employees is to be selected from the 18000 using systematic random sampling within what range of employee numbers will the first employee selected come from 1-34. Describe the difference between a statistic and a parameter. 1-35. Why is convenience sampling considered to be a nonstatistical sampling method 1-36. Describe how systematic random sampling could be used to select a random sample of 1000 customers who have a certificate of deposit at a commercial bank. Assume that the bank has 25000 customers who own a certificate of deposit. 1-37. Explain why a census does not necessarily have to involve a population of people. Use an example to illustrate. 1-38. If the manager at First City Bank surveys a sample of 100 customers to determine how many miles they live from the bank is the mean travel distance for this sample considered a parameter or a statistic Explain. 1-39. Explain the difference between stratified random sampling and cluster sampling. 1-40. Use Excel to generate five random numbers between 1 and 900. Business Applications 1-41. According to the U.S. Bureau of Labor Statistics the annual percentage increase in U.S. college tuition and fees in 1995 was 6.0 in 1999 it was 4.0 in 2004 it was 9.5 and in 2011 it was 5.4. Are these percentages statistics or parameters Explain. 1-42. According to an article in the Idaho Statesman a poll taken the day before elections in Germany showed Chancellor Gerhard Schroeder behind his challenger Angela Merkel by 6 to 8 percentage points. Is this a statistic or a parameter Explain. 1-43. Give the name of the kind of sampling that was most likely used in each of the following cases: a. a Wall Street Journal poll of 2000 people to determine the president’s approval rating b. a poll taken of each of the General Motors GM dealerships in Ohio in December to determine an estimate of the average number of Chevrolets not yet sold by GM dealerships in the United States c. a quality-assurance procedure within a Frito-Lay manufacturing plant that tests every 1000th bag of Fritos Corn Chips produced to make sure the bag is sealed properly d. a sampling technique in which a random sample from each of the tax brackets is obtained by the Internal Revenue Service to audit tax returns 1-44. Your manager has given you an Excel file that contains the names of the company’s 500 employees and has asked you to sample 50 employees from the list. You decide to take your sample as follows. First you assign a random number to each employee using Excel’s random number function Rand. Because the random number is volatile it recalculates itself whenever you modify the file you freeze the random numbers using the Copy—Paste Special—Values feature. You then sort by the random numbers in ascending order. Finally you take the first 50 sorted employees as your sample. Does this approach constitute a statistical or a nonstatistical sample Computer Applications 1-45. Sysco Foods is a statewide food distributor to restaurants universities and other establishments that prepare and sell food. The company has a very large warehouse in which the food is stored until it is pulled from the shelves to be delivered to the customers. The warehouse has 64 storage racks numbered 1-64. Each rack is three shelves high labeled A B and C and each shelf is divided into 80 sections numbered 1-80. 1- Exercises MyStatLab

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The Where Why and How of Data Collection END EXERCISES 1-3 Products are located by rack number shelf letter and section number. For example breakfast cereal is located at 43-A-52 rack 43 shelf A section 52. Each week employees perform an inventory for a sample of products. Certain products are selected and counted. The actual count is compared to the book count the quantity in the records that should be in stock. To simplify things assume that the company has selected breakfast cereals to inventory. Also for simplicity’s sake suppose the cereals occupy racks 1 through 5. a. Assume that you plan to use simple random sampling to select the sample. Use Excel to determine the sections on each of the five racks to be sampled. b. Assume that you wish to use cluster random sampling to select the sample. Discuss the steps you would take to carry out the sampling. c. In this case why might cluster sampling be preferred over simple random sampling Discuss. 1-46. United Airlines established a discount airline named Ted. The managers were interested in determining how flyers using Ted rate the airline service. They plan to question a random sample of flyers from the November 12 flights between Denver and Fort Lauderdale. A total of 578 people were on the flights that day. United has a list of the travelers together with their mailing addresses. Each traveler is given an identification number here from 001 to 578. Use Excel to generate a list of 40 flyer identification numbers so that those identified can be surveyed. 1-47. The National Park Service has started charging a user fee to park at selected trailheads and cross-country ski lots. Some users object to this fee claiming they already pay taxes for these areas. The agency has decided to randomly question selected users at fee areas in Colorado to assess the level of concern. a. Define the population of interest. b. Assume a sample of 250 is required. Describe the technique you would use to select a sample from the population. Which sampling technique did you suggest c. Assume the population of users is 4000. Use Excel to generate a list of users to be selected for the sample. 1-48. Mount Hillsdale Hospital has more than 4000 patient files listed alphabetically in its computer system. The office manager wants to survey a statistical sample of these patients to determine how satisfied they were with service provided by the hospital. She plans to use a telephone survey of 100 patients. a. Describe how you would attach identification numbers to the patient files for example how many digits and which digits would you use to indicate the first patient file b. Describe how the first random number would be obtained to begin a simple random sample method. c. How many random digits would you need for each random number you selected d. Use Excel to generate the list of patients to be surveyed. Data Types and Data Measurement Levels As you will see the statistical techniques deal with different types of data. The level of mea- surement may vary greatly from application to application. In general there are four types of data: quantitative qualitative time-series and cross-sectional. A discussion of each follows. Quantitative and Qualitative Data In some cases data values are best expressed in purely numerical or quantitative terms such as in dollars pounds inches or percentages. As an example a cell phone provider might collect data on the number of outgoing calls placed during a month by its customers. In another case a sports bar could collect data on the number of pitchers of beer sold weekly. In other situations the observation may signify only the category to which an item belongs. Categorical data are referred to as qualitative data. For example a bank might conduct a study of its outstanding real estate loans and keep track of the marital status of the loan customer—single married divorced or other. The same study also might examine the credit status of the customer—excellent good fair or poor. Still another part of the study might ask the customers to rate the service by the bank on a 1 to 5 scale with 1 very poor 2 poor 3 neutral 4 good and 5 very good. Note although the customers are asked to record a number 1 to 5 to indicate the service quality the data would still be considered qualitative because the numbers are just codes for the categories. Quantitative Data Measurements whose values are inherently numerical. Qualitative Data Data whose measurement scale is inherently categorical. Chapter Outcome 4.

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The Where Why and How of Data Collection Time-Series Data and Cross-Sectional Data Data may also be classified as being either time-series or cross-sectional. The data collected by the bank about its loan customers would be cross-sectional because the data from each customer relates to a fixed point in time. In another case if we sampled 100 stocks from the stock market and determined the closing stock price on March 15 the data would be considered cross-sectional because all measurements corresponded to one point in time. On the other hand Ford Motor Company tracks the sales of its F-150 pickup trucks on a monthly basis. Data values observed at intervals over time are referred to as time-series data. If we determined the closing stock price for a particular stock on a daily basis for a year the stock prices would be time-series data. Data Measurement Levels Data can also be identified by their level of measurement. This is important because the higher the data level the more sophisticated the analysis that can be performed. We shall discuss and give examples of four levels of data measurements: nominal ordinal interval and ratio. Figure 11 illustrates the hierarchy among these data levels with nominal data being the lowest level. Nominal Data Nominal data are the lowest form of data yet you will encounter this type of data many times. Assigning codes to categories generates nominal data. For example a survey question that asks for marital status provides the following responses: 1. Married 2. Single 3. Divorced 4. Other For each person a code of 1 2 3 or 4 would be recorded. These codes are nominal data. Note that the values of the code numbers have no specific meaning because the order of the categories is arbitrary. We might have shown it this way: 1. Single 2. Divorced 3. Married 4. Other With nominal data we also have complete control over what codes are used. For exam- ple we could have used 88. Single 11. Divorced 33. Married 55. Other All that matters is that you know which code stands for which category. Recognize also that the codes need not be numeric. We might use S Single D Divorced M Married O Other Time-Series Data A set of consecutive data values observed at successive points in time. Ordinal Data Ratio/Interval Data Categorical Codes ID Numbers Category Names Rankings Ordered Categories Lowest Level Basic Analysis Higher Level Mid-Level Analysis Highest Level Complete Analysis Measurements Nominal Data FIGURE 11 | Data Level Hierarchy Cross-Sectional Data A set of data values observed at a fixed point in time.

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The Where Why and How of Data Collection Ordinal Data Ordinal or rank data are one notch above nominal data on the measure- ment hierarchy. At this level the data elements can be rank-ordered on the basis of some relationship among them with the assigned values indicating this order. For example a typical market research technique is to offer potential customers the chance to use two unidentified brands of a product. The customers are then asked to indicate which brand they prefer. The brand eventually offered to the general public depends on how often it was the preferred test brand. The fact that an ordering of items took place makes this an ordinal measure. Bank loan applicants are asked to indicate the category corresponding to their household incomes: Under 20000 20000 to 40000 over 40000 1 2 3 The codes 1 2 and 3 refer to the particular income categories with higher codes assigned to higher incomes. Ordinal measurement allows decision makers to equate two or more observations or to rank-order the observations. In contrast nominal data can be compared only for equality. You cannot order nominal measurements. Thus a primary difference between ordinal and nominal data is that ordinal data can have both an equality and a greater than 7 or a less than 6 relationship whereas nominal data can have only an equality relationship. Interval Data If the distance between two data items can be measured on some scale and the data have ordinal properties 7 6 or the data are said to be interval data. The best example of interval data is the temperature scale. Both the Fahrenheit and Celsius tempera- ture scales have ordinal properties of “.” or “” and “5” In addition the distances between equally spaced points are preserved. For example 32°F 7 30°F and 80°C 7 78°C. The dif- ference between 32°F and 30°F is the same as the difference between 80°F and 78°F two degrees in each case. Thus interval data allow us to precisely measure the difference between any two values. With ordinal data this is not possible because all we can say is that one value is larger than another. Ratio Data Data that have all the characteristics of interval data but also have a true zero point at which zero means “none” are called ratio data. Ratio measurement is the highest level of measurement. Packagers of frozen foods encounter ratio measures when they pack their products by weight. Weight whether measured in pounds or grams is a ratio measurement because it has a unique zero point—zero meaning no weight. Many other types of data encountered in business environments involve ratio measurements for example distance money and time. The difference between interval and ratio measurements can be confusing because it involves the definition of a true zero. If you have 5 and your brother has 10 he has twice as much money as you. If you convert the dollars to pounds euros yen or pesos your brother will still have twice as much. If your money is lost or stolen you have no dollars. Money has a true zero. Likewise if you travel 100 miles today and 200 miles tomorrow the ratio of distance traveled will be 2/1 even if you convert the distance to kilometers. If on the third day you rest you have traveled no miles. Dis- tance has a true zero. Conversely if today’s temperature is 35°F 1.67°C and tomorrow’s is 70°F 21.11°C is tomorrow twice as warm as today The answer is no. One way to see this is to convert the Fahrenheit temperature to Celsius: The ratio will no longer be 21 12.641. Likewise if the temperature reads 0°F -17.59°C this does not imply that there is no temperature. It’s simply colder than 10°F -12.22°C Also 0°C 32°F is not the same temperature as 0°F. Thus temperature measured with either the Fahrenheit or Celsius scale an interval-level variable does not have a true zero. As was mentioned earlier a major reason for categorizing data by level and type is that the methods you can use to analyze the data are partially dependent on the level and type of data you have available.

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The Where Why and How of Data Collection EXAMPLE 1 CATEGORIZING DATA For many years U.S. News and World Report has published annual rankings based on various data collected from U.S. colleges and universities. Figure 12 shows a portion of the data in the file named Colleges and Universities. Each column cor- responds to a different variable for which data were collected. Before doing any statistical analyses with these data U.S. News and World Report employees need to determine the type and level for each of the factors. Limiting the effort to only those factors that are shown in Figure 12 this is done using the following steps: Step 1 Identify each factor in the data set. The factors or variables in the data set shown in Figure 12 are College State Public 1 Math V erbal appli. appli. new FT PT Name Private 2 SAT SAT rec’d. accepted. stud. under- under- enrolled grad grad Each of the 10 columns represents a different factor. Data might be missing for some colleges and universities. Step 2 Determine whether the data are time-series or cross-sectional. Because each row represents a different college or university and the data are for the same year the data are cross-sectional. Time-series data are measured over time—say over a period of years. Step 3 Determine which factors are quantitative data and which are qualitative data. Qualitative data are codes or numerical values that represent categories. Quantitative data are those that are purely numerical. In this case the data for the following factors are qualitative: College Name State Code for Public or Private College or University Data for the following factors are considered quantitative: Math SAT Verbal SAT new stud. enrolled appl. rec’d. appl. accepted PT undergrad FT undergrad FIGURE 12 | Data for U.S. Colleges and Universities Joe Gough/Shutterstock

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The Where Why and How of Data Collection Step 4 Determine the level of data measurement for each factor. The four levels of data are nominal ordinal interval and ratio. This data set has only nominal- and ratio-level data. The three nominal-level factors are College Name State Code for Public or Private College or University The others are all ratio-level data. Skill Development 1-49. For each of the following indicate whether the data are cross-sectional or time-series: a. quarterly unemployment rates b. unemployment rates by state c. monthly sales d. employment satisfaction data for a company 1-50. What is the difference between qualitative and quantitative data 1-51. For each of the following variables indicate the level of data measurement: a. product rating 1 excellent 2 good 3 fair 4  poor 5 very poor b. home ownership own rent other c. college grade point average d. marital status single married divorced other 1-52. What is the difference between ordinal and nominal data 1-53. Consumer Reports in its rating of cars indicates repair history with circles. The circles are either white black or half and half. To which level of data does this correspond Discuss. Business Applications 1-54. Verizon has a support center customers can call to get questions answered about their cell phone accounts. The manager in charge of the support center has recently conducted a study in which she surveyed 2300 customers. The customers who called the support center were transferred to a third party who asked the customers a series of questions. a. Indicate whether the data generated from this study will be considered cross-sectional or time-series. Explain why. b. One of the questions asked customers was approximately how many minutes they had been on hold waiting to get through to a support person. What level of data measurement is obtained from this question Explain. c. Another question asked the customer to rate the service on a scale of 1–7 with 1 being the worst possible service and 7 being the best possible service. What level of data measurement is achieved from this question Will the data be quantitative or qualitative Explain. 1-55. The following information can be found in the Murphy Oil Corporation Annual Report to Share- holders. For each variable indicate the level of data measurement. a. List of Principal Offices e.g. El Dorado Calgary Houston b. Income in millions of dollars from Continuing Operations c. List of Principal Subsidiaries e.g. Murphy Oil USA Inc. Murphy Exploration Production Company d. Number of branded retail outlets e. Petroleum products sold in barrels per day f. Major Exploration and Production Areas e.g. Malaysia Congo Ecuador g. Capital Expenditures measured in millions of dollars 1-56. You have collected the following information on 15 different real estate investment trusts REITs. Identify whether the data are cross-sectional or time-series. a. income distribution by region in 2012 b. per share diluted funds from operations FFO for the years 2006 to 2012 c. number of properties owned as of December 31 2012 d. the overall percentage of leased space for the 119 properties in service as of December 31 2012 e. dividends per share for the years 2006–2012 1-57. A loan manager for Bank of the Cascades has the responsibility for approving automobile loans. To assist her in this matter she has compiled data on 428 cars and trucks. These data are in the file called 2004-Automobiles. 1- Exercises MyStatLab

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The Where Why and How of Data Collection END EXERCISES 1-4 Column A Column B Column C Column D Column E Column F Account Number Caller Gender Account Holder Gender Past Due Amount Current Amount Due Was This a Billing Question Unique Tracking 1 Male 1 Male Numerical V alue Numerical V alue 3 Yes 2 Female 2 Female 4 No A small portion of the data is as follows: Indicate the level of data measurement for each of the variables in this data file. 1-58. Recently the manager of the call center for a large Internet bank asked his staff to collect data on a Account Number Caller Gender Account Holder Gender Past Due Amount Current Amount Due Was This a Billing Question 4348291 2 2 40.35 82.85 3 6008516 1 1 0 -129.67 4 17476479 1 2 0 76.38 4 13846306 2 2 0 99.24 4 21393711 1 1 0 37.98 3 random sample of the bank’s customers. Data on the following variables were collected and placed in a file called Bank Call Center: A Brief Introduction to Data Mining Data Mining—Finding the Important Hidden Relationships in Data What food products have an increased demand during hurricanes How do you win baseball games without star players Is my best friend the one to help me find a job What color car is least likely to be a “lemon” These and other interesting questions can and have been answered using data mining. Data mining consists of applying sophisticated statistical tech- niques and algorithms to the analysis of big data i.e. the wealth of new data that organiza- tions collect in many and varied forms. Through the application of data mining decisions can now be made on the basis of statistical analysis rather than on only managerial intuition and experience. The statistical techniques introduced in this text provide the basis for the more sophisticated statistical tools that are used by data mining analysts. Wal-Mart the nation’s largest retailer uses data mining to help it tailor product selec- tion based on the sales demographic and weather information it collects. While Wal-Mart managers might not be surprised that the demand for flashlights batteries and bottled water increased with hurricane warnings they were surprised to find that there was also an increase in the demand for strawberry Pop-Tarts before hurricanes hit. This knowledge allowed Wal- Mart to increase the availability of Pop-Tarts at selected stores affected by the hurricane alerts. The McKinsey Global Institute estimates that the full application of data mining to retailing could result in a potential increase in operating margins by as much as 60. Source: McKinsey Global Institute: Big Data: The Next Frontier for Innovation Competition and Productivity May 2011 by James Manyika Michael Chui Brad Brown Jacques Bughin Richard Dobbs Charles Roxburgh Angela Hung Byers. Chapter Outcome 5. a. Would you classify these data as time-series or cross-sectional Explain. b. Which of the variables are quantitative and which are qualitative c. For each of the six variables indicate the level of data measurement.

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The Where Why and How of Data Collection Data are everywhere and businesses are collecting more each day. Accounting and sales data are now captured and streamed instantly when transactions occur. Digital sensors in industrial equipment and automobiles can record and report data on vibration temperature physical location and the chemical composition of the surrounding air. But data are now more than numbers. Much of the data being collected today consists of words from Internet search engines such as Google searches and from pictures from social media postings on such platforms as Facebook. Together with the traditional numbers comprising quantitative data the availability of new unstructured qualitative data has led to a data explosion. IDC a technology research firm estimates that data are growing at a rate of 50 percent a year. All of these data—referred to as big data—have created a need not only for highly skilled data scientists who can mine and analyze it but also for managers who can make decisions using it. McKinsey Global Institute a consultancy firm believes that big data offer an oppor- tunity for organizations to create competitive advantages for themselves if they can under- stand and use the information to its full potential. They report that the use of big data “will become a key basis of competition and growth for individual firms.” This will create a need for highly trained data scientists and managers who can use data to support their decision making. Unfortunately McKinsey predicts that by 2018 there could be a shortage in the United States of 140000 to 190000 people with deep analytical skills as well as 1.5 million managers and analysts with the know-how needed to use big data to make meaningful and effective decisions. Source: McKinsey Global Institute: Big Data: The Next Frontier for Innovation Competition and Productivity May 2011 by James Manyika Michael Chui Brad Brown Jacques Bughin Richard Dobbs Charles Roxburgh Angela Hung Byers. The statistical tools you will learn in this course will provide you with a good first step toward preparing yourself for a career in data mining and business analytics.

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The Where Why and How of Data Collection 1 What Is Business Statistics Summary The two areas of statistics descriptive statistics and inferential statistics are introduced. Descriptive statistics includes visual tools such as charts and graphs and also the numerical measures such as the arithmetic average. The role of descriptive statistics is to describe data and help transform data into usable information. Inferential techniques are those that allow decision-makers to draw conclusions about a large body of data by examining a smaller subset of those data. Two areas of inference estimation and hypothesis testing are described. 2 Procedures for Collecting Data Summary Before data can be analyzed using business statistics techniques the data must be collected. The types of data collection reviewed are: experiments telephone surveys written questionnaires and direct observation and personal interviews. Data collection issues such as interviewer bias nonresponse bias selection bias observer bias and measurement error are covered. The concepts of internal validity and external validity are defned. Outcome 1. Know the key data collection methods. 3 Populations Samples and Sampling Techniques Outcome 2. Know the difference between a population and a sample. Outcome 3. Understand the similarities and differences between different sampling methods. 4 Data Types and Data Measurement Levels Summary The important concepts of population and sample are defned and examples of each are provided. Because many statistical applications involve samples emphasis is placed on how to select samples. Two main sampling categories are presented nonstatistical sampling and statistical sampling. The focus is on statistical sampling and four statistical sampling methods are discussed: simple random sampling stratifed random sampling cluster sampling and systematic random sampling. Summary This section discusses various ways in which data are classifed. For example data can be classifed as being either quantitative or qualitative. Data can also be cross-sectional or time-series. Another way to classify data is by the level of measurement. There are four levels from lowest to highest: nominal ordinal interval and ratio. Knowing the type of data you have is very important because the data type infuences the type of statistical procedures you can use. 5 A Brief Introduction to Data Mining Summary Because electronic data storage is so inexpensive organizations are collecting and storing greater volumes of data that ever before. As a result a relatively new feld of study called data mining has emerged. Data mining involves the art and science of delving into the data to identify patterns and conclusions that are not immediately evident in the data. This section briefy introduces the subject and discusses a few of the applications. Although data mining is not covered in depth in this text the concepts presented throughout the text form the basis for this important discipline. Outcome 4. Understand how to categorize data by type and level of measurement. Outcome 5. Become familiar with the concept of data mining and some of its applications. Conclusion Statistical analysis begins with data. You need to know how to collect data how to select samples from a population and the type and level of data you are using. Figure 13 summarizes the different sampling techniques presented in this chapter. Figure 14 gives a synopsis of the different data collection procedures and Figure 15 shows the different data types and measurement levels. Business statistics is a collection of procedures and techniques used by decision-makers to transform data into useful information. This chapter introduces the subject of business statistics. Included is a discussion of the different types of data and data collection methods. This chapter also describes the difference between populations and samples.

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The Where Why and How of Data Collection Time-Series Data Levels Data Type Data Timing Qualitative Nominal Ordinal Quantitative Interval Ratio Cross-Sectional FIGURE 15 | Data Classification Population N items Sampling Techniques Sample n items Nonrandom Sampling Convenience Sampling Judgment Sampling Ratio Sampling Simple Random Sampling Stratifed Random Sampling Systematic Random Sampling Cluster Sampling Random Sampling Sample n items Many possible samples FIGURE 13 | Sampling Techniques Experiments Data Collection Method Advantages Disadvantages Telephone Surveys Direct Observation Personal Interview Mail Questionnaires Written Surveys Provide controls Preplanned objectives Costly Time-consuming Requires planning Timely Relatively inexpensive Poor reputation Limited scope and length Inexpensive Can expand length Can use open-end questions Expands analysis opportunities No respondent bias Low response rate Requires exceptional clarity Potential observer bias Costly FIGURE 14 | Data Collection Techniques

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The Where Why and How of Data Collection Key Terms Arithmetic mean or average Bias Business intelligence Business statistics Census Closed-end questions Cluster sampling Convenience sampling Cross-sectional data Data Mining Demographic questions Experiment Experimental design External validity Internal validity Nonstatistical sampling techniques Open-end questions Population Qualitative data Quantitative data Sample Simple random sampling Statistical inference procedures Statistical sampling techniques Stratified random sampling Structured interview Systematic random sampling Time-series data Unstructured interview MyStatLab Chapter Exercises Conceptual Questions 1-59. Several organizations publish the results of presidential approval polls. Movements in these polls are seen as an indication of how the general public views presidential performance. Comment on these polls within the context of what was covered. 1-60. With what level of data is a bar chart most appropriately used 1-61. With what level of data is a histogram most appropriately used 1-62. Two people see the same movie one says it was average and the other says it was exceptional. What level of data are they using in these ratings Discuss how the same movie could receive different reviews. 1-63. The University of Michigan publishes a monthly measure of consumer confidence. This is taken as a possible indicator of future economic performance. Comment on this process within the context of what was covered. Business Applications 1-64. In a business publication such as The Wall Street Journal or Business W eek find a graph or chart representing time-series data. Discuss how the data were gathered and the purpose of the graph or chart. 1-65. In a business publication such as The Wall Street Journal or Business W eek find a graph or chart representing cross-sectional data. Discuss how the data were gathered and the purpose of the graph or chart. 1-66. The Oregonian newspaper has asked readers to e-mail and respond to the question “Do you believe police officers are using too much force in routine traffic stops” a. Would the results of this survey be considered a random sample b. What type of bias might be associated with a data collection system such as this Discuss what options might be used to reduce this bias potential. 1-67. The makers of Mama’s Home-Made Salsa are concerned about the quality of their product. The particular trait of concern is the thickness of the salsa in each jar. a. Discuss a plan by which the managers might determine the percentage of jars of salsa believed to have an unacceptable thickness by potential purchasers. 1 Define the sampling procedure to be used 2 the randomization method to be used to select the sample and 3 the measurement to be obtained. b. Explain why it would or wouldn’t be feasible or perhaps possible to take a census to address this issue. 1-68. A maker of energy drinks is considering abandoning can containers and going exclusively to bottles because the sales manager believes customers prefer drinking from bottles. However the vice president in charge of marketing is not convinced the sales manager is correct. a. Indicate the data collection method you would use. b. Indicate what procedures you would follow to apply this technique in this setting. c. State which level of data measurement applies to the data you would collect. Justify your answer. d. Are the data qualitative or quantitative Explain.

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The Where Why and How of Data Collection Statistical Data Collection McDonald’s Think of any well-known successful business in your community. What do you think has been its secret Competitive products or services Talented managers with vision Dedicated employees with great skills There’s no question these all play an important part in its success. But there’s more lots more. It’s “data.” That’s right data. The data collected by a business in the course of running its daily operations form the foundation of every decision made. Those data are analyzed using a variety of statistical techniques to provide decision makers with a succinct and clear picture of the company’s activities. The resulting statistical information then plays a key role in decision making whether those decisions are made by an accountant marketing manager or operations spe- cialist. To better understand just what types of business statistics organizations employ let’s take a look at one of the world’s most well-respected companies: McDonald’s. McDonald’s operates more than 30000 restaurants in more than 118 countries around the world. Total annual revenues recently surpassed the 20 billion mark. Wade Thomas vice presi- dent of U.S. Menu Management for McDonalds helps drive those sales but couldn’t do it without statistics. “When you’re as large as we are we can’t run the business on simple gut instinct. We rely heavily on all kinds of statistical data to help us determine whether our products are meeting customer expectations when products need to be updated and much more” says Wade. “The cost of making an educated guess is simply too great a risk.” McDonald’s restaurant owner/operators and managers also know the competitiveness of their individual restaurants depends on the data they collect and the statistical techniques used to ana- lyze the data into meaningful information. Each restaurant has a sophisticated cash register system that collects data such as indi- vidual customer orders service times and methods of payment to name a few. Periodically each U.S.–based restaurant undergoes a restaurant operations improvement process or ROIP study. A spe- cial team of reviewers monitors restaurant activity over a period of several days collecting data about everything from front-counter service and kitchen efficiency to drive-thru service times. The data are analyzed by McDonald’s U.S. Consumer and Business Insights group at McDonald’s headquarters near Chicago to help the res- taurant owner/operator and managers better understand what they’re doing well and where they have opportunities to grow. Steve Levigne vice president of Consumer and Busi- ness Insights manages the team that supports the company’s decision-making efforts. Both qualitative and quantitative data are collected and analyzed all the way down to the individual store level. “Depending on the audience the results may be rolled up to an aggregate picture of operations” says Steve. Software packages such as Microsoft Excel SAS and SPSS do most of the number crunching and are useful for preparing the graphical representa- tions of the information so decision makers can quickly see the results. Not all companies have an entire department staffed with spe- cialists in statistical analysis however. That’s where you come in. The more you know about the procedures for collecting and analyzing data and how to use them the better decision maker you’ll be regardless of your career aspirations. So it would seem there’s a strong relationship here—knowledge of statistics and your success. Discussion Questions: 1. You will recall that McDonald’s vice president of U.S. Menu Management Wade Thomas indicated that McDonald’s relied heavily on statistical data to determine in part if its products were meeting customer expectations. The narrative indicated that two important sources of data were the sophisticated register system and the restaurant operations improvement process ROIP. Describe the types of data that could be generated by these two methods and discuss how these data could be used to determine if McDonald’s products were meeting customer expectations. 2. One of McDonald’s uses of statistical data is to determine when products need to be updated. Discuss the kinds of data McDonald’s would require to make this determination. Also provide how these types of data would be used to determine when a product needed to be updated. 3. This video case presents the types of data collected and used by McDonald’s in the course of running its daily operations. For a moment imagine that McDonald’s did not collect these data. Attempt to describe how it might make a decision concerning for instance how much its annual advertising budget would be. 4. Visit a McDonald’s in your area. While there take note of the different types of data that could be collected using observation only. For each variable you identify determine the level of data measurement. Select three different variables from your list and outline the specific steps you would use to collect the data. Discuss how each of the variables could be used to help McDonald’s manage the restaurant. video

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The Where Why and How of Data Collection 1. Descriptive use charts graphs tables and numerical measures. 3. A bar chart is used whenever you want to display data that have already been categorized while a histogram is used to display data over a range of values for the factor under consideration. 5. Hypothesis testing uses statistical techniques to validate a claim. 13. statistical inference particularly estimation 17. written survey or telephone survey 19. An experiment is any process that generates data as its outcome. 23. internal and external validity 27. Advantages—low cost speed of delivery instant updating of data analysis disadvantages—low response and potential confusion about questions 29. personal observation data gathering 33. Part range Population size Sample size 18000 100 180 Thus the first person selected will come from employees 1 through 180. Once that person is randomly selected the second person will be the one numbered 100 higher than the first and so on. 37. The census would consist of all items produced on the line in a defined period of time. 41. parameters since it would include all U.S. colleges 43. a. stratified random sampling b. simple random sampling or possibly cluster random sampling c. systematic random sampling d. stratified random sampling 49. a. time-series b. cross-sectional c. time-series d. cross-sectional 51. a. ordinal—categories with defined order b. nominal—categories with no defined order c. ratio d. nominal—categories with no defined order 53. ordinal data 55. a. nominal data b. ratio data c. nominal data d. ratio data e. ratio data f. nominal data g. ratio data 61. interval or ratio data 67. a. Use a random sample or systematic random sample. b. The product is going to be ruined after testing it. You would not want to ruin the entire product that comes off the assembly line. Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. Berenson Mark L. and David M. Levine Basic Business Sta- tistics: Concepts and Applications 12th ed. Upper Saddle River NJ: Prentice Hall 2012. Cryer Jonathan D. and Robert B. Miller Statistics for Busi- ness: Data Analysis and Modeling 2nd ed. Belmont CA: Duxbury Press 1994. DeVeaux Richard D. Paul F. Velleman and David E. Bock Stats Data and Models 3rd ed. New York: Addison-Wesley 2012. Fowler Floyd J. Survey Research Methods 4th ed. Thousand Oaks CA: Sage Publications 2009. Hildebrand David and R. Lyman Ott Statistical Thinking for Managers 4th ed. Belmont CA: Duxbury Press 1998. John J. A. D. Whitiker and D. G. Johnson Statistical Thinking for Managers 2nd ed. Boca Raton FL: CRC Press 2005. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Scheaffer Richard L. William Mendenhall R. Lyman Ott and Kenneth G. Gerow Elementary Survey Sampling 7th ed. Brooks/Cole 2012. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. References

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The Where Why and How of Data Collection Nonstatistical Sampling Techniques Those methods of selecting samples using convenience judgment or other nonchance processes. Open-End Questions Questions that allow respondents the freedom to respond with any value words or statements of their own choosing. Population The set of all objects or individuals of interest or the measurements obtained from all objects or individuals of interest. Qualitative Data Data whose measurement scale is inherently categorical. Quantitative Data Measurements whose values are inherently numerical. Sample A subset of the population. Simple Random Sampling A method of selecting items from a population such that every possible sample of a specified size has an equal chance of being selected. Statistical Inference Procedures Procedures that allow a decision maker to reach a conclusion about a set of data based on a subset of that data. Statistical Sampling Techniques Those sampling methods that use selection techniques based on chance selection. Stratified Random Sampling A statistical sampling method in which the population is divided into subgroups called strata so that each population item belongs to only one stra- tum. The objective is to form strata such that the population values of interest within each stratum are as much alike as possible. Sample items are selected from each stratum using the simple random sampling method. Structured Interview Interviews in which the questions are scripted. Systematic Random Sampling A statistical sampling tech- nique that involves selecting every kth item in the popula- tion after a randomly selected starting point between 1 and k. The value of k is determined as the ratio of the population size over the desired sample size. Time-Series Data A set of consecutive data values observed at successive points in time. Unstructured Interview Interviews that begin with one or more broadly stated questions with further questions being based on the responses. Arithmetic Average or Mean The sum of all values divided by the number of values. Bias An effect that alters a statistical result by systematically distorting it different from a random error which may dis- tort on any one occasion but balances out on the average. Business Intelligence The application of tools and technolo- gies for gathering storing retrieving and analyzing data that businesses collect and use. Business Statistics A collection of procedures and techniques that are used to convert data into meaningful information in a business environment. Census An enumeration of the entire set of measurements taken from the whole population. Closed-End Questions Questions that require the respondent to select from a short list of defined choices. Cluster Sampling A method by which the population is divided into groups or clusters that are each intended to be mini-populations. A simple random sample of m clusters is selected. The items chosen from a cluster can be selected using any probability sampling technique. Convenience Sampling A sampling technique that selects the items from the population based on accessibility and ease of selection. Cross-Sectional Data A set of data values observed at a fixed point in time. Data Mining The application of statistical techniques and algorithms to the analysis of large data sets. Demographic Questions Questions relating to the respon- dents’ characteristics backgrounds and attributes. Experiment A process that produces a single outcome whose result cannot be predicted with certainty. Experimental Design A plan for performing an experiment in which the variable of interest is defined. One or more factors are identified to be manipulated changed or observed so that the impact or influence on the variable of interest can be measured or observed. External Validity A characteristic of an experiment whose results can be generalized beyond the test environment so that the outcomes can be replicated when the experiment is repeated. Internal Validity A characteristic of an experiment in which data are collected in such a way as to eliminate the effects of variables within the experimental environment that are not of interest to the researcher. Glossary

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Graphs Charts and Tables— Describing Your Data Quick Prep Links Review the definitions for nominal ordinal interval and ratio data in Sections 1–4. Examine the statistical software such as Excel that you will be using during this course to make sure you are aware of the procedures for constructing graphs and tables. For instance in Excel look at the Charts group on the Insert tab and the Pivot Table feature on the Insert tab. Look at popular newspapers such as USA Today and business periodicals such as Fortune Business Week or The Wall Street Journal for instances in which charts graphs or tables are used to convey information. Frequency Distributions and Histograms Bar Charts Pie Charts and Stem and Leaf Diagrams Line Charts and Scatter Diagrams Outcome 1. Construct frequency distributions both manually and with your computer. Outcome 2. Construct and interpret a frequency histogram. Outcome 3. Develop and interpret joint frequency distributions. Why you need to know We live in an age in which presentations and reports are expected to include high-quality graphs and charts that effectively transform data into information. Although the written word is still vital words become even more power- ful when coupled with an effective visual illustration of data. The adage that a picture is worth a thousand words is particularly relevant in business decision making. We are constantly bombarded with visual images and stimuli. Much of our time is spent watching television playing video games or working at a computer. These technolo- gies are advancing rapidly making the images sharper and more attractive to our eyes. Flat-panel high-definition televisions and high-resolution monitors represent significant improvements over the original technologies they replaced. However this phenomenon is not limited to video technology but has also become an important part of the way businesses communicate with customers employees suppliers and other constituents. When you graduate you will find yourself on both ends of the data analysis spectrum. On the one hand regardless of what you end up doing for a career you will almost certainly be involved in preparing reports and making presentations that require using visual descriptive statistical tools presented in this chapter. You will be on the “do it” end of the data analysis process. Thus you need to know how to use these statistical tools. On the other hand you will also find yourself reading reports or listening to presentations that others have made. In many instances you will be required to make important decisions or to reach conclusions based on the information in those reports or presentations. Thus you will be on the “use it” end of the data analysis process. You need to be knowledgeable about these tools to effectively screen and critique the work that others do for you. Charts and graphs are not just tools used internally by businesses. Business periodicals such as Fortune and Business Week use graphs and charts extensively in articles to help readers better understand key concepts. Many advertisements will even use graphs and charts effectively to convey their messages. Virtually every issue of The Wall Street Journal contains different graphs charts or tables that display data in an informative way. Outcome 4. Construct and interpret various types of bar charts. Outcome 5. Build a stem and leaf diagram. Outcome 6. Create a line chart and interpret the trend in the data. Outcome 7. Construct a scatter diagram and interpret it. MishAl/Shutterstock From Chapter 2 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved.

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Graphs Charts and Tables—Describing Your Data Thus you will find yourself to be both a producer and a consumer of the descriptive statistical techniques known as graphs charts and tables. You will create a competitive advantage for yourself through- out your career if you obtain a solid understanding of the techniques introduced in this text. This chapter introduces some of the most frequently used tools and techniques for describing data with graphs charts and tables. Although this analysis can be done manually we will provide output from Excel software showing that software can be used to perform the analysis easily quickly and with a finished quality that once required a graphic artist. Frequency Distributions and Histograms In today’s business climate companies collect massive amounts of data they hope will be useful for making decisions. Every time a customer makes a purchase at a store like Macy’s or the Gap data from that transaction is updated to the store’s database. Major retail stores like Walmart capture the number of different product categories included in each “market basket” of items purchased. Table 1 shows these data for all customer transactions for a single day at one Walmart store in Dallas. A total of 450 customers made purchases on the day in question. The first value in Table 1 4 indicates that the customer’s purchase included four different product categories for example food sporting goods photography supplies and dry goods. TABLE 1 | Product Categories per Customer at the Dallas Walmart 42588 1014834113 4 14454449544 10 7 11 4 10267 10546462324 5 54 1114192466762 3 65345653 1065774 3 82265 1199556531 7 66538433444764 9 1655447566956 10 4 758447466442 10 4 5 4 11879564284266 6 64657169159 1055 10 54757695321555 5 59532572464444 4 65855555255646 5 57 1022683135633 6 5453379445 10 6 105 9 43871843136755 5 474 116637944297 5 16683844193934 2 9557 10534776224 4 47354923432164 6 1814355 10444692 7 94536553465736 8 36157754666369 5 45 1015578916566 4 1065551656479 10 2 6 446 11954435 4626 7 35674546943369 4 3756 11448428242 3 651 105954514954 4

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Graphs Charts and Tables—Describing Your Data Although the data in Table 1 are easy to capture with the technology of today’s cash reg- isters in this form the data provide little or no information that managers could use to deter- mine the buying habits of their customers. However these data can be converted into useful information through descriptive statistical analysis. Frequency Distribution One of the first steps would be to construct a frequency distribution. The product data in Table 1 take on only a few possible values 1 2 3 c 11. The minimum number of product categories is 1 and the maximum number of categories in these data is 11. These data are called discrete data. When you encounter discrete data where the variable of interest can take on only a rea- sonably small number of possible values a frequency distribution is constructed by count- ing the number of times each possible value occurs in the data set. We organize these counts into a frequency distribution table as shown in Table 2. Now from this frequency distribu- tion we are able to see how the data values are spread over the different number of possible product categories. For instance you can see that the most frequently occurring number of product categories in a customer’s “market basket” is 4 which occurred 92 times. You can also see that the three most common numbers of product categories are 4 5 and 6. Only a very few times do customers purchase 10 or 11 product categories in their trip to the store. Consider another example in which a consulting firm surveyed random samples of residents in two cities Philadelphia and Knoxville. The firm is investigating the labor markets in these two communities for a client that is thinking of relocating its corporate offices to one of the two locations. Education level of the workforce in the two cities is a key factor in making the relocation decision. The consulting firm surveyed 160 randomly selected adults in Philadelphia and 330 adults in Knoxville and recorded the number of years of college attended. The responses ranged from zero to eight years. Table 3 shows the frequency distributions for each city. Suppose now we wished to compare the distribution for years of college for Philadelphia and Knoxville. How do the two cities’ distributions compare Do you see any difficulties in making this comparison Because the surveys contained different numbers of people it is dif- ficult to compare the frequency distributions directly. When the number of total observations differs comparisons are aided if relative frequencies are computed. Equation 1 is used to compute the relative frequencies. Table 4 shows the relative frequencies for each city’s distribution. This makes a com- parison of the two much easier. We see that Knoxville has relatively more people with- out any college 56.7 or with one year of college 18.8 than Philadelphia 21.9 Frequency Distribution A summary of a set of data that displays the number of observations in each of the distribution’s distinct categories or classes. Discrete Data Data that can take on a countable number of possible values. Relative Frequency The proportion of total observations that are in a given category. Relative frequency is computed by dividing the frequency in a category by the total number of observations. The relative frequencies can be converted to percentages by multiplying by 100. TABLE 2 | Dallas Walmart Product Categories Frequency Distribution Number of Product Catagories Frequency 125 229 342 492 583 671 735 819 929 10 18 11 7 Total 450 Chapter Outcome 1.

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Graphs Charts and Tables—Describing Your Data TABLE 3 |  Frequency Distributions of Years of College Education Philadelphia Knoxville Years of College Frequency Years of College Frequency 0 35 0 187 121 1 62 224 2 34 322 3 19 431 4 14 513 5 7 66 6 3 75 7 4 8 3 8 0 Total 160 Total 330 Relative Frequency Relative frequency 5 f n i f f 1 where: f i Frequency of the ith value of the discrete variable nf i k i f f 1 ∑ Total numbe m ro e e f observa r r ti a a ons k The number of different values for the discrete variable TABLE 4 | Relative Frequency Distributions of Years of College Philadelphia Knoxville Years of College Frequency Relative Frequency Frequency Relative Frequency 035 35160 0.219 187 187330 0.567 121 21160 0.131 62 62330 0.188 224 24160 0.150 34 34330 0.103 322 22160 0.138 19 19330 0.058 431 31160 0.194 14 14330 0.042 513 13160 0.081 7 7330 0.021 66 6160 0.038 3 3330 0.009 75 5160 0.031 4 4330 0.012 8 3 3160 0.019 0 0330 0.000 Total 160 330 and 13.1. At all other levels of education Philadelphia has relatively more people than Knoxville. The frequency distributions shown in Table 2 and Table 3 were developed from quantitative data. That is the variable of interest was numerical number of product catego- ries or number of years of college. However a frequency distribution can also be developed

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Graphs Charts and Tables—Describing Your Data TABLE 5 | TV Source Frequency Distribution TV Source Frequency DISH 80 Direct-TV 90 Cable 20 Other 10 Total 200 when the data are qualitative data or nonnumerical data. For instance if a survey asked homeowners how they get their TV signal the possible responses in this region are: DISH DirectTV Cable Other Table 5 shows the frequency distribution from a survey of 200 homeowners. EXAMPLE 1 FREQUENCY AND RELATIVE FREQUENCY DISTRIBUTIONS Real Estate Transactions In late 2008 the United States experienced a major economic decline thought to be due in part to the sub-prime loans that many lending institutions made during the previous few years. When the housing bubble burst many institutions experienced severe problems. As a result lenders became much more conservative in granting home loans which in turn made buying and selling homes more challenging. To demonstrate the mag- nitude of the problem in Kansas City the Association of Real Estate Brokers conducted a survey of 16 real estate agencies and collected data on the number of real estate transactions closed in December 2008. The following data were observed: 3001 1220 0210 2142 The real estate analysts can use the following steps to construct a frequency distribution and a relative frequency distribution for the number of real estate transactions. Step 1 List the possible values. The possible values for the discrete variable listed in order are 0 1 2 3 4. Step 2 Count the number of occurrences at each value. The frequency distribution follows: Transactions Frequency Relative Frequency 05 5/16 0.3125 14 4/16 0.2500 25 5/16 0.3125 31 1/16 0.0625 4 1 1/16 0.0625 Total 16 1.0000 Step 3 Determine the relative frequencies. The relative frequencies are determined by dividing each frequency by 16 as shown in the right-hand column above. Thus just over 31 of those responding reported no transactions during December 2008. END EXAMPLE TRY PROBLEM 1 EXAMPLE 2 FREQUENCY DISTRIBUTION FOR QUALITATIVE DATA Automobile Accidents State Farm Insurance recently surveyed a sample of the records for 15 policy holders to determine the make of the vehicle driven by the eldest member in the household. The following data reflect the results for 15 of the respondents: Ford Dodge Toyota Ford Buick Chevy Toyota Nissan Ford Chevy Ford Toyota Chevy BMW Honda Example 1 Developing Frequency and Relative Frequency Distributions for Discrete Data To develop a discrete data frequency distribution perform the following steps: List all possible values of the variable. If the variable is ordi- nal level or higher order the possible values from low to high. Count the number of occur- rences at each value of the vari- able and place this value in a column labeled “frequency.” To develop a relative frequency distribution do the following: Use Equation 1 and divide each frequency count by the total number of observations and place in a column headed “rela- tive frequency.” Elena Elisseeva/Shutterstock

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Graphs Charts and Tables—Describing Your Data The frequency distribution for this qualitative variable is found as follows: Step 1 List the possible values. For these sample data the possible values for the variable are BMW Buick Chevy Dodge Ford Honda Nissan Toyota. Step 2 Count the number of occurrences at each value. The frequency distribution is Car Company Frequency BMW 1 Buick 1 Chevy 3 Dodge 1 Ford 4 Honda 1 Nissan 1 Toyota 3 Total 15 END EXAMPLE TRY PROBLEM 8 BUSINESS APPLICATION FREQUENCY DISTRIBUTIONS ATHLETIC SHOE SURVEY In recent years a status symbol for many students has been the brand and style of athletic shoes they wear. Companies such as Nike and Adidas compete for the top position in the sport shoe market. A survey was recently conducted in which 100 college students at a southern state school were asked a number of questions including how many pairs of Nike shoes they currently own. The data are in a file called SportsShoes. The variable Number of Nike is a discrete quantitative variable. Figure 1 shows the frequency distribution output from Excel for the number of Nike shoes owned by those Excel 2010 Instructions: 1. Open File: SportsShoes.xlsx. 2. Enter the Possible Values for the Variable i.e. 0 1 2 3 4 etc. 3. Select the cells to contain the Frequency values. 4. Select the Formulas tab. 5. Click on the f x button. 6. Select the Statistics— FREQUENCY function. 7. Enter the range of data and the bin range the cells containing the possible number of shoes. 8. Press Ctrl-Shift-Enter to determine the frequency values. Minitab Instructions for similar results: 1. Open fle: SportsShoes.MTW.3.In Variables enter data column. 2. Choose Stat Tables Tally 4. Under Display check Counts. Individual Variables. 5. Click OK. FIGURE 1 | Excel 2010 Output—Nike Shoes Frequency Distribution Excel tutorials Excel Tutorial

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Graphs Charts and Tables—Describing Your Data surveyed. These frequency distributions show that although a few people own more than six pairs of Nike shoes the bulk of those surveyed own two or fewer pairs. Grouped Data Frequency Distributions In the previous examples the variable of interest was a discrete variable and the number of possible values for the variable was limited to only a few. However there are many instances in which the variable of interest will be either continuous e.g. weight time length or dis- crete and will have many possible outcomes e.g. age income stock prices yet you want to describe the variable using a frequency distribution. BUSINESS APPLICATION GROUPED DATA FREQUENCY DISTRIBUTIONS NETFLIX Netflix is one of the largest video rental companies in the United States. It rents movies that are sent to customers by mail and also provides a video streaming service via the Internet. Recently a distribution manager for Netflix conducted a survey of customers. Among the questions asked on the written survey was “How many DVD movies do you own” A total of 230 people completed the survey Table 6 shows the responses to the DVD ownership question. These data are discrete quantitative data. The values range from 0 to 30. The manager is interested in transforming these data into useful information by con- structing a frequency distribution. Table 7 shows one approach in which the possible values for the number of DVD movies owned is listed from 0 to 30. Although this frequency distribu- tion is a step forward in transforming the data into information because of the large number of possible values for DVD movies owned the 230 observations are spread over a large range making analysis difficult. In this case the manager might consider forming a grouped data frequency distribution by organizing the possible number of DVD movies owned into discrete categories or classes. To begin constructing a grouped frequency distribution sort the quantitative data from low to high. The sorted data is called a data array. Now define the classes for the variable of interest. Care needs to be taken when constructing these classes to ensure each data point is put into one and only one possible class. Therefore the classes should meet four criteria. Continuous Data Data whose possible values are uncountable and that may assume any value in an interval. TABLE 6 | DVD Movies Owned: Netflix Survey 9 4 13 10 51013141019 0 10 16 9 11 14 8 15 7 15 10 11976 12 12 14 15 16 15 14 10 13 91212101011 15 14 9 19 3 9 16 19 15 9 4 245623475 6 220083432 2 525226256 5 273516436 3 771627132 4 022462537 4 16 9 10 11 7 10 9 10 11 11 12 98979 17 8 13 14 13 10 6 12 5 14 7 13 12 9 6 10 157799 13 10 9 3 17 5 11969 15 8 11 13 41613 9 11 51213 0 333214020 3 715223213 2 333033311 13 24 24 17 17 15 25 20 15 20 21 23 25 17 13 22 18 17 30 21 18 21 17 16 25 14 15 24 21 15 Chapter Outcome 1.

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Graphs Charts and Tables—Describing Your Data First they must be mutually exclusive. Second they must be all-inclusive. Third if at all possible they should be of equal width. Fourth avoid empty classes if possible. Equal-width classes make analyzing and interpreting the frequency distribution easier. However there are some instances in which the presence of extreme high or low values makes it necessary to have an open-ended class. For example annual family incomes in the United States are mostly between 15000 and 200000. However there are some families with much higher family incomes. To best accommodate these high incomes you might consider having the highest income class be “over 200000” or “200000 and over” as a catchall for the high-income families. Empty classes are those for which there are no data values. If this occurs it may be because you have set up classes that are too narrow. Steps for Grouping Data into Classes There are four steps for grouping data such as that found in Table 6 into classes. Step 1 Determine the number of groups or classes to use. Although there is no absolute right or wrong number of classes one rule of thumb is to have between 5 and 20 classes. Another guideline for helping you determine how many classes to use is the 2 k Ú n rule where k the number Mutually Exclusive Classes Classes that do not overlap so that a data value can be placed in only one class. All-Inclusive Classes A set of classes that contains all the possible data values. Equal-Width Classes The distance between the lowest possible value and the highest possible value in each class is equal for all classes. TABLE 7 | Frequency Distribution of DVD Movies Owned DVD Movies Owned Frequency 08 18 222 322 411 513 612 714 85 919 10 14 11 9 12 8 13 12 14 8 15 12 16 6 17 7 18 2 19 3 20 2 21 4 22 1 23 1 24 3 25 3 26 0 27 0 28 0 29 0 30 1 Total 230

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Graphs Charts and Tables—Describing Your Data of classes and is defined to be the smallest integer so that 2 k Ú n where n is the number of data values. For example for n 230 the 2 k Ú n rule would suggest k 8 classes 2 8 256 Ú 230 while 2 7 128 6 230. This latter method was chosen for our example. Our preliminary step as specified previ- ously is to produce a frequency distribution from the data array as in Table 7. This will enhance our ability to envision the data structure and the classes. Remember these are only guidelines for the number of classes. There is no specific right or wrong number. In general use fewer classes for smaller data sets more classes for larger data sets. However using too few classes tends to condense data too much and information can be lost. Using too many classes spreads out the data so much that little advantage is gained over the original raw data. Step 2 Establish the class width. The minimum class width is determined by Equation 2. Class Width The distance between the lowest possible value and the highest possible value for a frequency class. For the Netflix data using eight classes we get W Largest value Smallest value Number of classes 30 0 8 375 . This means we could construct eight classes that are each 3.75 units wide to provide mutually exclusive and all-inclusive classes. However because our purpose is to make the data more understandable we suggest that you round up to a more convenient class width such as 4.0. If you do round the class width always round up. Step 3 Determine the class boundaries for each class. The class boundaries determine the lowest possible value and the highest pos- sible value for each class. In the Netflix example if we start the first class at 0 we get the class boundaries shown in the first column of the following table. Notice the classes have been formed to be mutually exclusive and all-inclusive. Step 4 Determine the class frequency for each class. The count for each class is known as a class frequency. As an example the number of observations in the first class is 60. DVD Movies Owned Classes Frequency 093 60 497 50 8911 47 12915 40 16919 18 20923 8 24927 6 28931 1 Total 230 Another step we can take to help analyze the Netflix data is to construct a relative frequency distribution a cumulative frequency distribution and a cumulative relative frequency distribution. Class Boundaries The upper and lower values of each class. Cumulative Frequency Distribution A summary of a set of data that displays the number of observations with values less than or equal to the upper limit of each of its classes. Cumulative Relative Frequency Distribution A summary of a set of data that displays the proportion of observations with values less than or equal to the upper limit of each of its classes. 2 W Largest value Smallest value Number of clas sses www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data The cumulative frequency distribution is shown in the “Cumulative Frequency” column. We can then form the cumulative relative frequency distribution as shown in the “Cumulative Relative Frequency” column. The cumulative relative frequency distribution indicates as an example that 85.7 of the sample own fewer than 16 DVD movies. EXAMPLE 3 FREQUENCY DISTRIBUTION FOR CONTINUOUS VARIABLES Emergency Response Communication Links One of the major efforts of the United States Office of Homeland Security has been to improve the communication between emergency respond- ers like the police and fire departments. The communications have been hampered by problems involving linking divergent radio and computer systems as well as communication protocols. While most cities have recognized the problem and made efforts to solve it Homeland Security recently funded practice exercises in 72 cities of different sizes throughout the United States. The resulting data already sorted but representing seconds before the systems were linked are as follows: 35 339 650 864 1025 1261 38 340 655 883 1028 1280 48 395 669 883 1036 1290 53 457 703 890 1044 1312 70 478 730 934 1087 1341 99 501 763 951 1091 1355 138 521 788 969 1126 1357 164 556 789 985 1176 1360 220 583 789 993 1199 1414 265 595 802 997 1199 1436 272 596 822 999 1237 1479 312 604 851 1018 1242 1492 Homeland Security wishes to construct a frequency distribution showing the times until the communication systems are linked. The frequency distribution is determined as follows: Step 1 Group the data into classes. The number of classes is arbitrary but typically will be between 5 and 20 depending on the volume of data. In this example we have n 72 data items. A common method of determining the number of classes is to use the 2 k ≥ n guideline. We get k 7 classes since 2 7 128 Ú 72 and 2 6 64 6 72. Step 2 Determine the class width. W Largest value Smallest value Number of cl lasses 1 492 35 7 208 1429 225 . ⇒ Note we have rounded the class width up from the minimum required value of 208.1429 to the more convenient value of 225. Example 3 Developing Frequency Distributions for Continuous Variables To develop a continuous data frequency distribution perform the following steps: Determine the desired number of classes or groups. One rule of thumb is to use 5 to 20 classes. The 2 k Ú n rule can also be used. Determine the minimum class width using W Largest Value Smallest Value Number of classes 5 Round the class width up to a more convenient value. Define the class boundaries making sure that the classes that are formed are mutually exclu- sive and all-inclusive. Ideally the classes should have equal widths and should all contain at least one observation. Determine the class frequency for each class. DVD Movies Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 093 60 0.261 60 0.261 497 50 0.217 110 0.478 8911 47 0.204 157 0.683 12915 40 0.174 197 0.857 16919 18 0.078 215 0.935 20923 8 0.035 223 0.970 24927 6 0.026 229 0.996 28931 1 0.004 230 1.000 Total 230

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Graphs Charts and Tables—Describing Your Data Step 3 Define the class boundaries. 0 and under 225 225 and under 450 450 and under 675 675 and under 900 900 and under 1125 1125 and under 1350 1350 and under 1575 These classes are mutually exclusive all-inclusive and have equal widths. Step 4 Determine the class frequency for each class. Time to Link Systems in seconds Frequency 0 and under 225 9 225 and under 450 6 450 and under 675 12 675 and under 900 13 900 and under 1125 14 1125 and under 1350 11 1350 and under 1575 7 This frequency distribution shows that most cities took between 450 and 1350 seconds 7.5 and 22.5 minutes to link their communications systems. END EXAMPLE TRY PROBLEM 5 Histograms Although frequency distributions are useful for analyzing large sets of data they are presented in table format and may not be as visually informative as a graph. If a frequency distribution has been developed from a quantitative variable a frequency histogram can be constructed directly from the frequency distribution. In many cases the histogram offers a superior format for transforming the data into useful information. Note: Histograms cannot be constructed from a frequency distribution in which the variable of interest is qualitative. However a similar graph called a bar chart discussed later in this chapter is used when qualitative data are involved. A histogram shows three general types of information: 1. It provides a visual indication of where the approximate center of the data is. Look for the center point along the horizontal axes in the histograms in Figure 2. Even though the shapes of the histograms are the same there is a clear difference in where the data are centered. 2. We can gain an understanding of the degree of spread or variation in the data. The more the data cluster around the center the smaller the variation in the data. If the data are spread out from the center the data exhibit greater variation. The examples in Figure 3 all have the same center but are different in terms of spread. 3. We can observe the shape of the distribution. Is it reasonably flat is it weighted to one side or the other is it balanced around the center or is it bell shaped BUSINESS APPLICATION CONSTRUCTING HISTOGRAMS CAPITAL CREDIT UNION Even for applications with small amounts of data such as the Netflix example constructing grouped data frequency distributions and histograms is a time- consuming process. Decision makers may hesitate to try different numbers of classes and different class limits because of the effort involved and because the “best” presentation of the data may be missed. We showed earlier that Excel provides the capability of constructing frequency distribu- tions. It is also quite capable of generating grouped data frequency distributions and histograms. Frequency Histogram A graph of a frequency distribution with the horizontal axis showing the classes the vertical axis showing the frequency count and for equal class widths the rectangles having a height equal to the frequency in each class. Excel Tutorial Excel tutorials Chapter Outcome 2. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Consider Capital Credit Union CCU in Mobile Alabama which recently began issuing a new credit card. Managers at CCU have been wondering how customers use the card so a sample of 300 customers was selected. Data on the current credit card balance rounded to the nearest dollar and the genders of the cardholders appear in the file Capital. As with the manual process the first step in Excel is to determine the number of classes. Recall that the rule of thumb is to use between 5 and 20 classes depending on the amount of data. Suppose we decide to use 10 classes. Next we determine the class width using Equation 2. The highest account balance in the sample is 1493.00. The minimum is 99.00. Thus the class width is W 1 493 00 99 00 139 40 . . . 10 which we round up to 150.00. 100 a b c 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 FIGURE 2 | Histograms Showing Different Centers FIGURE 3 | Histograms—Same Center Different Spread 100 a b c 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Our classes will be +90-+239.99 +240-+389.99 +390-+539.99 etc. The resulting histogram in Figure 4 shows that the data are centered in the class from 690 to 839.99. The customers vary considerably in their credit card balances but the distribution is quite symmetrical and somewhat bell shaped. CCU managers must decide whether the usage rate for the credit card is sufficient to warrant the cost of maintaining the credit card accounts. Issues with Excel If you use Excel to construct a histogram as indicated in the instruc- tions in Figure 4 the initial graph will have gaps between the bars. Because histograms illustrate the distribution of data across the range of all possible values for the quantitative variable histograms do not have gaps. Therefore to get the proper histogram format you need to close these gaps by setting the gap width to zero as indicated in the Excel instruc- tions shown in Figure 4. FIGURE 4 | Excel 2010 Histogram of Credit Card Balances 6. Put on a new worksheet ply and include the Chart Output. 7. Right mouse click on the bars and use the Format Data Series Options to set gap width to zero and add lines to the bars. 8. Convert the bins to actual class labels by typing labels in Column A. Note: The bin 239.99 is labeled 0–239.99. Excel 2010 Instructions: 1. Open fle: Capital.xlsx. 2. Set up an area on the worksheet for the bins defned as 239.99 389.99 etc. up to 1589.99. Be sure to include a label such as “Bins.” 3. On the Data tab click Data Analysis. 4. Select Histogram. 5. Input Range specifes the actual data values as the Credit Card Account Balance column and the bin range as the area defned in Step 2. Minitab Instructions for similar results: 1. Open fle: Capital.MTW. 2. Choose Graph Histogram. 3. Click Simple. 4. Click OK. 5. In Graph variables enter data column. 6. Click OK. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Emergency Response Time Distribution Frequency Emergency Response Times Seconds 300 250 200 150 100 50 0 0 30 60 90 120 150 180 210 240 300 270 This histogram indicates that the response times vary considerably. The center is somewhere in the range of 120 to 180 seconds. END EXAMPLE TRY PROBLEM 10 Example 4 Constructing Frequency Histograms To construct a frequency histo- gram perform the following steps: 1–4. Follow the steps for constructing a frequency distribution see Example 3. 5. Use the horizontal axis to represent classes for the variable of interest. Use the vertical axis to repre- sent the frequency in each class. 6. Draw vertical bars for each class or data value so that the heights of the bars correspond to the frequen- cies. Make sure there are no gaps between the bars. Note if the classes do not have equal widths the bar height should be adjusted to make the area of the bar proportional to the frequenc y . 7. Label the histogram appropriately. EXAMPLE 4 FREQUENCY HISTOGRAMS Emergency Response Times The Paris France director of Emergency Medical Response is interested in analyzing the time needed for response teams to reach their destinations in emer- gency situations after leaving their stations. She has acquired the response times for 1220 calls last month. To develop the frequency histogram perform the following steps: Steps 1–4 Construct a frequency distribution. Because response time is a continuous variable measured in seconds the data should be broken down into classes and the steps given in Example 3 should be used. The following frequency distribution with 10 classes was developed: Response Time Frequency Response Time Frequency 0 and under 30 36 180 and under 210 145 30 and under 60 68 210 and under 240 80 60 and under 90 195 240 and under 270 43 90 and under 120 180 270 and under 300 31 120 and under 150 260 Total 1220 150 and under 180 182 Step 5 Construct the axes for the histogram. The horizontal axis will be response time and the vertical axis will be frequency. Step 6 Construct bars with heights corresponding to the frequency of each class. Step 7 Label the histogram appropriately. This is shown as follows: Robert Wilson/Fotolia www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Relative Frequency Histograms and Ogives Histograms can also be used to display relative frequency distributions and cumulative rela- tive frequency distributions. A relative frequency histogram is formed in the same manner as a frequency histogram but relative frequencies rather than frequencies are used on the vertical axis. The cumulative relative frequency is presented using a graph called an ogive. Example 5 illustrates each of these graphical tools. EXAMPLE 5 RELATIVE FREQUENCY HISTOGRAMS AND OGIVES Emergency Response Times Continued Example 4 introduced the situation facing the emergency response manager in Paris. In that example she formed a frequency distribution for a sample of 1220 response times. She is now interested in graphing the relative frequencies and the ogive. To do so use the following steps: Step 1 Convert the frequency distribution into relative frequencies and cumulative relative frequencies. Response Time Frequency Relative Frequency Cumulative Relative Frequency 0 and under 30 36 361220 0.0295 0.0295 30 and under 60 68 681220 0.0557 0.0852 60 and under 90 195 1951220 0.1598 0.2451 90 and under 120 180 1801220 0.1475 0.3926 120 and under 150 260 2601220 0.2131 0.6057 150 and under 180 182 1821220 0.1492 0.7549 180 and under 210 145 1451220 0.1189 0.8738 210 and under 240 80 801220 0.0656 0.9393 240 and under 270 43 431220 0.0352 0.9746 270 and under 300 31 311220 0.0254 1.0000 1220 1.0000 Step 2 Construct the relative frequency histogram. Place the quantitative variable on the horizontal axis and the relative frequencies on the vertical axis. The vertical bars are drawn to heights corresponding to the relative frequencies of the classes. Ogive The graphical representation of the cumulative relative frequency. A line is connected to points plotted above the upper limit of each class at a height corresponding to the cumulative relative frequency. Robert Wilson/Fotolia Emergency Response Time Relative Frequency Distribution Relative Frequency Response Times seconds .25000 .20000 .15000 .10000 .05000 .00000 0 30 60 120 90 180 150 240 300 210 270 Note the relative frequency histogram has exactly the same shape as the frequency histogram. However the vertical axis has a different scale. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Step 3 Construct the ogive. Place a point above the upper limit of each class at a height corresponding to the cumulative relative frequency. Complete the ogive by drawing a line connecting these points. .00000 1.00000 .90000 .80000 .70000 .60000 .50000 .40000 .30000 .20000 .10000 Cumulative Relative Frequency Response Times Emergency Response Times Ogive 0 30 60 90 120 180 240 300 150 210 270 END EXAMPLE TRY PROBLEM 16 Joint Frequency Distributions Frequency distributions are effective tools for describing data. Thus far we have discussed how to develop grouped and ungrouped frequency distributions for one variable at a time. For instance in the Capital Credit Union example we were interested in customer credit card balances for all customers. We constructed a frequency distribution and histogram for that variable. However often we need to examine data that are characterized by more than one variable. This may involve constructing a joint frequency distribution for two variables. Joint frequency distributions can be constructed for qualitative or quantitative variables. EXAMPLE 6 JOINT FREQUENCY DISTRIBUTION Miami City Parking Parking is typically an issue in many large cities like Miami Flor- ida. Problems seem to occur for customers and employees both in locating a parking spot and in being able to quickly exit a lot at busy times. The parking manager for Miami City Parking has received complaints about the time required to exit garages in the downtown sector. To start analyzing the situation she has collected a small sample of data from 12 customers showing the type of payment cash or charge and the garage number Garage Number 1 2 or 3. One possibility is that using credit card payments increases exit times at the parking lots. The manager wishes to develop a joint frequency distribution to better understand the paying habits of those using her garages. To do this she can use the following steps: Step 1 Obtain the data. The paired data for the two variables for a sample of 12 customers are obtained. Customer Payment Method Parking Garage 1 Charge 2 2 Charge 1 3 Cash 2 4 Charge 2 5 Charge 1 Chapter Outcome 3. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Customer Payment Method Parking Garage 6 Cash 1 7 Cash 3 8 Charge 1 9 Charge 3 10 Cash 2 11 Cash 1 12 Charge 1 Step 2 Construct the rows and columns of the joint frequency table. The row variable will be the payment method and two rows will be used corresponding to the two payment methods. The column variable is parking garage number and it will have three levels because the data for this variable contain only the values 1 2 and 3. Note if a variable is continuous classes should be formed using the methods discussed in Example 3. Parking Garage 123 Payment Cash Step 3 Count the number of joint occurrences at each row level and each column level for all combinations of row and column values and place these frequencies in the appropriate cells. Parking Garage 1 2 3 Total Charge 4 2 1 7 Cash 2 2 1 5 Total 6 4 2 12 Step 4 Calculate the row and column totals see Step 3. The manager can now see that for this sample most people charged their parking fee seven people and Garage number 1 was used by most people in the sample used six people. Likewise four people used Garage number 1 and charged their parking fee. END EXAMPLE TRY PROBLEM 12 BUSINESS APPLICATION JOINT FREQUENCY DISTRIBUTION CAPITAL CREDIT UNION CONTINUED Recall that the Capital Credit Union discussed earlier was interested in evaluating the success of its new credit card. Figure 4 showed the frequency distribution and histogram for a sample of customer credit card balances. Although this information is useful the managers would like to know more. Specifically what does the credit card balance distribution look like for male versus female cardholders One way to approach this is to sort the data by the gender variable and develop frequency distributions and histograms for males and females separately. You could then make a visual comparison of the two to determine what if any difference exists between males and females. However an alternative approach is to jointly analyze the two variables: gender and credit card balance. Excel provides a means for analyzing two variables jointly. In Figure 4 we constructed the frequency distribution for the 300 credit card balances using 10 classes. The class width Example 6 Constructing Joint Frequency Distributions A joint frequency distribution is constructed using the following steps: Obtain a set of data consisting of paired responses for two variables. The responses can be qualitative or quantitative. If the responses are quantitative they can be discrete or continuous. Construct a table with r rows and c columns in which the number of rows represents the number of categories or numeric classes of one vari- able and the number of columns corresponds to the number of categories or numeric classes of the second variable. Count the number of joint occur- rences at each row level and each column level for all combina- tions of row and column values and place these frequencies in the appropriate cells. Compute the row and column totals which are called the marginal frequencies. If a joint relative frequency distribution is desired divide each cell frequency by the total number of paired observations. Excel Tutorial Excel tutorials www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data was set at 150. Figure 5 shows a table that is called a joint frequency distribution. This type of table is also called a cross-tabulation table. 1 The Capital Credit Union managers can use a joint frequency table to analyze the credit card balances for males versus females. For instance for the 42 customers with balances of 390 to 539 Figure 5 shows that 33 were males and 9 were females. Previously we discussed the concept of relative frequency proportions which Excel converts to percent- ages as a useful tool for making comparisons between two data sets. In this example comparisons between males and females would be easier if the frequencies were converted to proportions or percentages. The result is the joint relative frequency table shown in Figure 6. Notice that the percentages in each cell are percentages of the total 300 people in the survey. For example the 540-to-689 class had 20.33 61 of the 300 customers. The male customers with balances in the 540-to-689 range constituted 15 45 of the 300 customers whereas females with that balance level made up 5.33 16 of all 300 custom- ers. On the surface this result seems to indicate a big difference between males and females at this credit balance level. Suppose we really wanted to focus on the male versus female issue and control for the fact that there are far more male customers than female. We could compute the percentages differently. Rather than using a base of 300 the entire sample size we might instead be inter- ested in the percentages of the males who have balances at each level and the same measure for females. 2 There are many options for transferring data into useful information. Thus far we have introduced frequency distributions joint frequency tables and histograms. In the next section we discuss one of the most useful graphical tools: the bar chart. 1 In Excel the joint frequency distribution is developed using a tool called Pivot tables. Minitab Instructions for similar results: 1. Open fle: Capital.MTW. 2. Click on Data Code Numeric to Text. 3. Under Code data from columns select data column. 4. Under Into columns specify destination column: Classes. 5. In Original values defne each data class range. 6. In New specify code for each class. 7. Click OK. 8. Click on Stat Tables Cross Tabulation and Chi-Square. 9. Under Categorical Variables For rows enter Classes column and For columns enter Gender column. 10. Under Display check Counts. 11. Click OK. Excel 2010 Instructions: 1. Open fle: Capital.xlsx. 2. Place cursor anywhere in the data. 3. On the Insert tab click on PivotTable and click OK. 4. On the Options tab select Options in the PivotTable group. Select the Display tab and check Classic PivotTable layout. Click OK. 5. Drag Credit Card Account Balance to “Drop Row Fields Here” area. 6. Right-click in Credit Card Account Balance numbers and click Group. 7. Change Start at to 90. Change End to 1589. Change By to 150. 7. Drag Gender to “Drop Column Fields Here” area. 8. Drag Credit Card Account Balance to “Drop Value Fields Here” area. 9. Place cursor in the Data Item area right click and select Summarize Values By and select Count. FIGURE 5 | Excel 2010 Joint Frequency Distribution for Capital Credit Union 2 Such distributions are known as marginal distributions. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Excel 2010 Instructions: 1. Place cursor in the Gender numbers of the PivotTable. 2. Right-click and select Value Field Settings. 3. On the Show values as tab click on the down arrow and select of Grand Total. 4. Click OK. Minitab Instructions for similar results: 1. Open fle: Capital.MTW. 2. Steps 2–7 as in Figure 6. 3. Click on Stat Tables Cross Tabulation and Chi-square. 4. Under Categorical variables For rows enter Classes column and For columns enter Gender column. 5. Under Display check Total Percents. 6. Click OK. In Figure 6 we have used the Data Field Settings in the Excel PivotTable to represent the data as percentages. FIGURE 6 | Excel 2010 Joint Relative Frequencies for Capital Credit Union Skill Development 2-1. Given the following data develop a frequency distribution: 5 326 6 7 336 7 7 975 3 12 6 10 7 2 6 807 4 2-2. Assuming you have data for a variable with 2000 values using the 2 k Ú n guideline what is the smallest number of groups that should be used in developing a grouped data frequency distribution 2-3. A study is being conducted in which a variable of interest has 1000 observations. The minimum value in the data set is 300 points and the maximum is 2900 points. a. Use the 2 k Ú n guideline to determine the minimum number of classes to use in developing a grouped data frequency distribution. b. Based on the answer to part a determine the class width that should be used round up to the nearest 100 points. 2-4. Produce the relative frequency distribution from a sample size of 50 that gave rise to the following ogive: 0.0 0 100 200 300 Sales 400 500 600 0.2 0.4 0.6 Cumulative Relative Frequency 0.8 1.0 Ogive 2-5. You have the following data: 8 6 11 14 10 11972 8 9555 12 784 17 8 12788 7 10869 9 11 16 2 7 4 8445 5 9966 7 7954 5 14290 6 1 1 12 11 4 2- MyStatLab www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data a. Construct a frequency distribution for these data. Use the 2 k Ú n guideline to determine the number of classes to use. b. Develop a relative frequency distribution using the classes you constructed in part a. c. Develop a cumulative frequency distribution and a cumulative relative frequency distribution using the classes you constructed in part a. d. Develop a histogram based on the frequency distribution you constructed in part a. 2-6. Fill in the missing components of the following frequency distribution constructed for a sample size of 50: Class Frequency Relative Frequency Cumulative Relative Frequency 7.85 -6 0.12 -6 8.05 0.48 8.05 -6 4 -6 8.25 0.10 8.25 -6 2-7. The following cumulative relative frequency distribution summarizes data obtained in a study of the ending overages in dollars for the cash register balance at a business: Class Frequency Relative Frequency Cumulative Relative Frequency –60.00––40.00 2 0.04 0.04 –40.00––20.00 2 0.04 0.08 –20.00––00.00 8 0.16 0.24 00.00– 20.00 16 0.32 0.56 20.00– 40.00 20 0.40 0.96 40.00– 60.00 2 0.04 1.00 a. Determine the proportion of the days in which there were no shortages. b. Determine the proportion of the days the cash register was less than 20 off. c. Determine the proportion of the days the cash register was less than 40 over or at most 20 short. 2-8. You are given the following data: 6 10 6495 5 5 5762 5 5 5457 6 7 8684 7 5 5557 8 7 6754 6 4 4746 6 7 8676 7 8 5657 3 6 4744 a. Construct a frequency distribution for these data. b. Based on the frequency distribution develop a histogram. c. Construct a relative frequency distribution. d. Develop a relative frequency histogram. e. Compare the two histograms. Why do they look alike 2-9. Using the data from Problem 2-8 a. Construct a grouped data relative frequency distribution of the data. Use the 2 k Ú n guideline to determine the number of classes. b. Construct a cumulative frequency distribution of the data. c. Construct a relative frequency histogram. d. Construct an ogive. Business Applications 2-10. Burger King is one of the largest fast-food franchise operations in the world. Recently the district manager for Burger King in Las Vegas conducted a study in which she selected a random sample of sales receipts. She was interested in the number of line items on the receipts. For instance if a customer ordered two 1/4-pound hamburgers one side of fries and two soft drinks the number of line items would be five. The following data were observed: 7 5 76 5548 65 8 7 65 6294 45 8 4 96 6589 91 6 5 106 7655 56 8 7 68 6696 12 7 5 6 7 11 4434 14 11 2 5 5 8234 96 6 5 86 3645 8 10 a. Develop a frequency distribution for these data. Discuss briefly what the frequency distribution tells you about these sample data. b. Based on the results in part a construct a frequency histogram for these sample data. 2-11. In a survey conducted by AIG investors were asked to rate how knowledgeable they felt they were as investors. Both online and traditional investors were included in the survey. The survey resulted in the following data: Of the online investors 8 55 and 37 responded th ey were “savvy” “experienced” and “novice” res pecti v ely . Of the traditional investors the percentages were 4 29 and 67 respectively. Of the 600 investors surveyed 200 were traditional investors. a. Use the information to construct a joint frequency distribution. b. Use the information to construct a joint relative frequency distribution. c. Determine the proportion of investors who were both online investors and rated themselves experienced. d. Calculate the proportion of investors who were online investors. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 2-14. The bubble in U.S. housing prices burst in 2008 causing sales of houses to decline in almost every part of the country. Many homes were foreclosed because the owners could not make the payments. Below is a sample of 100 residential properties and the total balance on the mortgage at the time of foreclosure. 172229 211021 159205 247697 247469 176736 240815 195056 315097 257150 129779 207451 165225 178970 319101 87429 219808 242761 277389 213803 153468 205696 210447 179029 241331 117808 188909 376644 185523 168145 158094 135461 131457 263232 256262 240034 289973 302341 178684 226998 176440 268106 181507 118752 251009 196457 195249 195986 201680 233182 271552 123262 212411 246462 177673 103699 252375 192335 265992 232247 320004 213020 192546 295660 211876 265787 207443 203043 133014 289645 251560 302054 185381 284345 184869 237485 282506 278783 335920 199630 248272 232234 188833 168905 357612 241894 186956 114601 301728 251865 207040 221614 318154 156611 219730 201473 174840 196622 263686 159029 a. Using the 2 k Ú n guideline what is the minimum number of classes that should be used to display these data in a grouped data frequency distribution b. Referring to part a what should the class width be assuming you round the width up to nearest 1000 c. Referring to parts a and b develop a grouped data frequency distribution for these mortgage balance data. d. Based on your answer to part c construct and interpret a frequency histogram for the mortgage balance data. 2-15. Wageweb exhibits salary data obtained from surveys. It provides compensation information on more than 170 benchmark positions including finance positions. It recently reported that salaries of chief finance officers CFOs ranged from 127735 to 209981 before bonuses. Suppose the following data represent a sample of the annual salaries for 25 CFOs. Assume that data are in thousands of dollars. 173.1 171.2 141.9 112.6 211.1 156.5 145.4 134.0 192.0 185.8 168.3 131.0 214.4 155.2 164.9 123.9 161.9 162.7 178.8 161.3 182.0 165.8 213.1 177.4 159.3 a. Using 11 classes construct a cumulative frequency distribution. b. Determine the proportion of CFO salaries that are at least 175000. c. Determine the proportion of CFO salaries that are less than 205000 and at least 135000. 2-12. The sales manager for the Fox News TV station affiliate in a southern Florida city recently surveyed 20 advertisers and asked each one to rate the service of the station on the following scale: Very Good Good Fair Poor Very Poor 12 3 4 5 He also tracked the general time slot when the advertiser’s commercials were shown on the station. The following codes were used: 1 morning 2 afternoon 3 evening 4 various times The following sample data were observed: Rating Time Slot Rating Time Slot 2143 1122 3333 2133 1121 4411 2211 1153 2124 2234 a. Construct separate relative frequency distributions for each of the two variables. b. Construct a joint frequency distribution for these two variables. c. Construct a joint relative frequency distribution for these two variables. Write a short paragraph describing what the data imply. 2-13. A St. Louis–based shipping company recently selected a random sample of 49 airplane weight slips for crates shipped from an automobile parts supplier. The weights measured in pounds for the sampled crates are as follows: 89 83 97 101 86 89 86 91 84 89 87 93 86 90 86 92 92 88 88 92 86 93 80 93 77 98 94 95 94 88 95 87 99 98 90 91 87 89 89 96 88 94 95 79 94 86 92 94 85 a. Create a data array of the weights. b. Develop a frequency distribution using five classes having equal widths. c. Develop a histogram from the frequency distribution you created in part b. d. Develop a relative frequency and a cumulative relative frequency distribution for the weights using the same five classes created in part b. What percent of the sampled crates have weights greater than 96 pounds www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 2-16. One effect of the great recession was to lower the interest rate on fixed-rate mortgages. A sample of 30-year fixed-rate mortgage rates taken from financial institutions in the Pacific Northwest resulted in the following: 3.79 4.03 3.92 3.87 3.86 3.93 3.87 3.69 3.99 3.88 3.91 3.81 3.85 3.81 3.65 4.15 3.98 3.82 4.08 3.84 3.95 4.03 3.96 3.69 3.97 4.08 3.86 3.82 3.83 4.19 3.49 3.49 3.75 3.46 3.92 3.67 3.63 3.81 3.86 4.15 3.85 4.01 3.93 4.02 4.09 a. Construct a histogram with eight classes beginning at 3.46. b. Determine the proportion of 30-year fixed mortgage rates that are at least 3.76. c. Produce an ogive for the data. Computer Database Exercises 2-17. J.D. Power and Associates’ annual customer- satisfaction survey the Automotive Performance Execution and Layout APEAL Study SM in its 13th year was released on September 22 2008. The study measures owners’ satisfaction with the design content layout and performance of their new vehicles. A file titled APEAL2 contains the satisfaction ratings for 2008 for each make of car. a. Construct a histogram that starts at 710 and has class widths of 20 for the APEAL ratings. b. The past industry average APEAL rating was 866 for 2005. What does the 2008 data suggest in terms of the relative satisfaction with the 2008 models 2-18. The Franklin Tire Company is interested in demonstrating the durability of its steel-belted radial tires. To do this the managers have decided to put four tires on 100 different sport utility vehicles and drive them throughout Alaska. The data collected indicate the number of miles rounded to the nearest 1000 miles that each of the SUVs traveled before one of the tires on the vehicle did not meet minimum federal standards for tread thickness. The data file is called Franklin. a. Construct a frequency distribution and histogram using eight classes. Use 51 as the lower limit of the first class. b. The marketing department wishes to know the tread life of at least 50 of the tires the 10 that had the longest tread life and the longest tread life of these tires. Provide this information to the marketing department. Also provide any other significant items that point out the desirability of this line of steel- belted tires. c. Construct a frequency distribution and histogram using 12 classes using 51 as the lower limit of the first class. Compare your results with those in parts a and b. Which distribution gives the best information about the desirability of this line of steel-belted tires Discuss. 2-19. The California Golf Association recently conducted a survey of its members. Among other questions the members were asked to indicate the number of 18-hole rounds that they played last year. Data for a sample of 294 members are provided in the data file called Golf Survey. a. Using the 2 k Ú n guideline what is the minimum number of classes that should be used to display these data in a grouped data frequency distribution b. Referring to part a what should the class width be assuming you round the width up to the nearest integer c. Referring to parts a and b develop a grouped data frequency distribution for these golf data. d. Based on your answer to part c construct and interpret a frequency histogram for the data. 2-20. Ars Technia LLD published a news release Eric Bangeman “Dell Still King of Market Share” that presented the results of a study concerning the world market share for the major manufacturers of personal computers. It indicated that Dell held 17.9 of this market. The file titled PCMarket contains a sample of the market shares alluded to in the article. a. Construct a histogram from this set of data and identify the sample shares for each of the listed manufacturers. b. Excluding the data referred to as “other” determine the total share of the sample for manufacturers that have headquarters in the United States. 2-21. Orlando Florida is a well-known popular vacation destination visited by tourists from around the world. Consequently the Orlando International Airport is busy throughout the year. Among the variety of data collected by the Greater Orlando Airport Authority is the number of passengers by airline. The file Orlando Airport contains passenger data for December 2011. Suppose the airport manager is interested in analyzing the column labeled “Total” for this data. a. Using the 2 k Ú n guideline what is the minimum number of classes that should be used to display the data in the “Total” column in a grouped data frequency distribution b. Referring to part a what should the class width be assuming you round the width up to the nearest 1000 passengers c. Referring to parts a and b develop a grouped data frequency distribution for these airport data. d. Based on your answer to part c construct and interpret a frequency histogram for the data. 2-22. The manager of AJ’s Fitness Center a full-service health and exercise club recently conducted a survey of 1214 members. The objective of the survey was to www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 2-23. The file Danish Coffee contains data on individual coffee consumption in kg for 144 randomly selected Danish coffee drinkers. a. Construct a data array of the coffee consumption data. b. Construct a frequency distribution of the coffee consumption data. Within what class do more of the observations fall c. Construct a histogram of the coffee consumption data. Briefly comment on what the histogram reveals concerning the data. d. Develop a relative frequency distribution and a cumulative relative frequency distribution of the coffee data. What percentage of the coffee drinkers sampled consume 8.0 kg or more annually determine the satisfaction level of his club’s customers. In addition the survey asked for several demographic factors such as age and gender. The data from the survey are in a file called AJFitness. a. One of the key variables is “Overall Customer Satisfaction.” This variable is measured on an ordinal scale as follows: 5 very satisfied 4 satisfied 3 neutral 2 dissatisfied 1 very dissatisfied Develop a frequency distribution for this variable and discuss the results. b. Develop a joint relative frequency distribution for the variables “Overall Customer Satisfaction” and “Typical Visits Per Week.” Discuss the results. END EXERCISES 2-1 Bar Charts Pie Charts and Stem and Leaf Diagrams Bar Charts Section 1 introduced some of the basic tools for describing numerical variables both discrete and continuous when the data are in their raw form. However in many instances you will be working with categorical data or data that have already been summarized to some extent. In these cases an effective presentation tool is often a bar chart. BUSINESS APPLICATION DEVELOPING BAR CHARTS NEW CAR SALES The automobile industry is a significant part of the U.S economy. When car sales are up the economy is up and vice versa. Table 8 displays data showing the total number of cars sold in January 2012 by the eight largest automobile companies in the world. Although the table format is informative a graphical presentation is often desirable. Because the car sales data are characterized by car company a bar chart would work well in this instance. The bars on a bar chart can be vertical called a column bar chart or horizontal called a horizontal bar chart. Figure 7 illustrates an example of a column bar chart. The height of the bars corresponds to the number of cars sold by each company. This gives you an idea of the sales advantage held by General Motors in January 2012. One strength of the bar chart is its capability of displaying multiple variables on the same chart. For instance a bar chart can conveniently compare new car sales data for January 2012 and Bar Chart A graphical representation of a categorical data set in which a rectangle or bar is drawn over each category or class. The length or height of each bar represents the frequency or percentage of observations or some other measure associated with the category. The bars may be vertical or horizontal. The bars may all be the same color or they may be different colors depicting different categories. Additionally multiple variables can be graphed on the same bar chart. Chapter Outcome 4. TABLE 8 | January 2012 New Car Sales for the Top Eight Automobile Companies United States Car Company January 2012 Sales General Motors 167900 Ford 136300 Chrysler 101150 Toyota 125500 Honda 83000 Nissan 79300 Hyundai 42700 Mazda 24000 Source: Wall Street Journal Online February 1 2012. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 General Motors Ford Chrysler Toyota Honda Nissan Hyundai Mazda Automobile Company January 2012 New Car Sales FIGURE 7 | Bar Chart Showing January 2012 New Car Sales sales for the same month the previous year. Figure 8 is a horizontal bar chart that does just that. Notice that only General Motors had lower sales in January 2012 than in the previous January. People sometimes confuse histograms and bar charts. Although there are some similari- ties they are two very different graphical tools. Histograms are used to represent a frequency distribution associated with a single quantitative ratio- or interval-level variable. Refer to the histogram illustrations in Section 1. In every case the variable on the horizontal axis was numerical with values moving from low to high. The vertical axis shows the frequency count 0 100000 150000 200000 Automobile Company New Car Sales in U.S. 50000 Ford General Motors Toyota Chrysler Honda Nissan Hyundai Mazda 2012 Sales up 68 2012 Sales up 15 January 2011 Sales 2012 Sales up 10 2012 Sales up 9 2012 Sales up 8 2012 Sales up 44 2012 Sales up 7 2012 Sales down 6 January 2012 Sales FIGURE 8 | Bar Chart Comparing January 2011 and January 2012 New Cars Sold www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data or relative frequency for each numerical value or range of values. There are no gaps between the histogram bars. On the other hand bar charts are used when one or more variables of interest are categorical as in this case in which the category is “car company.” EXAMPLE 7 BAR CHARTS Investment Recommendations Fortune contained an article by David Stires called “The Best Stocks to Buy Now.” The article identified 40 companies as good investment opportunities. These companies were divided into five categories: Growth and Income Bar- gain Growth Deep V alue Small Wonders and Foreign V alue. For each company data for sev- eral key variables were reported including the price/earnings PE ratio based on the previous 12 months’ reported earnings. We are interested in constructing a bar chart of the PE ratios for the eight companies classified as Growth and Income. Step 1 Define the categories. Data for stock price and PE ratio for each of eight companies is shown as follows: Company Ticker Symbol PE Ratio Stock Price Abbott Labs ABT 21 49 Altria Group MO 14 65 Coca-Cola KO 21 42 Colgate-Palmolive CL 20 51 General Mills GIS 17 51 Pfizer PFE 13 29 Procter Gamble PG 21 53 Wyeth WYE 15 43 The category to be displayed is the company. Step 2 Determine the appropriate measure to be displayed. The measure of interest is the PE ratio. Step 3 Develop the bar chart. A column bar chart is developed by placing the eight companies on the horizontal axis and constructing bars whose heights correspond to the value of the company’s PE ratio. Each company is assigned a different-colored bar. The resulting bar chart is Abbott Labs ABT 0 5 10 15 Price/Earnings Ratio 20 21 14 21 20 17 13 21 15 25 Price/Earnings Ratio Altria Group MO Coca-Cola KO Colgate- Palmolive CL Company General Mills GIS Pfizer PFE Procter Gamble PG Wyeth WYE Example 7 Constructing Bar Charts A bar chart is constructed using the following steps: Define the categories for the variable of interest. For each category determine the appropriate measure or value. For a column bar chart locate the categories on the horizontal axis. The vertical axis is set to a scale corresponding to the values in the categories. For a horizon- tal bar chart place the categories on the vertical axis and set the scale of the horizontal axis in accordance with the values in the categories. Then construct bars either vertical or horizontal for each category such that the length or height corresponds to the value for the category. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Step 4 Interpret the results. The bar chart shows three companies with especially low PE ratios. These are Altria Group Pfizer and Wyeth. Thus of the eight recommended companies in the Growth and Income group these three have the lowest PE ratios. END EXAMPLE TRY PROBLEM 27 BUSINESS APPLICATION CONSTRUCTING BAR CHARTS BACH LOMBARD WILSON One of the most useful features of bar charts is that they can display multiple issues. Consider Bach Lombard Wilson a New England law firm. Recently the firm handled a case in which a woman was suing her employer a major electronics firm claiming the company gave higher starting salaries to men than to women. Consequently she stated even though the company tended to give equal-percentage raises to women and men the gap between the two groups widened. Attorneys at Bach Lombard Wilson had their staff assemble massive amounts of data. Table 9 provides an example of the type of data they collected. A bar chart is a more effective way to convey this information as Figure 9 shows. From this graph we can quickly see that in all years except 2009 the starting salaries for males did exceed those for females. The bar chart also illustrates that the general trend in starting salaries for both groups has been increasing though with a slight downturn in 2011. Do you think the information in Figure 9 alone is suf- ficient to rule in favor of the claimant in this lawsuit Bar charts like the one in Figure 9 that display two or more variables are referred to as cluster bar charts. Average Starting Salaries 80000 60000 40000 20000 70000 50000 30000 10000 0 2006 2007 2008 2009 2010 2011 2012 Female Male Males tend to have higher starting salaries. General upward trend in salaries. FIGURE 9 | Bar Chart of Starting Salaries TABLE 9 | Salary Data for Bach Lombard Wilson Year Males: Average Starting Salaries Females: Average Starting Salaries 2006 44456 41789 2007 47286 46478 2008 56234 53854 2009 57890 58600 2010 63467 59070 2011 61090 55321 2012 67543 64506 Excel Tutorial Excel tutorials www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Pie Charts Another graphical tool that can be used to transform data into information is the pie chart. EXAMPLE 8 PIE CHARTS Golf Equipment A survey was recently conducted of 300 golfers that asked questions about the impact of new technology on the game. One question asked the golfers to indicate which area of golf equipment is most responsible for improving an amateur golfer’s game. The following data were obtained: Equipment Frequency Golf ball 81 Club head material 66 Shaft material 63 Club head size 63 Shaft length 3 Don’t know 24 To display these data in pie chart form use the following steps: Step 1 Define the categories. The categories are the six equipment-response categories. Step 2 Determine the appropriate measure. The appropriate measure is the proportion of the golfers surveyed. The proportion for each category is determined by dividing the number of golfers in a category by the total sample size. For example for the category “golf ball” the percentage is 81300 0.27 27. Step 3 Construct the pie chart. The pie chart is constructed by dividing a circle into six slices one for each category such that each slice is proportional to the percentage of golfers in the category. Club Head Material 22 Shaf Material 21 Club Head Size 21 Golf Ball 27 Don’t Know 8 Shaf Length 1 Golf Equipment Impact END EXAMPLE TRY PROBLEM 28 Example 8 Constructing Pie Charts A pie chart is constructed using the following steps: Define the categories for the variable of interest. For each category determine the appropriate measure or value. The value assigned to each category is the proportion the category is to the total for all categories. Construct the pie chart by displaying one slice for each category that is proportional in size to the proportion the cat- egory value is to the total of all categories. Pie Chart A graph in the shape of a circle. The circle is divided into “slices” corresponding to the categories or classes to be displayed. The size of each slice is proportional to the magnitude of the displayed variable associated with each category or class. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Pie charts are sometimes mistakenly used when a bar chart would be more appropri- ate. For example a few years ago the student leaders at Boise State University wanted to draw attention to the funding inequities among the four public universities in Idaho. To do so they rented a large billboard adjacent to a major thoroughfare through downtown Boise. The billboard contained a large pie chart like the one shown in Figure 10 where each slice indicated the funding per student at a given university. However for a pie chart to be appro- priate the slices of the pie should represent parts of a total. But in the case of the billboard that was not the case. The amounts merely represented the dollars of state money spent per student at each university. The sum of the four dollar amounts on the pie chart was a meaningless number. In this case a bar chart like that shown in Figure 11 would have been more appropriate. Lewis and Clark College 5410 Idaho State University 6320 University of Idaho 7 143 Boise State University 5900 FIGURE 10 | Pie Chart: Per-Student Funding for Universities Boise State University 0 1000 2000 3000 4000 5000 6000 7 000 8000 Boise State University 5900 University of Idaho 7 143 Idaho State University 6320 Lewis and Clark College 5410 University of Idaho Idaho State University Lewis and Clark College FIGURE 11 | Bar Chart: Per-Student Funding for Universities www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Stem and Leaf Diagrams Another graphical technique useful for doing an exploratory analysis of quantitative data is called the stem and leaf diagram. The stem and leaf diagram is similar to the histogram introduced in Section 1 in that it displays the distribution for the quantitative variable. How- ever unlike the histogram in which the individual values of the data are lost if the vari- able of interest is broken into classes the stem and leaf diagram shows the individual data values. Although Excel does not have a stem and leaf procedure the PHStat add-ins to Excel do have a stem and leaf procedure. EXAMPLE 9 STEM AND LEAF DIAGRAMS Walk-In Health Clinic The administrator for the Walk-In Health Clinic in Sydney Australia is interested in performing an analysis of the number of patients who enter the clinic daily. One method for analyzing the data for a sample of 200 days is the stem and leaf diagram. The following data represent the number of patients on each of the 200 days: 113 112 63 127 110 129 142 115 192 94 165 121 105 140 85 93 105 140 93 126 183 118 67 104 162 110 76 109 91 132 88 96 132 80 144 112 57 139 123 124 172 149 198 114 88 111 133 117 138 134 53 147 108 109 153 89 159 99 130 93 161 118 115 117 128 98 125 184 134 132 117 127 166 72 122 109 124 92 82 69 110 128 151 67 142 177 135 121 143 89 160 115 138 79 104 76 89 110 44 140 117 103 59 109 145 117 162 108 141 139 148 175 107 117 87 87 150 152 80 168 88 127 131 85 143 101 137 111 128 147 110 81 111 149 154 90 150 117 101 116 153 176 112 147 87 177 190 66 62 154 143 122 176 153 97 106 86 62 146 98 134 135 127 118 109 143 146 152 140 95 102 137 158 69 122 135 136 129 91 136 135 86 131 154 132 59 136 85 142 137 155 190 120 154 102 109 97 157 144 149 The stem and leaf diagram is constructed using the following steps: Step 1 Sort the data from low to high. The lowest value is 44 patients and the highest value is 198 patients. Step 2 Split the values into a stem and leaf. Stem tens place leaf units place For example for the value 113 the stem is 11 and the leaf is 3. We are keeping one digit for the leaf. Step 3 List all possible stems from lowest to highest. Step 4 Itemize the leaves from lowest to highest and place next to the appropriate stems. Example 9 Constructing Stem and Leaf Diagrams To construct the stem and leaf diagram for a quantitative variable use the following steps: Sort the data from low to high. Analyze the data for the vari- able of interest to determine how you wish to split the values into a stem and a leaf. List all possible stems in a sin- gle column between the lowest and highest values in the data. For each stem list all leaves associated with the stem. Chapter Outcome 5. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 44 5 3 7 9 9 6 2 2 3 6 7 7 9 9 7 2 6 6 9 8 0 0 1 2 5 5 5 6 6 7 7 7 8 8 8 9 9 9 9 0 1 1 2 3 3 3 4 5 6 7 7 8 8 9 10 1 1 2 2 3 4 4 5 5 6 7 8 8 9 9 9 9 9 9 11 0 0 0 0 0 1 1 1 2 2 2 3 4 5 5 5 6 7 7 7 7 7 7 7 8 8 8 12 0 1 1 2 2 2 3 4 4 5 6 7 7 7 7 8 8 8 9 9 13 0 1 1 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 14 0 0 0 0 1 2 2 2 3 3 3 3 4 4 5 6 6 7 7 7 8 9 9 9 15 0 0 1 2 2 3 3 3 4 4 4 4 5 7 8 9 16 0 1 2 2 5 6 8 17 2 5 6 6 7 7 18 3 4 19 0 0 2 8 The stem and leaf diagram shows that most days have between 80 and 160 patients with the most frequent value in the 110- to 120-patient range. END EXAMPLE TRY PROBLEM 25 MyStatLab Skill Development 2-24. The following data reflect the percentages of employees with different levels of education: Education Level Percentage Less than high school graduate 18 High school graduate 34 Some college 14 College graduate 30 Graduate degree 4 Total 100 a. Develop a pie chart to illustrate these data. b. Develop a horizontal bar chart to illustrate these data. 2-25. Given the following data construct a stem and leaf diagram: 0.7 1.7 2.8 3.8 0.8 1.8 3.3 4.3 1.0 2.0 4.4 5.4 1.1 2.1 5.3 6.3 1.4 2.4 5.4 6.4 2.0 3.0 2-26. A university has the following number of students at each grade level. Freshman 3450 Sophomore 3190 Junior 2780 Senior 1980 Graduate 750 a. Construct a bar chart that effectively displays these data. b. Construct a pie chart to display these data. c. Referring to the graphs constructed in parts a and b indicate which you would favor as the most effective way of presenting these data. Discuss. 2-27. Given the following sales data for product category and sales region construct at least two different bar charts that display the data effectively: Region East West North South Product Type XJ-6 Model 200 300 50 170 X-15-Y Model 100 200 20 100 Craftsman 80 400 60 200 Generic 100 150 40 50 2-28. The 2010 Annual Report of Murphy Oil Corporation reports the following refinery yields in barrels per day by product category for the United States and the United Kingdom. United States 2010 Product Category Refinery Yields— barrels per day Gasoline 61128 Kerosene 11068 Diesel and Home Heating Oils 41305 Residuals 18082 Asphalt LPG and other 14802 Fuel and Loss 834 2- www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data United Kingdom 2010 Product Category Refinery Yields— barrels per day Gasoline 20889 Kerosene 11374 Diesel and Home Heating Oils 25995 Residuals 8296 Asphalt LPG and other 14799 Fuel and Loss 2810 a. Construct a pie chart to display United States refinery yields by product per day. Display the refinery yields data for each product category as a percentage of total refinery yields for all product categories. b. Construct a pie chart to display United Kingdom refinery yields by product per day. Display the refinery yields data for each product category as a percentage of total refinery yields for all product categories. c. Construct a bar chart that effectively compares United States and United Kingdom refinery yields by product category. 2-29. Boston Properties is a real estate investment trust REIT that owns first-class office properties in selected markets. According to its 2010 annual report its percentage of net operating income distribution by region as a percent of total income for the year ended December 31 2010 was Region 2010 Income Distribution New Y ork 41 Washington D.C. 23 Boston 21 San Francisco 12 Princeton 3 a. Construct a pie chart to display the percentage of net operating income distribution by region for 2010. b. Construct a bar chart to display the net operating income distribution by region for 2010. c. Briefly comment on which of the two charts you believe better summarizes and displays the data. 2-30. Hileman Services Company recently released the following data concerning its operating profits in billions for the five years: Year 2004 2005 2006 2007 2008 Profit 0.5 0.1 0.7 0.5 0.2 a. Construct a bar chart to graphically display these data. b. Construct a pie chart to graphically display these data. c. Select the display that most effectively displays the data and provide reasons for your choice. 2-31. DaimlerChrysler recently sold its Chrysler division to a private equity firm. Before the sale it reported its first- half revenues in billions as follows: Division Mercedes Chrysler Commercial Vehicles Financial Services Total Revenues 27.7 30.5 23.2 8.9 90.3 a. Produce a bar chart for these data. b. Determine the proportion of first-half revenues accounted for by its vehicle divisions. Business Applications 2-32. At the March meeting of the board of directors for the Graystone Services Company one of the regional managers put the following data on the overhead projector to illustrate the ratio of the number of units manufactured to the number of employees at each of Graystone’s five manufacturing plants: Plant Location Units Manufactured/Employees Bismarck ND 14.5 Boulder CO 9.8 Omaha NE 13.0 Harrisburg PA 17.6 Portland ME 5.9 a. Discuss whether a pie chart or a bar chart would be most appropriate to present these data graphically. b. Construct the chart you recommended in part a. 2-33. The first few years after the turn of the century saw a rapid increase in housing values followed by a rapid decline due in part to the sub-prime crisis. The following table indicates the increase in the number of homes valued at more than one million dollars before 2005. Year Number of 1 Million Homes 2000 394878 2001 495600 2002 595441 2003 714467 2004 1034386 Develop a horizontal bar chart to represent these data in graphical form. 2-34. The pharmaceutical industry is a very fast-growing segment of the U.S. and international economies. Recently there has been controversy over how studies are done to show that drugs are both safe and effective. One drug product Cymbalta which is an antidepressant was purported in a published abstract of an article in a medical journal to be superior to other competing products. Y et the article itself stated that no studies had actually been done to show such comparisons between Cymbalta and other competing products. In an August 2005 report in an article titled “Reading Fine Print Insurers Question Drug Studies” in The Wall Street Journal the following data www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data were presented showing the U.S. sales of antidepressant drugs by major brand. The sales data for the first half of 2005 are shown in the following table. Antidepressant Drug Sales First Half 2005 in Billions Effexor XR 1.29 Lexapro 1.03 Zoloft 1.55 Cymbalta 0.27 Other 0.97 Construct an appropriate graph to display these data. 2-35. The number of branded retail outlets in the United States and the United Kingdom for Murphy Oil Corporation as of December 31 of each year from 2001 to 2010 is shown below: Branded Retail Outlets 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 United States total 815 914 994 1127 1201 1164 1126 1154 1169 1215 United Kingdom 411 416 384 358 412 402 389 454 453 451 Develop a chart that effectively compares the number of branded retail outlets in North America with the number in the United Kingdom. 2-36. The 2011 Annual Report of the Procter Gamble Company reported the following net sales information for its six global business segments: Global Segment 2011 Net Sales Millions Beauty 20157 Grooming 8025 Health Care 12033 Snacks and Pet Care 3156 Fabric Care and Home Care 24837 Baby Care and Family Care 15606 a. Construct a bar chart that displays this information by global business segment for 2011. b. Construct a pie chart that displays each global business segment’s net sales as a percentage of total global segment net sales for 2011. 2-37. A fast-food restaurant monitors its drive-thru service times electronically to ensure that its speed of service is meeting the company’s goals. A sample of 28 drive- thru times was recently taken and is shown here. Speed of Service Time in Seconds 83 138 145 147 130 79 156 156 90 85 68 93 178 76 73 119 92 146 88 103 116 134 162 71 181 110 105 74 a. Construct a stem and leaf diagram of the speed of service times. b. What range of time might the restaurant say is the most frequent speed of service 2-38. A random sample of 30 customer records for a physician’s office showed the following time in days to collect insurance payments: Number of Days to Collect Payment 34 55 36 39 36 32 35 30 47 31 60 66 48 43 33 24 37 38 65 35 22 45 33 29 41 38 35 28 56 56 a. Construct a stem and leaf diagram of these data. b. Within what range of days are most payments collected 2-39. USA Today presented data to show that major airlines accounting for more than half of capacity were expected to be in bankruptcy court. The total seat capacity of major airlines was 858 billion at the time. For airlines expected to be in bankruptcy court the following data were presented: Airline Seat Capacity in Billions Airline United Delta Northwest U.S. Airways A TA Capacity 145 130 92 54 21 a. Construct a bar graph representing the contribution to the total seat capacity of the major airlines for the five airlines indicated. b. Produce a pie chart exhibiting the percentage of the total seat capacity for the five major airlines expected to be in bankruptcy court and the combined capacity of all others. c. Calculate the percentage of the total capacity of the airlines expected to be in bankruptcy court. Was USA Today correct in the percentage stated 2-40. Many of the world’s most successful companies rely on The NPD Group to provide global sales and marketing information that helps clients make more informed fact-based decisions to optimize their businesses. These customers need NPD help for insight on what is selling where and why so that they can understand and leverage the latest trends. They recently July 2009 released the following results of a survey intended to determine the market share distribution for the major corporations that make digital music devices: Corporation Apple SanDisk Creative Technology iRiver Samsung Market Share 74 6.4 3.9 3.6 2.6 www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data a. Generate a bar chart to display these data. b. Generate a pie chart to display these data. c. Which of the two displays most effectively presents the data Explain your answer. Computer Database Exercises 2-41. The Honda Ridgeline was among the highest-ranked compact pickups in J.D. Power and Associates’ annual customer-satisfaction survey. The study also found that models with high ratings have a tendency to stay on dealers’ lots a shorter period of time. As an example the Honda Ridgeline had stayed on dealers’ lots an average of 24 days. The file titled Honda contains 50 lengths of stay on dealers’ lots for Ridgeline trucks. a. Construct a stem and leaf display for these data. b. Determine the average length of stay on dealers’ lots for the Honda Ridgeline. Does this agree with the average obtained by J.D. Power and Associates Explain the difference. 2-42. The manager for Capital Educators Federal Credit Union has selected a random sample of 300 of the credit union’s credit card customers. The data are in a file called Capital. The manager is interested in graphically displaying the percentage of card holders of each gender. a. Determine the appropriate type of graph to use in this application. b. Construct the graph and interpret it. 2-43. Recently a study was conducted in which a random sample of hospitals was selected from each of four categories of hospitals: university related religious related community owned and privately owned. At issue is the hospital charges associated with outpatient gall bladder surgery. The following data are in the file called Hospitals: University Related Religious Affiliated Municipally Owned Privately Held 6120 4010 4320 5100 5960 3770 4650 4920 6300 3960 4575 5200 6500 3620 4440 5345 6250 3280 4900 4875 6695 3680 4560 5330 University Related Religious Affiliated Municipally Owned Privately Held 6475 3350 4610 5415 6250 3250 4850 5150 6880 3400 5380 6550 a. Compute the average charge for each hospital category. b. Construct a bar chart showing the averages by hospital category. c. Discuss why a pie chart would not in this case be an appropriate graphical tool. 2-44. Amazon.com has become one of the success stories of the Internet age. Its growth can be seen by examining its increasing sales volume in billions and the net income/loss during Amazon’s operations. A file titled Amazon contains these data for its first 13 years. a. Construct one bar graph illustrating the relationship between sales and income for each separate year of Amazon’s existence. b. Describe the type of relationship that exists between the years in business and Amazon’s sales volume. c. Amazon’s sales rose sharply. However its net income yielded losses which increased during the first few years. In which year did this situation reverse itself and show improvement in the net income balance sheet 2-45. In your capacity as assistant to the administrator at Freedom Hospital you have been asked to develop a graphical presentation that focuses on the insurance carried by the geriatric patients at the hospital. The data file Patients contains data for a sample of geriatric patients. In developing your presentation please do the following: a. Construct a pie chart that shows the percentage of patients with each health insurance payer. b. Develop a bar chart that shows total charges for patients by insurance payer. c. Develop a stem and leaf diagram for the length-of- stay variable. d. Develop a bar chart that shows the number of males and females by insurance carrier. END EXERCISES 2-2 Line Charts and Scatter Diagrams Line Charts Most of the examples that have been presented thus far have involved cross-sectional data or data gathered from many observations all taken at the same time. However if you have time- series data that are measured over time e.g. monthly quarterly annually an effective tool for presenting such data is a line chart. BUSINESS APPLICATION CONSTRUCTING LINE CHARTS MCGREGOR VINEYARDS McGregor Vineyards owns and operates a winery in the Sonoma Valley in northern California. At a recent company meeting the financial manager Chapter Outcome 6. Line Chart A two-dimensional chart showing time on the horizontal axis and the variable of interest on the vertical axis. Excel Tutorial Excel tutorials www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 5. Use the Layout tab in the Chart Tools to remove the Legend change the Chart Title add the Axis Titles and remove the grid lines. 6. Repeat Steps 2–5 for the Proft data. Excel 2010 Instructions: 1. Open fle: McGregor.xlsx. 2. Select the Sales dollars data to be graphed. 3. On the Insert tab click the Line chart. 4. Click the Line with Markers option. Minitab Instructions for similar results: 1. Open fle: McGregor.MTW. 2. Choose Graph Times Series Plot. 3. Select Simple. 4. Click OK. 5. In Series enter Sales and Proft columns. 6. Select Multiple Graphs. 7. Under Show Graph Variables select In separate panels of the same graph. 8. Click OK. OK. Sales Increasing but Profts Decreasing FIGURE 12 | Excel 2010 Output Showing McGregor Line Charts for Sales and Profits 5. Use the Layout tab in the Chart Tools to change the Chart Title add the Axis Titles remove the border and remove the grid lines. Excel 2010 Instructions: 1. Open fle: McGregor.xlsx. 2. Select the two variables Sales dollars and Proft to be graphed. 3. On the Insert tab click the Line chart. 4. Click the Line with Markers option. Minitab Instructions for similar results: 1. Open fle: McGregor.MTW. 2. Choose Graph Times Series Plot. 3. Select Multiple. 4. Click OK. 5. In Series enter Sales and Proft columns. 6. Select Multiple Graphs. 7. Under Show Graph Variables select Overlaid on the same graph. 8. Click OK. OK. FIGURE 13 | Excel 2010 Line Chart of McGregor Profit and Sales Using a Single Vertical Axis www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data expressed concern about the company’s profit trend over the past 20 weeks. He presented weekly profit and sales data to McGregor management personnel. The data are in the file McGregor. Initially the financial manager developed two separate line charts for this data: one for sales the other for profits. These are displayed in Figure 12. These line charts provide an indi- cation that although sales have been increasing the profit trend is downward. But to fit both Excel graphs on one page he had to compress the size of the graphs. This “flattened” the lines somewhat masking the magnitude of the problem. What the financial manager needed is one graph with both profits and sales. Figure 13 shows his first attempt. This is better but there still is a problem: The sales and profit vari- ables are of different magnitudes. This results in the profit line being flattened out to almost a straight line. The profit trend is hidden. To overcome this problem the financial manager needed to construct his graph using two scales one for each variable. Figure 14 shows the improved graph. We can now clearly see that although sales are moving steadily higher profits are headed downhill. For some reason costs are rising faster than revenues and this graph should motivate McGregor Vineyards to look into the problem. EXAMPLE 10 LINE CHARTS Grogan Builders Grogan Builders produces mobile homes in Alberta Canada. The owners are planning to expand the manufacturing facilities. To do so requires additional financing. In preparation for the meeting with the bankers the owners have assembled data on total annual sales for the past 10 years. These data are shown as follows: 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 1426 1678 2591 2105 2744 3068 2755 3689 4003 3997 Example 10 Constructing Line Charts A line chart also commonly called a trend chart is developed using the following steps: Identify the time-series vari- able of interest and determine the maximum value and the range of time periods covered in the data. Construct the horizontal axis for the time periods using equal spacing between time periods. Construct the vertical axis with a scale appropriate for the range of values of the time-series variable. Plot the points on the graph and connect the points with straight lines. 5. Move graph to separate page. 6. Select Proft Line on graph and rick click. 7. Click on the Format Data Series. 8. Click on Secondary Axis. 9. Click on Layout and add titles as desired. Excel 2010 Instructions: 1. Open fle: McGregor.xlsx. 2. Select data from the proft and sales column. 3. Click on Insert. 4. Click on Line Chart. Minitab Instructions for similar results: 1. Open fle: McGregor.MTW. 2. Choose Graph Times Series Plot. 3. Select Multiple. 4. Click OK. 5. In Series enter Sales and Proft columns. 6. Select Multiple Graphs. 8. Click OK. Two vertical axes: Left Sale Right Profts Profts and Sales moving in different directions. FIGURE 14 | Excel 2010 Line Chart for Sales and Profits Using Two Vertical Axes and Different Scales www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data The owners wish to present these data in a line chart to effectively show the company’s sales growth over the 10-year period. To construct the line chart the following steps are used: Step 1 Identify the time-series variable. The time-series variable is units sold measured over 10 years with a maximum value of 4003. Step 2 Construct the horizontal and vertical axes. The horizontal axis will have the 10 time periods equally spaced. The vertical axis will start at zero and go to a value exceeding 4003. We will use 4500. The vertical axis will also be divided into 500-unit increments. Step 3 Plot the data values on the graph and connect the points with straight lines. Scatter Diagrams In Section 1 we introduced a set of statistical procedures known as joint frequency distribu- tions that allow the decision maker to examine two variables at the same time. Another pro- cedure used to study two quantitative variables simultaneously is the scatter diagram or the scatter plot. There are many situations in which we are interested in understanding the bivariate rela- tionship between two quantitative variables. For example a company would like to know the relationship between sales and advertising. A bank might be interested in the relationship between savings account balances and credit card balances for its customers. A real estate agent might wish to know the relationship between the selling price of houses and the number of days that the houses have been on the market. The list of possibilities is almost limitless. Regardless of the variables involved there are several key relationships we are looking for when we develop a scatter diagram. Figure 15 shows scatter diagrams representing some key bivariate relationships that might exist between two quantitative variables. Elsewhere the text introduces a statistical technique called regression analysis that focuses on the relationship between two variables. These variables are known as dependent and independent variables. BUSINESS APPLICATION CREATING SCATTER DIAGRAMS PERSONAL COMPUTERS Can you think of any product that has increased in quality and capability as rapidly as personal computers Not that many years ago an 8-MB RAM system with a 486 processor and a 640-KB hard drive sold in the mid-2500 range. Now the same money would buy a 3.0 GHz or faster machine with a 100+ GB hard drive and 512-MB RAM or more A few years ago we examined various Web sites looking for the best prices on personal computers. The data file called Personal Computers contains data on several characteristics Chapter Outcome 7. Scatter Diagram or Scatter Plot A two-dimensional graph of plotted points in which the vertical axis represents values of one quantitative variable and the horizontal axis represents values of the other quantitative variable. Each plotted point has coordinates whose values are obtained from the respective variables. Independent Variable A variable whose values are thought to impact the values of the dependent variable. The independent variable or explanatory variable is often within the direct control of the decision maker. On a scatter plot the independent variable or explanatory variable is graphed on the x axis. Dependent Variable A variable whose values are thought to be a function of or dependent on the values of another variable called the independent variable. On a scatter plot the dependent variable is placed on the y axis and is often called the response variable. END EXAMPLE TRY PROBLEM 47 0 500 4000 4500 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 1000 1500 2000 2500 3000 3500 Mobile Homes Sold Year Grogan Builders Annual Sales www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data including processor speed hard drive capacity RAM whether a monitor is included and price for 13 personal computers. Of particular interest is the relationship between the computer price and processing speed. Our objective is to develop a scatter diagram to graphically depict what if any relationship exists between these two variables. The dependent variable is price and the independent variable is processor speed. Figure 16 shows the Excel scatter diagram output. The relationship between processor speed and price is somewhat curvilinear and positive. y x a Linear y x d Curvilinear y x b Linear y x e No Relationship y x c Curvilinear y x f No Relationship FIGURE 15 | Scatter Diagrams Showing Relationships Between x and y Excel Tutorial Excel tutorials 3. On Insert tab click Scatter and then click Scatter with only Markers option. 4. Move chart to separate page. 5. Use the Layout tab of the Chart Tools to add titles and remove grid lines. Excel 2010 Instructions: 1. Open fle: Personal Computers.xlsx. 2. Select data for chart Processor GHz and Price. Hint: Use Crtl key to select just the two desired columns. Minitab Instructions for similar results: 1. Open fle: Personal Computers.MTW. 2. Choose Graph Scatterplot. 3. Select Simple. 4. Click OK. 5. In Y enter Price column. In X enter Processor Speed column. 6. Click OK. FIGURE 16 | Excel 2010 Scatter Diagrams for Personal Computer Data www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data EXAMPLE 11 SCATTER DIAGRAMS F ortune’s Best Eight Companies Each year F ortune magazine surveys employees regarding job satisfaction to try to determine which companies are the “best” companies to work for in the United States. F ortune also collects a variety of data associated with these companies. For example the table here shows data for the top eight companies on three variables: number of U.S. employees number of training hours per year per employee and total revenue in millions of dollars. Company U.S. Employees Training Hr/Yr Revenues Millions Southwest Airlines 24757 15 3400 Kingston Technology 552 100 1300 SAS Institute 3154 32 653 Fel-Pro 2577 60 450 TD Industries 976 40 127 MBNA 18050 48 3300 W.L. Gore 4118 27 1200 Microsoft 14936 8 8700 To better understand these companies we might be interested in the relationship between number of U.S. employees and revenue and between training hours and U.S. employees. To construct these scatter diagrams we can use the following steps: Step 1 Identify the two variables of interest. In the first case one variable is U.S. employees and the second is revenue. In the second case one variable is training hours and the other is U.S. employees. Step 2 Identify the dependent and independent variables. In each case think of U.S. employees as the independent x variable. Thus Case 1: y revenue vertical axis x U.S. employees horizontal axis Case 2: y training hours vertical axis x U.S. employees horizontal axis Step 3 Establish the scales for the vertical and horizontal axes. The maximum value for each variable is revenue +8700 U.S.employees 24757 training hours 100 Step 4 Plot the joint values for the two variables by placing a point in the x y space. Example 11 Constructing Scatter Diagrams A scatter diagram is a two-dimen- sional graph showing the joint val- ues for two quantitative variables. It is constructed using the following steps: Identify the two quantitative variables and collect paired responses for the two variables. Determine which variable will be placed on the vertical axis and which variable will be placed on the horizontal axis. Often the vertical axis can be considered the dependent vari- able y and the horizontal axis can be considered the independ- ent variable x. Define the range of values for each variable and define the appropriate scale for the x and y axes. Plot the joint values for the two variables by placing a point in the xy space. Do not connect the points. 0 30000 25000 20000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Revenue Millions U.S. Employees Scatter Diagram 15000 10000 5000 0 General Positive Relationship www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Descriptive Statistics and Data Mining The purpose of data mining and the descriptive statistics techniques discussed in the chapter are essentially the same—that is to take large amounts of data and arrange it so some under- lying pattern is more easily identifiable to the decision maker. In fact several of the tools are the same only under different names. A few examples should suffice. Pareto Charts We once had the CEO of a Fortune 500 company tell us most companies don’t get serious about quality until they face a crisis. The crisis his company faced was run- ning out of money. Pareto charts are a commonly used tool of quality but they are just histo- grams with a different name. END EXAMPLE TRY PROBLEM 46 Number 48 32 23 8 4 Percent 41.7 41.7 27.8 20.0 7.0 3.5 Cum 69.6 89.6 96.5 100.0 Misshapen Causes All Others Cracks Surface Scars Incomplete 50 40 30 20 10 0 Number Pareto Chart of Causes 0 30000 25000 20000 20 40 60 80 100 120 Training Hours U.S. Employees Scatter Diagram 15000 10000 5000 0 General Negative Relationship www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 2- Skill Development 2-46. The following data represent 11 observations of two quantitative variables: x contact hours with client y profit generated from client. x yxyx y x y 45 2345 54 3811 34 -700 24 1975 56 4200 24 2406 45 3457 32 206 26 278 23 3250 47 2478 a. Construct a scatter plot of the data. Indicate whether the plot suggests a linear or nonlinear relationship between the dependent and independent variables. b. Determine how much influence one data point will have on your perception of the relationship between the independent and dependent variables by deleting the data point with the smallest x value. What appears to be the relationship between the dependent and independent variables 2-47. You have the following sales data for the past 12 months. Develop a line graph for these data. Month Jan Feb Mar Apr May Jun Sales 200 230 210 300 320 290 Month Jul Aug Sep Oct Nov Dec Sales 300 360 400 410 390 450 MyStatLab 2-48. The following data have been selected for two variables y and x. Construct a scatter plot for these two variables and indicate what type of relationship if any appears to be present. yx 100 23.5 250 17.8 70 28.6 130 19.3 190 15.9 250 19.1 40 35.3 2-49. The year-end dollar value in millions of deposits for Bank of the Ozarks Inc. for the years 1997–2010 are shown below Year 1997 1998 1999 2000 2001 2002 2003 Deposits 296 529 596 678 678 790 1062 Year 2004 2005 2006 2007 2008 2009 2010 Deposits 1380 1592 2045 2057 2341 2029 2541 Develop a chart that effectively displays the deposit data over time. The Pareto Principle named after Alfredo Pareto but identified in quality circum- stances by Joseph Juran has led business decision makers to realize the majority of sick days taken by employees are taken by a minority of employees or the majority of customer complaints are filed by a minority of customers or a majority of quality problems arise from a minority of possible causes. Consider the previous Pareto chart taken from a popular quality-control text. The example involves a company trying to determine the reason for complaints about its shipments to customers. The company was able to identify 20 possible types of complaints. This should look familiar. It is a histogram although it is called a Pareto Chart and shows 3 out of 20 possible causes account for almost 90 of the complaints. This form of histogram will show a company where to concentrate its efforts to reduce customer complaints. Charts like this are also easier to use for employees who are not comfortable using more complicated statistical tools. Scatter Diagrams Numerous police departments use a version of scatter diagrams to determine where to concentrate their efforts. The area of the diagram is a map of the city and the locations of crimes are plotted often using a color code to identify the type of crime. By identifying clusters of crimes rather than spreading their efforts equally throughout the city police are able to concentrate patrol and reduction efforts where the plots indicate clusters are happening. Numerous cities such as San Francisco have found such scatter diagrams to be effective tools in combating crime. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 2-50. VanAuker Properties’ controller collected the following data on annual sales and the years of experience of members of his sales staff: Sales K: 200 191 135 236 305 183 50 192 184 73 Years: 10 459 12 6 2 76 2 a. Construct a scatter plot representing these data. b. Determine the kind of relationship that exists if any between years of experience and sales. c. Approximate the increase in sales that accrues with each additional year of experience for a member of the sales force. Business Applications 2-51. Amazon.com celebrated its 13th anniversary in July 2007. Its growth can be seen by examining its increasing sales volume in billions as reported by Hoovers Inc. Sales 0.0005 0.0157 0.1477 0.6098 1.6398 Year 1995 1996 1997 1998 1999 Sales 2.7619 3.1229 3.9329 5.2637 6.9211 Year 2000 2001 2002 2003 2004 Sales 8.490 10.711 14.835 Year 2005 2006 2007 a. Construct a line plot for Amazon’s sales. b. Describe the type of relationship that exists between the years in business and Amazon’s sales volume. c. In which year does it appear that Amazon had the sharpest increase in sales 2-52. In July 2005 Greg Sandoval of the Associated Press authored a study of the video game industry that focused on the efforts of the industry to interest women in the games. In that study he cited another report by the Entertainments Software Association that indicated that the percentage of women who played video games in 2004 was 43 whereas only 12.5 of the software developers were female. Sandoval also presented the following data showing the U.S. computer/video game sales: Year Sales Billions 1996 3.80 1997 4.30 1998 5.70 1999 6.10 2000 6.00 2001 6.30 2002 6.95 2003 7.00 2004 7.30 Construct a line chart showing these computer/video game sales data. Write a short statement describing the graph. 2-53. The recent performance of U.S. equity markets has increased the popularity of dividend-paying stocks for some investors. Shown below are the diluted net earnings per common share and the dividends per common share for the Procter Gamble Company PG for the years 1996–2011. Year Diluted Net Earnings per Common Share Dividends per Common Share 1996 1.00 0.40 1997 1.14 0.45 1998 1.28 0.51 1999 1.29 0.57 2000 1.23 0.64 2001 1.03 0.70 2002 1.54 0.76 2003 1.85 0.82 2004 2.32 0.93 2005 2.66 1.03 2006 2.64 1.15 2007 3.04 1.28 2008 3.64 1.45 2009 4.26 1.64 2010 4.11 1.80 2011 3.93 1.97 a. Construct a line chart of diluted net earnings per common share for the years shown. b. Construct a line chart of dividends per common share for the years shown. c. Construct the appropriate chart for determining whether there is a relationship between diluted net earnings per common share and dividends per common share for the years shown. Briefly comment on the nature of any relationship you believe your chart reveals. 2-54. Business Week Reed Stanley et al. “Open Season on Big Oil” September 26 2005 reported on data provided by A. G. Edwards Sons concerning the profits billions for 10 of the largest integrated oil and gas companies over the period from 1999 to 2005. Year 1999 2000 2001 2002 2003 2004 2005 Profit Billions 33.3 62.5 58.3 41.7 66.7 91.7 118.0 a. Produce a line plot of the profit versus the year. b. Describe the types of relationships that exist between years and profits during the specified time period. c. Which of the relationships would you use to project the companies’ profits in the year 2006 Explain your answer. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Computer Database Exercises 2-55. Major League Baseball MLB is played in 30 North American cities including Toronto Canada. Having a team in a city is generally considered to provide an economic boost to the community. Although winning is the stated goal for all teams the business side of baseball has to do with attendance. The data file MLB Attendance-2008 contains data for both home and road game attendance for all 30 MLB teams for 2008. Of interest is the relationship between average home attendance and average road attendance. Using the 2008 attendance data construct the appropriate graph to help determine the relationship between these two variables and discuss the implications of the graph. 2-56. In the October 17 2005 issue of Fortune a special advertising section focused on private jets. Included in the section was an article about “fractional” jet ownership in which wealthy individuals and companies share ownership in private jets. The idea is that the expensive airplanes can be better utilized if more than one individual or company has an ownership stake. AvData Inc. provided data showing the number of fractional ownerships since 1986. These data are in the file called JetOwnership. Using these data develop a line chart that displays the trend in fractional ownership between 1986 and 2004. Discuss. 2-57. Starting in 2005 a chain of events including the war in Iraq Hurricane Katrina and the expanding economies in India and China lead to a sharp increase in fuel costs. As a result the U.S. airline industry has been hit hard financially with many airlines declaring bankruptcy. Some airlines are substituting smaller planes on certain routes in an attempt to reduce fuel costs. As an analyst for one of the major airlines you have been asked to analyze the relationship between passenger capacity and fuel consumption per hour. Data for 19 commonly flown planes is presented in the file called Airplanes. Develop the appropriate graph to illustrate the relationship between fuel consumption per hour and passenger capacity. Discuss. 2-58. Japolli Bakery tracks sales of its different bread products on a daily basis. The data at the bottom of this page show sales for 22 consecutive days at one of its retail outlets in Nashville. Develop a line chart that displays these data. The data are also located in a data file called Japolli Bakery. Discuss what if any conclusions you might be able to reach from the line chart. 2-59. Energy prices have been a major source of economic and political debate in the United States and around the world. Consumers have recently seen gasoline prices both rise and fall rapidly and the impact of fuel prices has been blamed for economic problems in the United States at different points in time. Although no longer doing so for years the California Energy Commission published yearly gasoline prices. The data found in the file called Gasoline Prices reflect the average price of regular unleaded gasoline in the state of California for the years between 1970 and 2005. The first price column is the actual average price of gasoline during each of Japolli Bakery Day of Week White Wheat Multigrain Black Cinnamon Raisin Sourdough French Light Oat Friday 436 456 417 311 95 96 224 Saturday 653 571 557 416 129 140 224 Sunday 496 490 403 351 114 108 228 Monday 786 611 570 473 165 148 304 Tuesday 547 474 424 365 144 104 256 Wednesday 513 443 380 317 100 92 180 Thursday 817 669 622 518 181 152 308 Friday 375 390 299 256 124 88 172 Saturday 700 678 564 463 173 136 248 Sunday 597 502 457 383 140 144 312 Monday 536 530 428 360 135 112 356 Tuesday 875 703 605 549 201 188 356 Wednesday 421 433 336 312 100 104 224 Thursday 667 576 541 438 152 144 304 Friday 506 461 406 342 135 116 264 Saturday 470 352 377 266 84 92 172 Sunday 748 643 599 425 153 148 316 Monday 376 367 310 279 128 104 208 Tuesday 704 646 586 426 174 160 264 Wednesday 591 504 408 349 140 120 276 Thursday 564 497 415 348 107 120 212 Friday 817 673 644 492 200 180 348 www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data END EXERCISES 2-3 those years. The second column is the average price adjusted for inflation with 2005 being the base year. a. Construct an appropriate chart showing the actual average price of gasoline in California over the years between 1970 and 2005. b. Add to the graph developed in part a the data for the adjusted gasoline prices. c. Based on the graph from part b what conclusions might be reached about the price of gasoline over the years between 1970 and 2005 2-60. Federal flood insurance underwritten by the federal government was initiated in 1968. This federal flood insurance coverage has according to USA Today “How You Pay for People to Build in Flood Zones” September 21 2005 more than tripled in the past 15 years. A file titled Flood contains the amount of federal flood insurance coverage for each of the years from 1990 to 2004. a. Produce a line plot for these data. b. Describe the type of relationship between the year and the amount of federal flood insurance. c. Determine the average increase per year in federal flood insurance. 2-61. The Office of Management and Budget keeps data on many facets of corporations. One item that has become a matter of concern is the number of applications for patents submitted compared to the backlog of applications that have not been processed by the end of the year. A file titled Patent provides data extracted from a USA Today article that addresses the problem. a. Construct the two line plots on the same axes. b. Determine the types of relationships that exist between the years and the two patent-related variables. c. During which years did the backlog of applications at the end of the year equal approximately the same number of patent applications 2-62. The sub-prime mortgage crisis that hit the world economy also impacted the real estate market. Both new and existing home sales were affected. A file titled EHSales contains the number of existing homes sold in millions from September of 2007 to September 2008. a. Construct a line plot for these data. b. The data file also contains data concerning the median selling price thousands. Construct a graph containing the line plot for both the number of sales tens of thousands and the median thousands price of these sales for the indicated time period. c. Describe the relationship between the two line plots constructed in part b. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 1 Frequency Distributions and Histograms 2 Bar Charts Pie Charts and Stem and Leaf Diagrams Outcome 4. Construct and interpret various types of bar charts. Outcome 5. Build a stem and leaf diagram. 3 Line Charts and Scatter Diagrams Outcome 6. Create a line chart and interpret the trend in the data. Outcome 7. Create a scatter plot and interpret it. Summary When you are working with time-series data and you are interested in displaying the pattern in the data over time the chart that is used is called a line chart. The vertical axis displays the value of the time-series variable while the horizontal axis contains the time increments. The points are plotted and are usually connected by straight lines. In other cases you may be interested in the relationship between two quantitative variables the graphical tool that is used is called a scatter diagram. The variable judged to be the dependent variable is placed on the vertical axis and independent variable goes on the horizontal axis. The joint values are plotted as points in the two-dimensional space. Do not connect the points with lines. Conclusion There are many types of charts graphs and tables that can be used to display data. The technique that is used often depends on the type and level of data you have. In cases where multiple graphs or charts can apply you should select the one that most effectively displays the data for your application. Figure 17 summarizes the different graphical options that are presented in this chapter. The old adage states that a picture is worth a thousand words. In many ways this applies to descriptive statistics. The use of graphs charts and tables to display data in a way that helps decision-makers better understand the data is one of the major applications of business statistics. This chapter has introduced many of the most frequently used graphical techniques using examples and business applications. Summary A frequency distribution is used to determine the number of occurrences in your data that fall at each possible data value or within defned ranges of possible values. It represents a good summary of the data and from a frequency distribution you can form a graph called a histogram. This histogram gives a visual picture showing how the data are distributed. You can use the histogram to see where the data’s center is and how spread out the data are. It is often helpful to convert the frequencies in a frequency distribution to relative frequencies and to construct a relative frequency distribution and a relative frequency histogram. Another option is to convert the frequency distribution to a cumulative frequency distribution and then a graph called an ogive. Finally if you are analyzing two variables simultaneously you may want to construct a joint frequency distribution. Outcome 1. Construct frequency distributions both manually and with your computer. Outcome 2. Construct and interpret a frequency histogram. Outcome 3. Develop and interpret joint frequency distributions. Summary If your data are discrete or are nominal or ordinal level three charts introduced in this section are often considered. These are bar charts pie charts and stem and leaf diagrams. A bar chart can be arranged with the bars vertical or horizontal. A single bar chart can be used to describe two or more variables. In situations where you wish to show how the parts making up a total are distributed a pie chart is often used. The “slices” of the pie are many times depicted as the percentage of the total. A lesser used graphical tool that provides a quick view of how the data are distributed is the stem and leaf diagram. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Equations Cumulative Relative Frequency Distribution Quantitative Qualitative Categorical Nominal/Ordinal Time Series Cross-Sectional Grouped or Ungrouped Discrete or Continuous Interval/Ratio Line Chart Bar Chart Vertical Frequency Distribution Bar Chart Vertical or Horizontal Relative Frequency Distribution Joint Frequency Distribution Frequency Distribution Relative Frequency Distribution Joint Frequency Distribution Data Type Data Class Histogram Pie Chart Stem and Leaf Diagram Scatter Diagram Ogive FIGURE 17 | Summary: Descriptive Statistical Techniques 1 Relative Frequency Relative frequency 5 f n i 2 Class Width W Largest value Smallest value Number of clas sses Key Terms All-inclusive classes Bar chart Class boundaries Class width Continuous data Cumulative frequency distribution Cumulative relative frequency distribution Chapter Exercises MyStatLab Conceptual Questions 2-63. Discuss the advantages of constructing a relative frequency distribution as opposed to a frequency distribution. 2-64. What are the characteristics of a data set that would lead you to construct a bar chart 2-65. What are the characteristics of a data set that would lead you to construct a pie chart 2-66. Discuss the differences in data that would lead you to construct a line chart as opposed to a scatter plot. Dependent variable Discrete data Equal-width classes Frequency distribution Frequency histogram Independent variable Line chart Mutually exclusive classes Ogive Pie chart Relative frequency Scatter diagram or scatter plot www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Business Applications 2-67. USA T oday reported Anthony Breznican and Gary Strauss “Where Have All the Moviegoers Gone” June 23 2005 that in the summer of 2005 ticket sales to movie theaters had fallen for 17 straight weeks the industry’s longest losing streak since 1985. To determine the long- term trends in ticket sales the following data representing the number of admissions in billions were obtained from the National Association of Theatre Owners: Year 1987 1988 1989 1990 1991 1992 Admissions 1.09 1.08 1.26 1.19 1.14 1.17 Year 1993 1994 1995 1996 1997 1998 Admissions 1.24 1.28 1.26 1.34 1.39 1.48 Year 1999 2000 2001 2002 2003 2004 Admissions 1.47 1.42 1.49 1.63 1.57 1.53 Year 2005 2006 2007 Admissions 1.38 1.41 1.40 a. Produce a line plot of the data. b. Describe any trends that you detect. 2-68. The following data represent the commuting distances for employees of the Pay-and-Carry Department store. a. The personnel manager for Pay-and-Carry would like you to develop a frequency distribution and histogram for these data. b. Develop a stem and leaf diagram for these data. c. Break the data into three groups under 3 miles 3 and under 6 miles and 6 and over. Construct a pie chart to illustrate the proportion of employees in each category. Commuting Distance miles 3.5 2.0 4.0 2.5 0.3 1.0 12.0 17.5 3.0 3.5 6.5 9.0 3.0 4.0 9.0 16.0 3.5 0.5 2.5 1.0 0.7 1.5 1.4 12.0 9.2 8.3 1.0 3.0 7.5 3.2 2.0 1.0 3.5 3.6 1.9 2.0 3.0 1.5 0.4 6.4 11.0 2.5 2.4 2.7 4.0 2.0 2.0 3.0 d. Referring to part c construct a bar chart to depict the proportion of employees in each category. 2-69. Anyone attending college realizes tuition costs have increased rapidly. In fact tuition had risen at a faster pace than inflation for more than two decades. Data showing costs for both private and public colleges for selected years are shown below. Year 1984 1989 1994 1999 2004 Private College Tuition 9202 12146 13844 16454 19710 Public College Tuition 2074 2395 3188 3632 4694 a. Construct one bar graph illustrating the relationship between private and public university tuition for the displayed years. b. Describe the tuition trend for both private and public college tuition. 2-70. A recent article in USA Today used the following data to illustrate the decline in the percentage of men who receive college and advanced degrees: Bachelor Doctorate 1989 2003 2014 1989 2003 2014 Men 47 43 40 64 57 49 Women 53 57 60 36 43 51 Education Department projection. a. Use one graph that contains two bar charts each of which represents the kind of degree received to display the relationship between the percentages of men and women receiving each type of degree. b. Describe any trends that might be evident. 2-71. The Minnesota State Fishing Bureau has contracted with a university biologist to study the length of walleyes fish caught in Minnesota lakes. The biologist collected data on a sample of 1000 fish caught and developed the following relative frequency distribution: Class Length inches Relative Frequency f i 8 to less than 10 0.22 10 to less than 12 0.15 12 to less than 14 0.25 14 to less than 16 0.24 16 to less than 18 0.06 18 to less than 20 0.05 20 to less than 22 0.03 a. Construct a frequency distribution from this relative frequency distribution and then produce a histogram based on the frequency distribution. b. Construct a pie chart from the relative frequency distribution. Discuss which of the two graphs the pie chart or the histogram you think is more effective in presenting the fish length data. 2-72. A computer software company has been looking at the amount of time customers spend on hold after their call is answered by the central switchboard. The company would like to have at most 2 of the callers wait two minutes or more. The company’s calling service has provided the following data showing how long each of last month’s callers spent on hold: Class Number Less than 15 seconds 456 15 to less than 30 seconds 718 30 to less than 45 seconds 891 45 to less than 60 seconds 823 60 to less than 75 seconds 610 75 to less than 90 seconds 449 90 to less than 105 seconds 385 105 to less than 120 seconds 221 Continued www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Class Number 120 to less than 150 seconds 158 150 to less than 180 seconds 124 180 to less than 240 seconds 87 More than 240 seconds 153 a. Develop a relative frequency distribution and ogive for these data. b. The company estimates it loses an average of 30 in business from callers who must wait two minutes or more before receiving assistance. The company thinks that last month’s distribution of waiting times is typical. Estimate how much money the company is losing in business per month because people have to wait too long before receiving assistance. 2-73. The regional sales manager for American Toys Inc. recently collected data on weekly sales in dollars for the 15 stores in his region. He also collected data on the number of salesclerk work hours during the week for each of the stores. The data are as follows: Store Sales Hours 1 23300 120 2 25600 135 3 19200 96 4 10211 102 5 19330 240 6 35789 190 7 12540 108 8 43150 234 9 27886 140 10 54156 300 11 34080 254 12 25900 180 13 36400 270 14 25760 175 15 31500 256 a. Develop a scatter plot of these data. Determine which variable should be the dependent variable and which should be the independent variable. b. Based on the scatter plot what if any conclusions might the sales manager reach with respect to the relationship between sales and number of clerk hours worked Do any stores stand out as being different Discuss. Computer Database Exercises 2-74. The Energy Information Administration published a press release on September 26 2005 Paula Weir and Pedro Saavedra “Two Multi-Phase Surveys That Combine Overlapping Sample Cycles at Phase I”. The file titled Diesel contains the average on-highway diesel prices for each of 53 weeks from September 27 2004 to September 26 2005. It also contains equivalent information for the state of California recognized as having the highest national prices. a. Construct a chart containing line plots for both the national average and California’s diesel prices. Describe the relationship between the diesel prices in California and the national average. b. In what week did the California average diesel price surpass 3.00 a gallon c. Determine the smallest and largest price paid in California for a gallon of diesel. At what weeks did these occur Use this information to project when California gas prices might exceed 4.00 assuming a linear trend between California diesel prices and the weeks in which they occurred. 2-75. A recent article in USA Today reported that Apple had 74 of the digital music device market according to researcher The NPD Group. The NPD Group provides global sales and marketing information that helps clients make more informed fact-based decisions to optimize their businesses. The data in the file titled Digital provide the brand of digital devices owned by a sample of consumers that would produce the market shares alluded to in the article. Produce a pie chart that represents the market shares obtained from the referenced sample. Indicate the market shares and the identity of those manufacturers in the pie chart. 2-76. The file Home-Prices contains information about single-family housing prices in 100 metropolitan areas in the United States. a. Construct a frequency distribution and histogram of 1997 median single-family home prices. Use the 2 k Ú n guideline to determine the appropriate number of classes. b. Construct a cumulative relative frequency distribution and ogive for 1997 median single- family home prices. c. Repeat parts a and b but this time use 1.5 times as many class intervals as recommended by the 2 k Ú n guideline. What was the impact of using more class intervals 2-77. Elliel’s Department Store tracks its inventory on a monthly basis. Monthly data for the years 2008–2012 are in the file called Elliels. a. Construct a line chart showing the monthly inventory over the five years. Discuss what this graph implies about inventory. b. Sum the monthly inventory figures for each year. Then present the sums in bar chart form. Discuss whether you think this is an appropriate graph to describe the inventory situation at Elliels. 2-78. The Energy Information Administration EIA surveys the price of diesel fuel. The EIA-888 is a survey of diesel fuel outlet prices from truck stops and service stations across the country. It produces estimates of national and regional prices. The diesel fuel prices that are released are used by the trucking industry to make rate adjustments in hauling contracts. The file titled Diesel contains the www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data average on-highway diesel prices for each of 53 weeks from September 27 2004 to September 26 2005. a. Construct a histogram with 11 classes beginning at 1.85. b. Are there any data points that are unusually larger than the rest of the data In which classes do these points occur What is the interpretation of this phenomenon video Video Case 2 Drive-Thru Service Times  McDonald’s When you’re on the go and looking for a quick meal where do you go If you’re like millions of people every day you make a stop at McDonald’s. Known as “quick service restaurants” in the industry not “fast food” companies such as McDon- ald’s invest heavily to determine the most efficient and effective ways to provide fast high-quality service in all phases of their business. Drive-thru operations play a vital role. It’s not surprising that attention is focused on the drive-thru process. After all more than 60 of individual restaurant revenues in the United States come from the drive-thru experience. Yet understanding the process is more complex than just counting cars. Marla King professor at the company’s international training center Hamburger University got her start 25 years ago working at a McDonald’s drive-thru. She now coaches new restaurant owners and managers. “Our stated drive-thru service time is 90 seconds or less. We train every man- ager and team member to understand that a quality customer expe- rience at the drive-thru depends on them” says Marla. Some of the factors that affect customers’ ability to complete their purchases within 90 seconds include restaurant staffing equipment layout in the restaurant training efficiency of the grill team and frequency of customer arrivals to name a few. Also customer order patterns also play a role. Some customers will just order drinks whereas others seem to need enough food to feed an entire soccer team. And then there are the special orders. Obviously there is plenty of room for variability here. Yet that doesn’t stop the company from using statistical tech- niques to better understand the drive-thru action. In particular McDonald’s uses graphical techniques to display data and to help transform the data into useful information. For restaurant manag- ers to achieve the goal in their own restaurants they need training in proper restaurant and drive-thru operations. Hamburger Uni- versity McDonald’s training center located near Chicago Illi- nois satisfies that need. In the mock-up restaurant service lab managers go through a “before and after” training scenario. In the “before” scenario they run the restaurant for 30 minutes as if they were back in their home restaurants. Managers in the train- ing class are assigned to be crew customers drive-thru cars spe- cial-needs guests such as hearing impaired indecisive clumsy or observers. Statistical data about the operations revenues and service times are collected and analyzed. Without the right train- ing the restaurant’s operations usually start breaking down after 10-15 minutes. After debriefing and analyzing the data col- lected the managers make suggestions for adjustments and head back to the service lab to try again. This time the results usually come in well within standards. “When presented with the quanti- tative results managers are pretty quick to make the connections between better operations higher revenues and happier custom- ers” Marla states. When managers return to their respective restaurants the train- ing results and techniques are shared with staff who are charged with implementing the ideas locally. The results of the training eventually are measured when McDonald’s conducts a restaurant operations improvement process study or ROIP. The goal is sim- ple: improved operations. When the ROIP review is completed statistical analyses are performed and managers are given their results. Depending on the results decisions might be made that require additional financial resources building construction staff training or reconfiguring layouts. Yet one thing is clear: Statis- tics drive the decisions behind McDonald’s drive-through service operations. Discussion Questions: 1. After returning from the training session at Hamburger University a McDonald’s store owner selected a random sample of 362 drive-thru customers and carefully measured the time it took from when a customer entered the McDonald’s property until the customer received the order at the drive-thru window. These data are in the file called McDonald’s Drive-Thru Waiting Times. Note the owner selected some customers during the breakfast period others during lunch and others during dinner. Construct any appropriate graphs and charts that will effectively display these drive-thru data. Prepare a short discussion indicating the conclusions that this store owner might reach after reviewing the graphs and charts you have prepared. 2. Referring to question 1 suppose the manager comes away with the conclusion that his store is not meeting the 90-second customer service goal. As a result he plans to dig deeper into the problem by collecting more data from the drive-thru process. Discuss what other measures you would suggest the manager collect. Discuss how these data could be of potential value in helping the store owner understand his problem. 3. Visit a local McDonald’s that has a drive-thru facility. Randomly sample 20 drive-thru customers and collect the following data: a. the total time from arrival on the property to departure from the drive-thru window b. the time from when customers place the order until they receive their order and exit the drive-thru process c. the number of cars in the line when the sampled vehicle enters the drive-thru process d. Using the data that you have collected construct appropri- ate graphs and charts to describe these data. Write a short report discussing the data www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Server Downtime After getting outstanding grades in high school and scoring very high on his ACT and SAT tests Clayton Haney had his choice of colleges but wanted to follow his parents’ legacy and enrolled at Northwestern University. Clayton soon learned that there is a big difference between getting high grades in high school and being a good student. Although he was recognized as being quite bright and very quick to pick up on things he had never learned how to study. As a result after slightly more than two years at Northwestern Clayton was asked to try his luck at another university. To the cha- grin of his parents Clayton decided that college was not for him. After short stints working for a computer manufacturer and as a manager for a Blockbuster video store Clayton landed a job working for EDS. EDS contracts to support information technol- ogy implementation and application for companies in the United States and throughout the world. Clayton had to train himself in virtually all aspects of personal computers and local area networks and was assigned to work for a client in the Chicago area. Clayton’s first assignment was to research the downtime on one of the client’s primary network servers. He was asked to study the downtime data for the month of April and to make a short pres- entation to the company’s management. The downtime data are in a file called Server Downtime. These data are also shown in Table C-1-A. Although Clayton is very good at solving computer prob- lems he has had no training or experience in analyzing data so he is going to need some help. Required Tasks: 1. Construct a frequency distribution showing the number of times during the month that the server was down for each downtime cause category. 2. Develop a bar chart that displays the data from the frequency distribution in part a. 3. Develop a histogram that displays the downtime data. TABLE C-1-A |  Date Problem Experienced Downtime Minutes 04/01 Lockups 25 04/02 Lockups 35 04/05 Memory Errors 10 04/07 Lockups 40 04/09 Weekly Virus Scan 60 04/09 Lockups 30 04/09 Memory Errors 35 04/09 Memory Errors 20 04/12 Slow Startup 45 04/12 Weekly Virus Scan 60 04/13 Memory Errors 30 04/14 Memory Errors 10 04/19 Manual Re-start 20 04/20 Memory Errors 35 04/20 Weekly Virus Scan 60 04/20 Lockups 25 04/21 Memory Errors 35 04/22 Memory Errors 20 04/27 Memory Errors 40 04/28 Weekly Virus Scan 60 04/28 Memory Errors 15 04/28 Lockups 25 4. Develop a pie chart that breaks down the percentage of total downtime that is attributed to each downtime cause during the month. 5. Prepare a short written report that discusses the downtime data. Make sure you include the graphs and charts in the report. Hudson Valley Apples Inc. As a rule Stacey Fredrick preferred to work in the field rather than do “office” work in her capacity as a midlevel manager with Hudson Valley Apples Inc. a large grower and processor of apples in the state of New Y ork. However after just leaving a staff meeting at which she was asked to prepare a report of apple consumption in the United States Stacey was actually looking forward to spend- ing some time at her computer “crunching some numbers.” Arden Golchein senior marketing manager indicated that he would e-mail her a data file that contained apple consumption data from 1970 through 2009 and told her that he wanted a very nice report using graphs charts and tables to describe apple consumption. When she got to her desk the e-mail was waiting and she saved the file under the name Hudson Valley Apples. Stacey had done quite a bit of descriptive analysis in her previous job with the New York State Department of Agriculture so she had several ideas for types of graphs and tables that she might construct. She began by creating a list of the tasks that she thought would be needed. Required Tasks: 1. Construct a line chart showing the total annual availability of apples. 2. Construct one line chart that shows two things: the annual availability of fresh apples and the annual availability of processed apples. 3. Construct a line chart that shows the annual availability for each type of processed apples. 4. Construct a histogram for the total annual availability of apples. 5. Write a short report that discusses the historical pattern of apple availability. The report will include all pertinent charts and graphs. www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Welco Lumber Company—Part A Gene Denning wears several hats at the Welco Lumber Company including process improvement team leader shipping manager and assistant human resources manager. Welco Lumber makes cedar fencing materials at its Naples Idaho facility employing about 160 people. More than 75 of the cost of the finished cedar fence boards is in the cedar logs that the company buys on the open market. Therefore it is very important that the company get as much fin- ished product as possible from each log. One of the most important steps in the manufacturing process is referred to as the head rig. The head rig is a large saw that breaks down the logs into slabs and cants. Figure C-3-A shows the concept. From small logs with diameters of 12 inches or less one cant and four or fewer usable slabs are obtained. From larger logs multiple cants and four slabs are obtained. Finished fence boards can be produced from both the slabs and the cants. At some companies the head rig cutting operation is automated and the cuts are made based on a scanner system and computer algorithms. However at Welco Lumber the head rig is operated manually by operators who must look at a log as it arrives and determine how best to break the log down to get the most finished product. In addition the operators are responsible for making sure that the cants are “centered” so that maximum product can be gained from them. Recently Gene Denning headed up a study in which he video- taped 365 logs being broken down by the head rig. All three opera- tors April Sid and Jim were involved. Each log was marked as to its true diameter. Then Gene observed the way the log was broken down and the degree to which the cants were properly centered. He then determined the projected value of the finished product from each log given the way it was actually cut. In addition he also determined what the value would have been had the log been cut in the optimal way. Data for this study is in a file called Welco Lumber. A portion of the data is shown in Table C-3-A. You have been asked to assist Gene by analyzing these data using graphs charts and tables as appropriate. He wishes to focus on the lost profit to the company and whether there are differences among the operators. Also do the operators tend to do a better job on small logs than on large logs In general he is hoping to learn as much as possible from this study and needs your help with the analysis. TABLE C-3-A | Head Rig Data—Welco Lumber Company 5-Nov-06 Through 21-Nov-06 Head Rig Log Study Baseline Log Operator Log Size Large/Small Log Correct Cut Yes or No Error Category Actual Value Potential Value Potential Gain 1 Sid 15 Large No Excessive Log Breakdown 59.00 65.97 6.97 2 Sid 17 Large No Excessive Log Breakdown 79.27 85.33 6.06 3 Sid 11 Small Yes No Error 35.40 35.40 0.00 4 Sid 11 Small No Off Center Cant 31.61 35.40 3.79 5 Sid 14 Large No Reduced V alue Cut 47.67 58.86 11.19 6 Sid 17 Large Yes No Error 85.33 85.33 0.00 7 Sid 8 Small Yes No Error 16.22 16.22 0.00 8 Sid 11 Small Yes No Error 35.40 35.40 0.00 9 Sid 9 Small Yes No Error 21.54 21.54 0.00 10 Sid 9 Small No Off Center Cant 18.92 21.54 2.62 11 Sid 10 Small Yes No Error 21.54 21.54 0.00 12 Sid 8 Small Yes No Error 16.22 16.22 0.00 13 Sid 10 Small No Off Center Cant 25.71 28.97 3.26 14 Sid 12 Small Yes No Error 41.79 41.79 0.00 15 Sid 11 Small Yes No Error 35.40 35.40 0.00 Slabs Slabs Cant FIGURE C-3-A | Log Breakdown at the Head Rig www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 3. a. 2 k Ú n or 2 10 1024 Ú 1000. Thus use k 10 classes. b. w High - Low Classes 2900 - 300 10 2600 10 260 round to 300 5. a. 2.833 which is rounded to 3. b. Divide the number of occurrences frequency in each class by the total number of occurrences. c. Compute a running sum for each class by adding the frequency for that class to the frequencies for all classes above it. d. Classes form the horizontal axis and the frequency forms the vertical axis. Bars corresponding to the frequency of each class are developed. 7. a. 1 - 0.24 0.76 b. 0.56 - 0.08 0.48 c. 0.96 - 0.08 0.86 9. a. b. cumulative frequencies: 2 27 53 59 60 c. cumulative relative frequencies: 0.0333 0.4500 0.8833 0.9833 1.000 d. ogive 13. a. The weights are sorted from smallest to largest to create the data array. b. c. The histogram can be created from the frequency distrib ution. d. 10.20 15. a. w Largest - smallest number of classes 214.4 - 105.0 11 9.945 S w 10. b. 8 of the 25 or 0.32 of the salaries are at least 175000 c. 18 of the 25 or 0.72 having salaries that are at most 205000 but at least 135000 19. a. 9 classes b. w High - Low Classes 32 - 10 9 22 9 2.44 round up to 3.0 c. The frequency distribution with nine classes and a class width of 3.0 will depend on the starting point for the first class. This starting value must be at or below the minimum value of 10. d. The distribution is mound shaped and fairly symmetrical. It appears that the center is between 19 and 22 rounds per year but the rounds played are quite spread out around the center. 21. a. 2 5 32 and 2 6 64. Therefore 6 classes are chosen. b. The maximum value is 602708 and the minimum value is 160 from the Total column. The difference is 602708 - 160 602548. The class width would be 6025486 100424.67. Rounding up to the nearest 1000 produces a class width of 101000. c. d. The vast majority of airlines had fewer than 101000 monthly passengers. 23. a. Order the observations coffee consumption from smallest to largest. b. Using the 2 k 7 n guideline the number of classes k would be 0.9 and w 110.1 - 3.528 0.825 which is rounded to 0. 9. The class with the largest number is 6.2–7.0 kg of coffee. c. The histogram can be created from the frequency distri- bution. The classes are shown on the horizontal axis and the frequency on the vertical axis. d. 8.33 1100 - 91.672 of the coffee drinkers sampled consume 8.0 kg or more annually. 29. a. The pie chart categories are the regions and the measure is the region’s percentage of total income. b. The bar chart categories are the regions and the measure for each category is the region’s percentage of total income. c. The bar chart however makes it easier to compare per- centages across regions. 31. b. 1 - 0.0985 0.9015 33. The bar chart is skewed indicating that the number of 1 million houses is growing rapidly. The growth is exponential rather than linear. 35. A bar chart can be used to make the comparison. 37. a. The stem unit is 10 and the leaf unit is 1. b. between 70 and 79 seconds 41. a. Leaf unit 1.0 b. Slightly skewed to the left. The center is between 24 and 26. c. x 2428 50 48.56 Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. Class Frequency Relative Frequency 2–3 2 0.0333 4–5 25 0.4167 6–7 26 0.4333 8–9 6 0.1000 10–11 1 0.0167 Weight Classes Frequency 77–81 3 82–86 9 87–91 16 92–96 16 97–101 5 Total 49 Class Frequency 0 6 101000 33 101000 6 202000 2 202000 6 303000 2 303000 6 404000 1 404000 6 505000 2 505000 6 606000 1 www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data 51. b. curvilinear c. The largest difference in sales occurred between 2010 and 2011. That difference was almost 14 billions. 55. A slight positive linear relationship 61. b. Both relationships seem to be linear in nature. c. This occurred in 1998 1999 and 2001. 67. b. It appears that there is a positive linear relationship between the attendance and the year. 73. a. The independent variable is hours and the dependent variable is sales. b. It appears that there is a positive linear relationship. 43. a. b. The largest average charges occur at university-related hospitals and the lowest average appears to be in religious-affiliated hospitals. c. A pie chart showing how that total is divided among the four hospital types would not be useful or appropriate. 47. The sales have trended upward over the past 12 months. 49. The line chart shows that year-end deposits have been increas- ing since 1997 but have increased more sharply since 2002 and have leveled off between 2006 and 2007 and recently become more volatile. University Related Religious Affiliated Municipally Owned Privately Held 6398 3591 4613 5191 Continuous Data Data whose possible values are uncountable and that may assume any value in an interval. Cumulative Frequency Distribution A summary of a set of data that displays the number of observations with values less than or equal to the upper limit of each of its classes. Cumulative Relative Frequency Distribution A summary of a set of data that displays the proportion of observations with values less than or equal to the upper limit of each of its classes. Dependent Variable A variable whose values are thought to be a function of or dependent on the values of another vari- able called the independent variable. On a scatter plot the dependent variable is placed on the y axis and is often called the response variable. All-Inclusive Classes A set of classes that contains all the pos- sible data values. Bar Chart A graphical representation of a categorical data set in which a rectangle or bar is drawn over each category or class. The length or height of each bar represents the fre- quency or percentage of observations or some other meas- ure associated with the category. The bars may be vertical or horizontal. The bars may all be the same color or they may be different colors depicting different categories. Addi- tionally multiple variables can be graphed on the same bar chart. Class Boundaries The upper and lower values of each class. Class Width The distance between the lowest possible value and the highest possible value for a frequency class. Glossary Berenson Mark L. and David M. Levine Basic Business Sta- tistics: Concepts and Applications 12th ed. Upper Saddle River NJ: Prentice Hall 2012. Cleveland William S. “Graphs in Scientific Publications” The American Statistician 38 November 1984 pp. 261–269. Cleveland William S. and R. McGill “Graphical Perception: Theory Experimentation and Application to the Develop- ment of Graphical Methods” Journal of the American Statis- tical Association 79 September 1984 pp. 531–554. Cryer Jonathan D. and Robert B. Miller Statistics for Busi- ness: Data Analysis and Modeling 2nd ed. Belmont CA: Duxbury Press 1994. DeVeaux Richard D. Paul F. Velleman and David E. Bock Stats Data and Models 3rd ed. New York: Addison-Wesley 2012. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. Tufte Edward R. Envisioning Information Cheshire CT: Graphics Press 1990. Tufte Edward R. The Visual Display of Quantitative Informa- tion 2nd ed. Cheshire CT: Graphics Press 2001. Tukey John W. Exploratory Data Analysis Reading MA: Addison-Wesley 1977. References www.downloadslide.com

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Graphs Charts and Tables—Describing Your Data Mutually Exclusive Classes Classes that do not overlap so that a data value can be placed in only one class. Ogive The graphical representation of the cumulative relative frequency. A line is connected to points plotted above the upper limit of each class at a height corresponding to the cumulative relative frequency. Pie Chart A graph in the shape of a circle. The circle is divided into “slices” corresponding to the categories or classes to be displayed. The size of each slice is proportional to the magnitude of the displayed variable associated with each category or class. Relative Frequency The proportion of total observations that are in a given category. Relative frequency is computed by dividing the frequency in a category by the total number of observations. The relative frequencies can be converted to percentages by multiplying by 100. Scatter Diagram or Scatter Plot A two-dimensional graph of plotted points in which the vertical axis represents values of one quantitative variable and the horizontal axis repre- sents values of the other quantitative variable. Each plotted point has coordinates whose values are obtained from the respective variables. Discrete Data Data that can take on a countable number of possible values. Equal-Width Classes The distance between the lowest pos- sible value and the highest possible value in each class is equal for all classes. Frequency Distribution A summary of a set of data that dis- plays the number of observations in each of the distribu- tion’s distinct categories or classes. Frequency Histogram A graph of a frequency distribution with the horizontal axis showing the classes the verti- cal axis showing the frequency count and for equal class widths the rectangles having a height equal to the frequency in each class. Independent Variable A variable whose values are thought to impact the values of the dependent variable. The inde- pendent variable or explanatory variable is often within the direct control of the decision maker. On a scatter plot the independent variable or explanatory variable is graphed on the x axis. Line Chart A two-dimensional chart showing time on the hori- zontal axis and the variable of interest on the vertical axis. www.downloadslide.com

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1 Measures of Center and Location 2 Measures of Variation 3 Using the Mean and Standard Deviation Together Outcome 3. Compute the range interquartile range variance and standard deviation and know what these values mean. Outcome 4. Compute a z score and the coefficient of variation and understand how they are applied in decision-making situations. Outcome 5. Understand the Empirical Rule and Tchebysheff’s Theorem Quick Prep Links Review the definitions for nominal ordinal interval and ratio data. Examine the statistical software such as Excel used during this course to identify the tools for computing descriptive measures. For instance in Excel look at the function wizard and the descriptive statistics tools on the Data tab under Data Analysis. Review the material on frequency histograms paying special attention to how histograms help determine where the data are centered and how the data are spread around the center. Describing Data Using Numerical Measures Outcome 1. Compute the mean median mode and weighted mean for a set of data and understand what these values represent. Outcome 2. Construct a box and whisker graph and interpret it. Why you need to know Suppose you are the advertising manager for a major airline and you want to develop an ad campaign touting how much cheaper your fares are than the competition’s. You must be careful that your claims are valid. First the Federal Trade Commission FTC is charged with regulating advertising and requires that advertising be truthful. Second customers who can show that they were misled by an incorrect claim about prices could sue your company. You need to use statistical procedures to determine the validity of any claim you might want to make about your prices. Graphs and charts provide effective tools for transforming data into information however they are only a starting point. Graphs and charts do not reveal all the information contained in a set of data. To make your descriptive toolkit complete you need to become familiar with key descriptive measures that are widely used in fully describing data. You might start by sampling travel routes from your company and from the competition. You could then determine the price of a round-trip flight for each route for your airline and your competitors. You might graph the data for each company as a histogram but a clear comparison using only this graph might be difficult. Instead you could compute the summary flight price measures for the various airlines and show these values side by side perhaps in a bar chart. Thus to effectively describe data you will need to combine the graphical tools with the numerical measures introduced in this text. 1 Measures of Center and Location Histograms are an effective way of converting quantitative data into use- ful information. The histogram provides a visual indication of where data are centered and how much spread there is in the data around the center. Okea/Fotolia LLC From Chapter 3 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Describing Data Using Numerical Measures However to fully describe a quantitative variable we also need to compute measures of its center and spread. These measures can then be coupled with the histogram to give a clear picture of the variable’s distribution. This section focuses on measures of the center of data. Section 2 introduces measures of the spread of data. Parameters and Statistics Depending on whether we are working with a population or a sample a numerical measure is known as either a parameter or a statistic. Population Mean There are three important measures of the center of a set of data. The first of these is the mean or average of the data. To find the mean we sum the values and divide the sum by the number of data values as shown in Equation 1. Parameter A measure computed from the entire population. As long as the population does not change the value of the parameter will not change. Statistic A measure computed from a sample that has been selected from a population. The value of the statistic will depend on which sample is selected. Mean A numerical measure of the center of a set of quantitative measures computed by dividing the sum of the values by the number of values in the data. Population Mean ∑ x N i i N 1 1 where: m Population mean 1mu2 N Population size x i ith individual value of variable x The population mean is represented by the Greek symbol m pronounced “mu.” The formal notation in the numerator for the sum of the x values reads xx i i i N i 1 Sum all values where goes from ∑ → 1 1to N In other words we are summing all N values in the population. Because you almost always sum all the data values to simplify notation in this text we generally will drop the subscripts after the first time we introduce a formula. Thus the for- mula for the population mean will be written as ∑ x N BUSINESS APPLICATION POPULATION MEAN FOSTER CITY HOTEL The manager of a small hotel in Foster City California was asked by the corporate vice president to analyze the Sunday night registration information for the past eight weeks. Data on three variables were collected: x 1 Total number of rooms rented x 2 Total dollar revenue from the room rentals x 3 Number of customer complaints that came from guests each Sunday These data are shown in Table 1. They are a population because they include all data that interest the vice president. Figure 1 shows the frequency histogram for the number of rooms rented. If the manager wants to describe the data further she can locate the center of the data by finding the bal- ance point for the histogram. Think of the horizontal axis as a plank and the histogram bars as weights proportional to their area. The center of the data would be the point at which the plank would balance. As shown in Figure 1 the balance point seems to be about 15 rooms. Population Mean The average for all values in the population computed by dividing the sum of all values by the population size. Chapter Outcome 1. www.downloadslide.com

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Describing Data Using Numerical Measures Eyeing the histogram might yield a reasonable approximation of the center. However computing a numerical measure of the center directly from the data is preferable. The most frequently used measure of the center is the mean. The population mean for number of rooms rented is computed using Equation 1 as follows: ∑ ++ + + ++ + x N 22 13 10 16 23 13 11 13 8 121 8 15 125 . Thus the average number of rooms rented on Sunday for the past eight weeks is 15.125. This is the true balance point for the data. Take a look at Table 2 where we calculate what is called a deviation 1x i -m2 by subtracting the mean from each value x i . TABLE 1 | Foster City Hotel Data Week Rooms Rented Revenue Complaints 1 22 1870 0 2 13 1590 2 3 10 1760 1 4 16 2345 0 5 23 4563 2 6 13 1630 1 7 11 2156 0 8 13 1756 0 TABLE 2 | Centering Concept of the Mean Using Hotel Data x x2μ 5 Deviation 22 22 - 15.125 6.875 13 13 - 15.125-2.125 10 10 - 15.125-5.125 16 16 - 15.125 0.875 23 23 - 15.125 7.875 13 13 - 15.125-2.125 11 11 - 15.125-4.125 13 13 - 15.125 -2.125 Σ1x-m2 0.000 d Sum of deviations equals zero. Number of Occurrences 5 to 10 11 to 15 16 to 20 21 to 25 Approximate Balance Point Rooms Rented 5 4 3 2 1 0 FIGURE 1 |  Balance Point Rooms Rented at Foster City Hotel www.downloadslide.com

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Describing Data Using Numerical Measures Note that the sum of the deviations of the data from the mean is zero. This is not a coinci- dence. For any set of data the sum of the deviations around the mean will be zero. EXAMPLE 1 COMPUTING THE POPULATION MEAN United Airlines As the airline industry becomes increas- ingly competitive in an effort to increase profits many airlines are reducing flights. Therefore the supply of idled airplanes has increased. United Airlines headquartered in Chicago has decided to expand its fleet. Suppose United selects additional planes from a list of 17 possible planes including such models as the Boeing 747-400 the Air Bus 300-B4 and the DC 9-10. At a recent meeting the chief operating officer asked a member of his staff to determine the mean fuel consumption rate per hour of operation for the population of 17 planes. Step 1 Collect data for the quantitative variable of interest. The staff member was able to determine for each of the 17 planes the hourly fuel consumption in gallons for a flight between Chicago and New York City. These data are recorded as follows: Airplane Fuel Consumption 1galhr2 B747-400 3529 L-1011-100/200 2215 DC-10-10 2174 A300-B4 1482 A310-300 1574 B767-300 1503 B767-200 1377 B757-200 985 B727-200 1249 MD-80 882 B737-300 732 DC-9-50 848 B727-100 806 B737-100/200 1104 F-100 631 DC-9-30-11 804 DC-9-10 764 Step 2 Add the data values. ∑x 3529 2215 2174 ... 764 22 659 Step 3 Divide the sum by the number of values in the population using Equation 1. ∑x N 22 659 17 1 332 9 . The mean number of gallons of fuel consumed per hour on these 17 planes is 1332.9. END EXAMPLE BUSINESS APPLICATION POPULATION MEAN FOSTER CITY HOTEL CONTINUED In addition to collecting data on the number of rooms rented on Sunday nights the Foster City Hotel manager also collected data on the room- rental revenue generated and the number of complaints on Sunday nights. Excel can quite easily be used to calculate numerical measures such as the mean. Because these data are the population of all nights of interest to the hotel manager she can compute the population mean How to do it Example 1 Computing the Population Mean When the available data consti- tute the population of interest the population mean is computed using the following steps: 1. Collect the data for the variable of interest for all items in the population. The data must be quantitative. 2. Sum all values in the population 1Σx i 2. 3. Divide the sum1Σx i 2. by the number of values N in the population to get the popula- tion mean. The formula for the population mean is ∑ x N m Bruce Leibowitz/ Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures m revenue per night. The population mean is m +2208.75 as shown in the Excel output in Figure 2. Likewise the mean number of complaints is m 0.75 per night. Note there are other measures shown in the figures. We will discuss several of these later in the chapter. Now for these eight Sunday nights the manager can report to the corporate vice presi- dent that the mean number of rooms rented is 15.13. This level of business generated a mean nightly revenue of 2208.75. The number of complaints averaged 0.75 less than 1 per night. These values are the true means for the population and are therefore called parameters. Sample Mean The data for the Foster City Hotel constituted the population of interest. Thus m 15.13 rooms rented is the parameter measure. However if the data constitute a sample rather than a population the mean for the sample sample mean is computed using Equation 2. Sample Mean The average for all values in the sample computed by dividing the sum of all sample values by the sample size. Sample Mean x x n i i n 5 51 ∑ 2 where: x Sample mean 1pronounced xbar2 n Sample size Excel Tutorial Excel tutorials Notice Equation 2 is the same as Equation 1 except that we sum the sample values not the population values and divide by the sample size not the population size. FIGURE 2 |  Excel 2010 Output Showing Mean Revenue for the Foster City Hotel Mean Rooms Rented 15.13 Mean Revenue 2208.75 Mean Complaints 0.75 Minitab Instructions for similar results: 1. Open fle: Foster.MTW. 2. Choose Stat Basic Statistics Display Descriptive Statistics. 3. In Variables enter columns Rooms Rented Revenue and Complaints. 4. Click Statistics. 5. Check required statistics. 6. Click OK. OK. Excel 2010 Instructions: 1. Open fle: Foster.xlsx. 2. Select the Data tab. 3. Click on Data Analysis Descriptive Statistics. 4. Defne data range for the desired variables. 5. Check Summary Statistics. 6. Name new Output Sheet. 7. On Home tab adjust decimal places as desired. Chapter Outcome 1. www.downloadslide.com

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Describing Data Using Numerical Measures The notation for the sample mean is x. Sample descriptors statistics are usually assigned a Roman character. Recall that population values usually are assigned a Greek character. EXAMPLE 2 COMPUTING A SAMPLE MEAN Professor Salaries A newspaper reporter in Wisconsin collected a sample of seven univer- sity professors and determined their annual salaries. As part of her story she wished to be able to report the mean salary. The following steps are used to calculate the sample mean salaries for professors in Wisconsin. Step 1 Collect the sample data. 5x i 6 5Professor Salaries6 5+144000 +98000 +204000 +177000 +155000 +316000 +1000006 Step 2 Add the values in the sample. Σx +144000 + +98000 + +204000 + +177000 + +155000 + +316000 + +100000 +1194000 Step 3 Divide the sum by the sample size Equation 2. x x n 55 5 ∑ . 1 194 000 7 170 571 43 Therefore the mean salary for the sample of seven professors is 170571.43. END EXAMPLE The Impact of Extreme Values on the Mean The mean population or sample is the balance point for data so using the mean as a measure of the center generally makes sense. However the mean does have a potential disadvantage: The mean can be affected by extreme values. There are many instances in business when this may occur. For example in a population or sample of income data there likely will be extremes on the high end that will pull the mean upward from the center. Example 3 illustrates how an extreme value can affect the mean. In these situations a second measure called the median may be more appropriate. EXAMPLE 3 IMPACT OF EXTREME VALUES Professor Salaries Suppose the sample of professor salaries see Example 2 had been slightly different. If the salary recorded as 316000 had actually been 1000000 must also be the college’s football coach how would the mean be affected We can see the impact as follows: Step 1 Collect the sample data. x i Professor Salaries 144000 98000 204000 +177000 +155000 +1000000 +1000006 extreme value Step 2 Add the values. Σx +144000 + 98000 + 204000 + 177000 + 155000 + 1000000 + 100000 +1878000 Step 3 Divide the sum by the number of values in the sample. x x n 55 5 ∑ . 1 878 000 7 268 285 71 Recall in Example 2 the sample mean was 170571.43. www.downloadslide.com

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Describing Data Using Numerical Measures With only one value in the sample changed the mean is now substantially higher than before. Because the mean is affected by extreme values it may be a misleading measure of the data’s center. In this case the mean is larger than all but one of the starting salaries. END EXAMPLE TRY PROBLEM 15 Median Another measure of the center is called the median. The median is found by first arranging data in numerical order from smallest to largest. Data that are sorted in order are referred to as a data array. Equation 3 is used to find the index point corresponding to the median value for a set of data placed in numerical order from low to high. Median The median is a center value that divides a data array into two halves. We use u ∼ to denote the population median and M d to denote the sample median. Data Array Data that have been arranged in numerical order. Median Index in 5 1 2 3 where: i The index of the point in the data array corresponding to the median value n Sample size If i is not an integer round its value up to the next highest integer. This next highest integer then is the position of the median in the data array. If i is an integer the median is the average of the values in position i and position i + 1. For instance suppose a personnel manager has hired 10 new employees. The ages of each of these employees sorted from low to high is listed as follows: 23 25 25 34 35 45 46 47 52 54 Using Equation 3 to find the median index we get in 55 5 1 2 1 2 10 5 Since the index is an integer the median value will be the average of the 5th and 6th values in the data set. Thus the median is M d 55 35 45 2 40 + Consider another case in which customers at a restaurant are asked to rate the service they received on a scale of 1 to 100. A total of 15 customers were asked to provide the ratings. The data sorted from low to high are presented as follows: 60 68 75 77 80 80 80 85 88 90 95 95 95 95 99 Using Equation 3 we get the median index: in 55 5 1 2 1 2 15 7 5 . Since the index is not an integer we round 7.5 up to 8. Thus the median 1M d 2 is the 8th data value from either end. In this case M d 85 Chapter Outcome 1. www.downloadslide.com

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Describing Data Using Numerical Measures EXAMPLE 4 COMPUTING THE MEDIAN Professor Salaries Consider again the example involving the news reporter in Wisconsin and the sample salary data in Example 2. The median for these data is computed using the following steps: Step 1 Collect the sample data. 5x i 6 5Professor Salaries6 5+144000 +98000 +204000 +177000 +155000 +316000 +1000006 Step 2 Sort the data from smallest to largest forming a data array. 5x i 6 5+98000 +100000 +144000 +155000 +177000 +204000 +3160006 Step 3 Calculate the median index. Using Equation 3 we get i 1 2 73 5 .. Rounding up the median is the fourth value from either end of the data array. Step 4 Find the median. 5x i 6 5+98000 +100000 +144000 +155000 +177000 +204000 +3160006 fourth value M d The median salary is 155000. The notation for the sample median is M d . Note if the number of data values in a sample or population is an even number the median is the average of the two middle values. END EXAMPLE TRY PROBLEM 2 Skewed and Symmetric Distributions Data in a population or sample can be either symmetric or skewed depending on how the data are distributed around the center. In the original professor salary example Example 2 the mean for the sample of seven managers was 170571.43. In Example 4 the median salary was 155000. Thus for these data the mean and the median are not equal. This sample data set is right skewed because xM d 55 . . 170 571 43 155 000 Figure 3 illustrates examples of right-skewed left-skewed and symmetric distributions. The greater the difference between the mean and the median the more skewed the distribu- tion. Example 5 shows that an advantage of the median over the mean is that the median is not affected by extreme values. Thus the median is particularly useful as a measure of the center when the data are highly skewed. 1 Symmetric Data Data sets whose values are evenly spread around the center. For symmetric data the mean and median are equal. Skewed Data Data sets that are not symmetric. For skewed data the mean will be larger or smaller than the median. Right-Skewed Data A data distribution is right skewed if the mean for the data is larger than the median. Left-Skewed Data A data distribution is left skewed if the mean for the data is smaller than the median. 1 Excel can be used to calculate a skewness statistic. The sign on the skewness statistic implies the direction of skewness. The higher the absolute value the more the data are skewed. FIGURE 3 |  Skewed and Symmetric Distributions Frequency x a Right-Skewed Median Mean Frequency x b Left-Skewed Mean Median Frequency x c Symmetric Mean Median www.downloadslide.com

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Describing Data Using Numerical Measures EXAMPLE 5 IMPACT OF EXTREME VALUES ON THE MEDIAN Professor Salaries Continued In Example 3 when we substituted a 1000000 sal- ary for the professor who had a salary of 316000 the sample mean salary increased from 170571.43 to 268285.71. What will happen to the median The median is determined using the following steps: Step 1 Collect the sample data. The sample salary data including the extremely high salary are 5x i 6 5Professor Salary6 5+144000 +98000 +204000 +177000 +155000 +1000000 +1000006 Step 2 Sort the data from smallest to largest forming a data array. 5x i 6 5+98000 +100000 +144000 +155000 +177000 +204000 +10000006 Step 3 Calculate the median index. Using Equation 3 we get i55 1 2 73 5 .. Rounding up the median is the fourth value from either end of the data array. Step 4 Find the median. 5x i 6 5+98000 +100000 +144000 +155000 +177000 +204000 +10000006 fourth value M d The median professor salary is 155000 the same value as in Example 4 when the high sal- ary was not included in the data. Thus the median is not affected by the extreme values in the data. END EXAMPLE TRY PROBLEM 2 Mode The mean is the most commonly used measure of central location followed closely by the median. However the mode is another measure that is occasionally used as a measure of central location. A data set may have more than one mode if two or more values tie for the most frequently occurring value. Example 6 illustrates this concept and shows how the mode is determined. EXAMPLE 6 DETERMINING THE MODE Smoky Mountain Pizza The owners of Smoky Moun- tain Pizza are planning to expand their restaurant to include an open-air patio. Before finalizing the design the managers want to know what the most frequently occurring group size is so they can organize the seating arrangements to best meet demand. They wish to know the mode which can be calcu- lated using the following steps: Step 1 Collect the sample data. A sample of 20 groups was selected at random. These data are 5x i 6 5people6 52 4 1 2 3 2 4 2 3 6 8 4 2 1 7 4 2 4 4 36 Step 2 Organize the data into a frequency distribution. x i Frequency 1 2 2 6 3 3 4 6 Mode The mode is the value in a data set that occurs most frequently. Chapter Outcome 1. Beth V an Trees/Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures x i Frequency 5 0 6 1 7 1 8 1 Total 20 Step 3 Determine the values that occurs occur most frequently. In this case there are two modes because the values 2 and 4 each occurred six times. Thus the modes are 2 and 4. END EXAMPLE TRY PROBLEM 2 A common mistake is to state the mode as being the frequency of the most frequently occurring value. In Example 6 you might be tempted to say that the mode 6 because that was the highest frequency. Instead there were two modes 2 and 4 each of which occurred six times. If no value occurs more frequently than any other the data set is said to not have a mode. The mode might be particularly useful in describing the central location value for clothes sizes. For example shoes come in full and half sizes. Consider the following sample data that have been sorted from low to high: 5x6 57.5 8.0 8.5 9.0 9.0 10.0 10.0 10.0 10.5 10.5 11.0 11.56 The mean for these sample data is x x n 55 5 5 ∑+ ++ 75 80 115 12 115 50 12 963 .. . . . Although 9.63 is the numerical average the mode is 10 because more people wore that size shoe than any other. In making purchasing decisions a shoe store manager would order more shoes at the modal size than at any other size. The mean isn’t of any particular use in her pur- chasing decision. Applying the Measures of Central Tendency The cost of tuition is an important factor that most students and their families consider when deciding where to attend college. The data file Colleges and Universities contains data for a sample of 718 colleges and universities in the United States. The cost of out-of-state tuition is one of the variables in the data file. Suppose a guidance counselor who will be advising students about college choices wishes to conduct a descriptive analysis for this quantitative variable. Figure 4 shows a frequency histogram generated using Excel. This histogram is a good place to begin the descriptive analysis since it allows the analyst to get a good indication of the center value the spread around the center and the general shape of the distribution of out- of-state tuition for these colleges and universities. Given that the file contains 718 colleges and universities using the 2 k Ú n rule the guidance counselor used k 10 classes. The least expensive school in the file is CUNY–Medgar Evers College in New York at 2600 and the most expensive is Franklin and Marshall in Pennsylvania at 24940. Based on this histogram in Figure 4 what would you conclude about the distribution of college tuition Is it skewed right or left The analysis can be extended by computing appropriate descriptive measures for the out-of-state tuition variable. Specifically we want to look at measures of central location. Figure 5 shows the Excel output with descriptive measures for out-of-state tuition. First focus on the primary measures of central location: mean and median. These are Mean +9933.38 Median +9433 These statistics provide measures of the center of the out-of-state tuition variable. The mean tuition value was 9933.38 whereas the median was 9433. Because the mean exceeds the Excel tutorials Excel Tutorial www.downloadslide.com

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Describing Data Using Numerical Measures median we conclude that the data are right skewed—the same conclusion you should have reached by looking at the histogram in Figure 4. Issues with Excel In many instances data files will have “missing values.” That is the val- ues for one or more variables may not be available for some of the observations. The data may have been lost or they were not measured when the data were collected. Many times when you receive data like this the missing values will be coded in a special way. For example the code NA might be used or a -99 might be entered to signify that the datum for that observation is missing. Statistical software packages typically have flexible procedures for dealing with missing data. However Excel does not contain a missing-value option. If you attempt to use certain data analysis options in Excel such as Descriptive Statistics in the presence of nonnumeric 1NA2 data you will get an error message. When that happens you must clear the missing values generally by deleting all rows with missing values. In some instances you can save the good data in the row by using Edit-Clear-All for the cell in question. However a bigger problem exists when the missing value has been coded as an arbitrary numeric value 1-992. In this case unless you go into the data and clear these values Excel will use the -99 values in the computations as if they are real values. The result will be incorrect calculations. Also if a data set contains more than one mode Excel’s Descriptive Statistics tool will only show the first mode in the list of modes and will not warn you that multiple modes exist. For instance if you look at Figure 5 Excel has computed a mode +6550. If you examine these data you will see that a tuition of 6550 occurred 14 times. This is the most frequently occurring value. However had other tuition values occurred 14 times too Excel’s Descriptive Statistics tool would not have so indicated. Excel 2010 does however have a function MODE.MULT that will display multiple modes when they exist. FIGURE 4 |  Excel 2010 Frequency Histogram of College Tuition Prices Minitab Instructions for similar results: 1. Open fle: Colleges and Universities.MTW. 2. Choose Graph Histogram. 3. Click Simple. 4. Click OK. 5. In Graph variables enter data column out-of-state tuition. 6. Click OK. Excel 2010 Instructions: 1. Open fle Colleges and Universities.xlsx. 2. Set up an area in the worksheet for the bins upper limit of each class as 4750 7000 etc. Be sure to label the column of upper limits as “Bins.” 3. On the Data tab click Data Analysis Histogram. 4. Input Range specifes the actual data values as the out-of-state tuition. 5. Put on a new worksheet and include Chart Output. 6. Right click on the bars and use the Format Data Series Options to set gap width to zero and add lines to bars. 7. Convert the bins in column A of the histogram output sheet to actual class labels. Note the bin labeled 4750 is changed to “under 4750.” 8. Click on Layout and set titles as desired. www.downloadslide.com

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Describing Data Using Numerical Measures Other Measures of Location Weighted Mean The arithmetic mean is the most frequently used measure of central loca- tion. Equations 1 and 2 are used when you have either a population or a sample. For instance the sample mean is computed using x x n xx x x n n 55 ∑ ++ + + 12 3 In this case each x value is given an equal weight in the computation of the mean. However in some applications there is reason to weight the data values differently. In those cases we need to compute a weighted mean. Equations 4 and 5 are used to find the weighted mean or weighted average for a popula- tion and for a sample respectively. Weighted Mean The mean value of data values that have been weighted according to their relative importance. Weighted Mean for a Population w ii i wx w ∑ ∑ 4 FIGURE 5 |  Excel 2010 Descriptive Statistics for College Tuition Data Excel 2010 Instructions: 1. Open fle: Colleges and Universities.xlsx. 2. Select the Data tab. 3. Click on Data Analysis Descriptive Statistics. 4. Defne data range for the desired variables. 5. Check Summary Statistics. 6. Name new Output Sheet. 7. On Home tab adjust decimal places as desired. Note the Skewness statistic is a small positive number indicating a slight amount of positive right skew to the tuition data. The higher the absolute value of the Skewness statistic the greater the skewness in the data. Minitab Instructions for similar results: 1. Open fle: Colleges and Universities.MTW. 2. Choose Stat Basic Statistics Display Descriptive Statistics. 3. In Variables enter column out-of-state tuition. 4. Click Statistics. 5. Check required statistics. 6. Click OK. OK. Mean Median Mode Chapter Outcome 1. www.downloadslide.com

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Describing Data Using Numerical Measures EXAMPLE 7 CALCULATING A WEIGHTED POPULATION MEAN Myers Associates Recently the law firm of Myers Associates was involved in litigating a discrimination suit concerning ski instructors at a ski resort in Colorado. One ski instructor from Germany had sued the operator of the ski resort claiming he had not received equitable pay com- pared with the other ski instructors from Norway and the United States. In preparing a defense the Myers attorneys planned to compute the mean annual income for all seven Norwegian ski instruc- tors at the resort. However because these instructors worked different numbers of days during the ski season a weighted mean needed to be computed. This was done using the following steps: Step 1 Collect the desired data and determine the weight to be assigned to each data value. In this case the variable of interest was the income of the ski instructors. The population consisted of seven Norwegian instructors. The weights were the number of days that the instructors worked. The following data and weights were determined: x i Income: +7600 +3900 +5300 +4000 +7200 +2300 +5100 w i Days: 50 30 40 25 60 15 50 Step 2 Multiply each weight by the data value and sum these. Σw i x i 15021+76002 + 13021+39002+ c +15021+51002 +1530500 Step 3 Sum the weights for all values the weights are the days. Σw i 50 + 30 + 40 + 25 + 60 + 15 + 50 270 Step 4 Compute the weighted mean. Divide the weighted sum by the sum of the weights. Because we are working with the population the result will be the population weighted mean. W ii i wx w ∑ ∑ . 1 530 500 270 5 668 52 Thus taking into account the number of days worked the Norwegian ski instructors had a mean income of 5668.52. END EXAMPLE TRY PROBLEM 8 One weighted-mean example that you are probably very familiar with is your college grade point average GPA. At most schools A 4 points B 3 points and so forth. Each course has a certain number of credits usually 1 to 5. The credits are the weights. Your GPA is computed by summing the product of points earned in a course and the credits for the course and then dividing this sum by the total number of credits earned. Percentiles In some applications we might wish to describe the location of the data in terms other than the center of the data. For example prior to enrolling at your university you took the SAT or ACT test and received a percentile score in math and verbal skills. Percentiles The pth percentile in a data array is a value that divides the data set into two parts. The lower segment contains at least p and the upper segment contains at least 1 100 - p2 of the data. The 50th percentile is the median. Weighted Mean for a Sample x wx w w ii i 5 ∑ ∑ 5 where: w i The weight of the ith data value x i The ith data value www.downloadslide.com

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Describing Data Using Numerical Measures If you received word that your standardized exam score was at the 90th percentile it means that you scored as high as or higher than 90 of the other students who took the exam. The score at the 50th percentile would indicate that you were at the median where at least 50 scored at or below and at least 50 scored at or above your score. 2 To illustrate how to manually approximate a percentile value consider a situation in which you have 309 customers enter a bank during the course of a day. The time rounded to the nearest minute that each customer spends in the bank is recorded. If we wish to approxi- mate the 10th percentile we would begin by first sorting the data in order from low to high then assign each data value a location index from 1 to 309 and next determine the location index that corresponds to the 10th percentile using Equation 6. 2 More rigorously the percentile is that value or set of values such that at least p of the data is as small or smaller than that value and at least 1100 - p2 of the data is at least as large as that value. For introductory courses a convention has been adopted to average the largest and smallest values that qualify as a certain percentile. This is why the median was defined as it was earlier for data sets with an even number of data values. Percentile Location Index in 5 p 100 6 where: p Desired percent n Number of values in the data set If i is not an integer round up to the next highest integer. The next integer greater than i corresponds to the position of the pth percentile in the data set. If i is an integer the pth percentile is the average of the values in position i and position i + 1. Thus the index value associated with the 10th percentile is in 55 5 p 100 10 100 309 30 90 . Because i 30.90 is not an integer we round to the next highest integer which is 31. The 10th percentile corresponds to the value in the 31st position from the low end of the sorted data. EXAMPLE 8 CALCULATING PERCENTILES Henson Trucking The Henson Trucking Company is a small company in the business of moving people from one home to another within the Dallas Texas area. Historically the owners have charged the customers on an hourly basis regardless of the distance of the move within the Dallas city limits. However they are now considering adding a surcharge for moves over a certain distance. They have decided to base this charge on the 80th percentile. They have a sample of travel-distance data for 30 moves. These data are as follows: 13.5 8.6 16.2 21.4 21.0 23.7 4.1 13.8 20.5 9.6 11.5 6.5 5.8 10.1 11.1 4.4 12.2 13.0 15.7 13.2 13.4 13.1 21.7 14.6 14.1 12.4 24.9 19.3 26.9 11.7 The 80th percentile can be computed using these steps. Step 1 Sort the data from lowest to highest 4.1 4.4 5.8 6.5 8.6 9.6 10.1 11.1 11.5 11.7 12.2 12.4 13.0 13.1 13.2 13.4 13.5 13.8 14.1 14.6 15.7 16.2 19.3 20.5 21.0 21.4 21.7 23.7 24.9 26.9 How to do it Example 8 Calculating Percentiles To calculate a specific percentile for a set of quantitative data you can use the following steps: 1. Sort the data in order from the lowest to highest value. 2. Determine the percentile loca- tion index i using Equation 6. in p 100 5 where p Desired percent n Number of values in the data set 3. If i is not an integer then round to next highest integer. The pth percentile is located at the rounded index position. If i is an integer the pth percentile is the average of the values at location index positions i and i + 1. Gilles Lougassi/Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures Step 2 Determine percentile location index i using Equation 6. The 80th percentile location index is in 55 5 p 100 80 100 30 24 Step 3 Locate the appropriate percentile. Because i 24 is an integer value the 80th percentile is found by averaging the values in the 24th and 25th positions. These are 20.5 and 21.0. Thus the 80th percentile is 120.5 + 21.022 20.75 therefore any distance exceeding 20.75 miles will be subject to a surcharge. END EXAMPLE TRY PROBLEM 7 Quartiles Another location measure that can be used to describe data is quartiles. The first quartile corresponds to the 25th percentile. That is it is the value at or below which there is at least 25 one quarter of the data and at or above which there is at least 75 of the data. The third quartile is also the 75th percentile. It is the value at or below which there is at least 75 of the data and at or above which there is at least 25 of the data. The second quartile is the 50th percentile and is also the median. A quartile value can be approximated manually using the same method as for percentiles using Equation 6. For the 309 bank customer-service times mentioned earlier the location of the first-quartile 25th percentile index is found after sorting the data as in 55 5 p 100 25 100 309 77 25 . Because 77.25 is not an integer value we round up to 78. The first quartile is the 78th value from the low end of the sorted data. Issues with Excel The procedure that Excel uses to compute quartiles is not standard. There- fore the quartile and percentile values from Excel will be slightly different from those we find manually using Equation 6. For example referring to Example 8 when Excel is used to compute the 80th percentile for the moving distances the value returned is 20.60 miles. This is slightly dif- ferent from the 20.75 we found in Example 8. Equation 6 is generally accepted by statisticians to be correct. However Excel will give reasonably close percentile and quartile values. Box and Whisker Plots A descriptive tool that many decision makers like to use is called a box and whisker plot or a box plot. The box and whisker plot incorporates the median and the quartiles to graphically display quantitative data. It is also used to identify outliers that are unusually small or large data values that lie mostly by themselves. EXAMPLE 9 CONSTRUCTING A BOX AND WHISKER PLOT Chevron Corporation A demand analyst for Chevron Corporation has recently performed a study at one of the com- pany’s stores in which he asked customers to set their trip odometer to zero when they filled up. Then when the custom- ers returned for their next fill-up he recorded the miles that had been driven and wishes to construct a box and whisker plot as part of a presentation to describe the driving patterns of Chevron customers between fill-ups. The sorted sample data showing the miles between fill-ups is as follows: 231 236 241 242 242 243 243 243 248 248 249 250 251 251 252 252 254 255 255 256 256 257 259 260 260 260 260 262 262 264 265 265 265 266 268 268 270 276 277 277 280 286 300 324 345 Quartiles Quartiles in a data array are those values that divide the data set into four equal-sized groups. The median corresponds to the second quartile. Box and Whisker Plot A graph that is composed of two parts: a box and the whiskers. The box has a width that ranges from the first quartile Q 3 to the third quartile Q 3 . A vertical line through the box is placed at the median. Limits are located at a value that is 1.5 times the difference between Q 1 and Q 3 below Q 1 and above Q 3 . The whiskers extend to the left to the lowest value within the limits and to the right to the highest value within the limits. Chapter Outcome 2. Laura Gangi Pond/Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures The box and whisker plot is computed using the following steps: Step 1 Sort the data from low to high. Step 2 Calculate the 25th percentile Q 1 the 50th percentile median and the 75th percentile Q 3 . The location index for Q 1 is in 55 5 p 100 25 100 45 11 25 . Thus Q 1 will be the 12th value which is 250 miles. The median location is in 55 5 p 100 50 100 45 22 5 . In the sorted data the median is the 23rd value which is 259 miles. The third-quartile location is in 55 5 p 100 75 100 45 33 75 . Thus Q 3 is the 34th data value. This is 266 miles. Step 3 Draw the box so the ends correspond to Q 1 and Q 3 . 230 Q 1 Q 3 240 250 260 270 280 290 300 310 320 330 340 350 Step 4 Draw a vertical line through the box at the median. 230 Q 1 Q 3 240 250 260 270 Median 280 290 300 310 320 330 340 350 Step 5 Compute the upper and lower limits. The lower limit is computed as Q 1 - 1.51Q 3 - Q 1 2. This is Lower Limit 250 - 1.51266 - 2502 226 The upper limit is Q 3 + 1.51Q 3 - Q 1 2. This is Upper Limit 266 + 1.51266 - 2502 290 Any value outside these limits is identified as an outlier. How to do it Example 9 Constructing a Box and Whisker Plot A box and whisker plot is a graphi- cal summary of a quantitative variable. It is constructed using the following steps: 1. Sort the data values from low to high. 2. Use Equation 6 to find the 25th percentile 1Q 1 first quartile2 the 50th percentile 1Q 2 median2 and the 75th percentile 1Q 3 third quartile2. 3. Draw a box so that the ends of the box are at Q 1 and Q 3 . This box will contain the middle 50 of the data values in the population or sample. 4. Draw a vertical line through the box at the median. Half the data values in the box will be on either side of the median. 5. Calculate the interquartile range 1IQR Q 3 - Q 1 2. The interquartile range will be discussed more fully in Section 2. Compute the lower limit for the box and whisker plot as Q 1 - 1.51Q 3 - Q 1 2. The upper limit is Q 3 + 1.51Q 3 - Q 1 2. Any data values outside these limits are referred to as outliers. 6. Extend dashed lines called the whiskers from each end of the box to the lowest and highest value within the limits. 7. Any value outside the limits outlier found in step 5 is marked with an asterisk . www.downloadslide.com

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Describing Data Using Numerical Measures Step 6 Draw the whiskers. The whiskers are drawn to the smallest and largest values within the limits. 230 Q 1 Q 3 240 250 260 270 Median 280 290 300 310 320 330 340 350 Lower Limit 226 Upper Limit 290 Outliers Step 7 Plot the outliers. The outliers are plotted as values outside the limits. END EXAMPLE TRY PROBLEM 5 Data-Level Issues You need to be very aware of the level of data you are working with before computing the numerical measures introduced in this chapter. A common mistake is to compute means on nominal-level data. For example a major electronics manufacturer recently surveyed a sam- ple of customers to determine whether they preferred black white or colored stereo cases. The data were coded as follows: 1 black 2 white 3 colored A few of the responses are Color code 51 1 3 2 1 2 2 2 3 1 1 1 3 2 2 1 26 Using these codes the sample mean is x x n 5 55 ∑ 30 17 1 765 . As you can see reporting that customers prefer a color somewhere between black and white but closer to white would be meaningless. The mean should not be used with nominal data. This type of mistake tends to happen when people use computer software to perform their calculations. Asking Excel or other statistical software to compute the mean median and so on for all the variables in the data set is very easy. Then a table is created and before long the meaningless measures creep into your report. Don’t let that happen. There is also some disagreement about whether means should be computed on ordinal data. For example in market research a 5- or 7-point scale is often used to measure customers’ atti- tudes about products or TV commercials. For example we might set up the following scale: 1 Strongly agree 2 Agree 3 Neutral 4 Disagree 5 Strongly disagree Customer responses to a particular question are obtained on this scale from 1 to 5. For a sam- ple of n 10 people we might get the following responses to a question: Response 52 2 1 3 3 1 5 2 1 36 www.downloadslide.com

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Describing Data Using Numerical Measures The mean rating is 2.3. We could then compute the mean for a second issue and compare the means. However what exactly do we have First when we compute a mean for a scaled vari- able we are making two basic assumptions: 1. We are assuming the distance between a rating of 1 and 2 is the same as the distance between 2 and 3. We are also saying these distances are exactly the same for the second issue’s variable to which you wish to compare it. Although from a numerical standpoint this is true in terms of what the scale is measuring is the difference between strongly agree and agree the same as the difference between agree and neutral If not is the mean really a meaningful measure 2. We are also assuming people who respond to the survey have the same definition of what “strongly agree” means or what “disagree” means. When you mark a 4 disagree on your survey are you applying the same criteria as someone else who also marks a 4 on the same issue If not then the mean might be misleading. Although these difficulties exist with ordinal data we see many examples in which means are computed and used for decision purposes. In fact we once had a dean who focused on one particular question on the course evaluation survey that was administered in every class each semester. This question was “Considering all factors of importance to you how would you rate this instructor” 1 Excellent 2 Good 3 Average 4 Poor 5 Very poor The dean then had his staff compute means for each class and for each professor. He then listed classes and faculty in order based on the mean values and he based a major part of the performance evaluation on where a faculty member stood with respect to mean score on this one question. By the way he carried the calculations for the mean out to three decimal places In general the median is the preferred measure of central location for ordinal data instead of the mean. Figure 6 summarizes the three measures of the center that have been discussed in this section. Skill Development 3-1. A random sample of 15 articles in Fortune revealed the following word counts per article: 5176 6005 5052 5310 4188 4132 5736 5381 4983 4423 5002 4573 4209 5611 4568 Compute the mean median first quartile and third quartile for these sample data. 3-2. The following data reflect the number of defects produced on an assembly line at the Dearfield Electronics Company for the past 8 days. 30201352 51300133 43184240 a. Compute the mean number of defects for this population of days. 3-1: Exercises MyStatLab FIGURE 6 |  Descriptive Measures of the Center Descriptive Measure Computation Method Data Level Advantages/ Disadvantages Mean Sum of values divided by the number of values Ratio Interval Numerical center of the data Sum of deviations from the mean is zero Sensitive to extreme values Median Middle value for data that have been sorted Ratio Interval Ordinal Not sensitive to extreme values Computed only from the center values t use information from all the data Mode Values that occur most frequently in the data Ratio Interval Ordinal Nominal May not refect the center May not exist t have multiple modes www.downloadslide.com

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Describing Data Using Numerical Measures b. Compute the median number of defects produced for this population of days. c. Determine if there is a mode number of defects and if so indicate the mode value. 3-3. A European cereal maker recently sampled 20 of its medium-size oat cereal packages to determine the weights of the cereal in each package. These sample data measured in ounces are as follows: 14.7 16.3 14.3 14.2 18.7 13.2 13.1 14.4 16.2 12.8 13.6 17.1 14.4 11.5 15.5 15.9 13.8 14.2 15.1 13.5 Calculate the first and third quartiles for these sample data. 3-4. The time in seconds that it took for each of 16 vehicles to exit a parking lot in downtown Cincinnati is 106 153 169 116 135 78 51 129 100 141 72 101 130 125 128 139 Compute the mean median first quartile and third quartile for the sample data. 3-5. A random sample of the miles driven by 20 rental car customers is shown as follows: 90 85 100 150 125 75 50 100 75 60 35 90 100 125 75 85 50 100 50 80 Develop a box and whisker plot for the sample data. 3-6. Examine the following data: 23 65 45 19 35 28 39 100 50 26 25 27 24 17 12 106 23 19 39 70 20 18 44 31 a. Compute the quartiles. b. Calculate the 90th percentile. c. Develop a box and whisker plot. d. Calculate the 20th and the 30th percentiles. 3-7. Consider the following data that represent the commute distances for students who attend Emory University: 3.1 4.7 8.4 11.6 12.1 13.0 13.4 16.1 17.3 20.8 22.8 24.3 26.2 26.6 26.7 31.2 32.2 35.8 35.8 39.8 a. Determine the 80th percentile. b. Determine numbers that are the 25th and 75th percentiles. c. Determine a number that qualifies as a median for these data. 3-8. A professor wishes to develop a numerical method for giving grades. He intends to base the grade on homework two midterms a project and a final examination. He wishes the final exam to have the largest influence on the grade. He wants the project to have 10 each midterm to have 20 and the homework to have 10 of the influence of the semester grade. a. Determine the weights the professor should use to produce a weighted average for grading purposes. b. For a student with the following grades during the quarter calculate a weighted average for the course: Instrument Final Project Midterm 1 Midterm 2 Homework Percentage Grade 64 98 67 63 89 c. Calculate an unweighted average of these five scores and discuss why the weighted average would be preferable here. Business Applications 3-9. The manager for the Jiffy Lube in Saratoga Florida has collected data on the number of customers who agreed to purchase an air filter when they were also having their oil changed. The sample data are shown as follows: 21 19 21 19 19 20 18 12 20 19 17 14 21 22 25 21 22 23 10 19 25 14 17 18 a. Compute the mean median and mode for these data. b. Indicate whether the data are skewed or symmetrical. c. Construct a box and whisker plot for these data. Referring to your answer in part b does the box plot support your conclusion about skewness Discuss. 3-10. During the past few years there has been a lot of discussion about the price of university textbooks. The complaints have come from many places including students faculty parents and even government officials. The publishing companies have been called on to explain why textbooks cost so much. Recently one of the major publishing companies was asked to testify before a congressional panel in Washington D.C. As part of the presentation the president of the company organized his talk around four main areas: production costs author royalties marketing costs and bookstore markup. He used one of his company’s business statistics texts as an example when he pointed out the production costs—including editing proofing printing binding inventory holding and distribution— come to about 32 per book sold. Authors receive 12 per copy for the hundreds of hours of creative work in writing the book and supplementary materials. Marketing costs are pegged at about 5 per copy sold and go to pay for the book sales force and examination copies sent to professors. The book is then sold to bookstores for 70 per copy a markup on costs of about 40 to cover overhead and the publishing costs associated with many upper-division low-market texts that lose money for the company. Once university bookstores purchase the book they mark it up place www.downloadslide.com

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Describing Data Using Numerical Measures Speed of Service time in seconds 83 138 145 147 130 79 156 156 90 85 68 93 178 76 73 119 92 146 88 103 116 134 162 71 181 110 105 74 a. Compute the mean median and mode for these sample data. b. Indicate whether the data are symmetrical or skewed. c. Construct a box and whisker plot for the sample data. Does the box and whisker plot support your conclusions in part b concerning the symmetry or skewness of these data 3-14. Todd Lindsey Associates a commercial real estate company located in Boston owns six office buildings in the Boston area that it leases to businesses. The lease price per square foot differs by building due to location and building amenities. Currently all six buildings are fully leased at the prices shown here. Price per Square Foot Number of Square Feet Building 1 75 125000 Building 2 85 37500 Building 3 90 77500 Building 4 45 35000 Building 5 55 60000 Building 6 110 130000 a. Compute the weighted average mean price per square foot for these buildings. b. Why is the weighted average price per square foot preferred to a simple average price per square foot in this case 3-15. Business Week recently reported that L. G. Philips LCD Co. would complete a new factory in Paju South Korea. It will be the world’s largest maker of liquid-crystal display panels. The arrival of the plant means that flat-panel LCD televisions would become increasingly affordable. The average retail cost of a 20″ LCD television in 2000 was 5139. To obtain what the average retail cost of a 37″ LCD was in 2008 a survey yielded the following data in U.S.: 606.70 558.12 625.82 533.70 464.37 511.15 400.56 538.20 531.64 632.14 474.86 567.46 588.39 528.78 610.32 564.71 912.68 475.87 545.25 589.15 a. Calculate the mean cost for these data. b. Examine the data presented. Choose an appropriate measure of the center of the data justify the choice and calculate the measure. it on the shelf and sell it to the student. If books go unsold they are returned to the publisher for a full refund. The following data reflect the dollar markup on the business statistics text for a sample of 20 college bookstores: 33 32 42 31 31 37 37 34 47 31 42 29 36 32 25 29 47 26 32 40 a. Compute the mean markup on the business statistics text by university bookstores in the sample. b. Compute the median markup. c. Determine the mode markup. d. Write a short paragraph discussing the statistics computed in parts a–c. 3-11. The Xang Corporation operates five clothing suppliers in China to provide merchandise for Nike. Nike recently sought information from the five plants. One variable for which data were collected was the total money in U.S. dollars the company spent on medical support for its employees in the first three months of the year. Data on number of employees at the plants are also shown. These data are as follows: Medical 7400 14400 12300 6200 3100 Employees 123 402 256 109 67 a. Compute the weighted mean medical payments for these five plants using number of employees as the weights. b. Explain why Nike would desire that a weighted average be computed in this situation rather than a simple numeric average. 3-12. The Tru-Green Lawn Company provides yard care services for customers throughout the Denver area. The company owner recently tracked the time his field employees spent at a sample of customer locations. He was hoping to use these data to help him with his scheduling and to establish billing rates. The following sample data in minutes were recorded: 31 27 29 22 24 30 28 21 29 26 22 17 17 20 38 10 38 25 27 23 23 13 17 34 25 29 22 22 14 11 29 26 29 29 37 32 27 26 18 22 Describe the central tendency of these data by computing the mean median and mode. Based on these measures can you conclude that the distribution of time spent at customer locations is skewed or symmetric 3-13. Eastern States Bank and Trust monitors its drive- thru service times electronically to ensure that its speed of service is meeting the company’s goals. A sample of 28 drive-thru times was recently taken and is shown here. www.downloadslide.com

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Describing Data Using Numerical Measures c. The influence an observation has on a statistic may be calculated by deleting the observation and calculating the difference between the original statistic and the statistic with the data point removed. The larger the difference the more influential the data point. Identify the data points that have the most and the least influence in the calculation of the sample mean. 3-16. The following table exhibits base salary data obtained from a survey of over 170 benchmark positions including finance positions. It reports the salaries of a sample of 25 chief finance officers for mid-sized firms. Assume the data are in thousands of dollars. 173.1 171.2 141.9 112.6 211.1 156.5 145.4 134.0 192.0 185.8 168.3 131.0 214.4 155.2 164.9 123.9 161.9 162.7 178.8 161.3 182.0 165.8 213.1 177.4 159.3 a. Calculate the mean salary of the CFOs. b. Based on measures of the center of the data determine if the CFO salary data are skewed. c. Construct a box and whisker plot and summarize the characteristics of the CFO salaries that it reveals. 3-17. The Federal Deposit Insurance Corporation FDIC insures deposits in banks and thrift institutions for up to 250000. Before the banking crisis of late 2008 there were 8885 FDIC–insured institutions with deposits of 6826804000000 there were 7436 in late 2011 with deposits of 7966700000000. a. Calculate the average deposits per bank for FDIC– insured institutions during both time periods. b. Describe the relationship between the two averages calculated in part a. Can you provide a reason for the difference c. Would the two averages be considered to be parameters or statistics Explain. Computer Database Exercises 3-18. Each year Business Week publishes information and rankings of master of business administration MBA programs. The data file MBA Analysis contains data on several variables for eight reputable MBA programs. The variables include pre– and post–MBA salary percentage salary increase undergraduate GPA average Graduate Management Admission Test GMAT score annual tuition and expected annual student cost. Compute the mean and median for each of the variables in the database and write a short report that summarizes the data. Include any appropriate charts or graphs to assist in your report. 3-19. Dynamic random-access memory DRAM memory chips are made from silicon wafers in manufacturing facilities through a very complex process called wafer fabs. The wafers are routed through the fab machines in an order that is referred to as a recipe. The wafers may go through the same machine several times as the chip is created. The data file DRAM Chips contains a sample of processing times measured in fractions of hours at a particular machine center for one chip recipe. a. Compute the mean processing time. b. Compute the median processing time. c. Determine what the mode processing time is. d. Calculate the 80th percentile for processing time. 3-20. Japolli Bakery tracks sales of its different bread products on a daily basis. The data for 22 consecutive days at one of its retail outlets in Nashville are in a file called Japolli Bakery. Calculate the mean mode and median sales for each of the bread categories and write a short report that describes these data. Use any charts or graphs that may be helpful in more fully describing the data. 3-21. Before the sub-prime loan crisis and the end of the “housing bubble” in 2008 the value of houses was escalating rapidly as much as 40 a year in some areas. In an effort to track housing prices the National Association of Realtors developed the Pending Home Sales Index PHSI a new leading indicator for the housing market. An index of 100 is equal to the average level of contract activity during 2001 the first year to be analyzed. The index is based on a large national sample representing about 20 of home sales. The file titled Pending contains the PHSI from December 2010 to December 2011. a. Determine the mean and median for the PHSI from December 2010 through December 2011. Specify the shape of the PHSI’s distribution. b. The PHSI was at 91.5 in December 2010 and it was at 100.1 in November of 2011. Determine the average monthly increase in the PHSI for this period. c. Using your answer to part b suggest a weighting scheme to calculate the weighted mean for the months between December 2010 and November 2011. Use the scheme to produce the weighted average of the PHSI in this time period. d. Does the weighted average seem more appropriate here Explain. 3-22. Homeowners and businesses pay taxes on the assessed value of their property. As a result property taxes can be a problem for elderly homeowners who are on a fixed retirement income. Whereas these retirement incomes remain basically constant because of rising real estate prices the property taxes in many areas of the country have risen dramatically. In some cases homeowners are required to sell their homes because they can’t afford the taxes. In Phoenix Arizona government officials are considering giving certain elderly homeowners a property tax reduction based on income. One proposal calls for all homeowners over the age of 65 with incomes at or below the 20th percentile to get a reduction in property taxes. A random sample of 50 people over the age of 65 was selected and the household income as reported on the most current federal tax return was recorded. These data are also in the file called Property Tax Incomes. www.downloadslide.com

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Describing Data Using Numerical Measures b. Are the data skewed or symmetric c. Approximately what percent of the data values are between 2900 and 3250 3-24. USA Today reported a survey made by Nationwide Mutual Insurance that indicated the average amount of time spent to resolve identity theft cases was slightly more than 88 hours. The file titled Theft contains data that would produce this statistic. a. Construct a stem and leaf display. Indicate the shape of data displayed by the stem and leaf display. b. Use measures that indicate the shape of the distribution. Do these measures give results that agree with the shape shown in part a c. Considering your answers to part a and b indicate which measure you would recommend using to indicate the center of the data. Use these data to establish the income cutoff point to qualify for the property tax cut. 35303 56855 7928 26006 28278 54215 38850 15733 29786 65878 46658 62874 49427 19017 46007 32367 31904 35534 66668 37986 10669 54337 8858 45263 37746 14550 8748 58075 23381 11725 45044 55807 54211 42961 62682 32939 38698 11632 66714 31869 57530 59233 14136 8824 42183 58443 34553 26805 16133 61785 3-23. Suppose a random sample of 137 households in Detroit was taken as part of a study on annual household spending for food at home. The sample data are contained in the file Detroit Eats. a. For the sample data compute the mean and the median and construct a box and whisker plot. 2 Measures of Variation BUSINESS APPLICATION MEASURING VARIATION USING THE RANGE FLEETWOOD MOBILE HOMES Consider the situation involving two manufacturing facilities for Fleetwood Mobile Homes. The division vice president asked the two plant managers to record the number of mobile homes produced weekly over a five-week period. The resulting sample data are shown in Table 3. Instead of reporting these raw data the managers reported only the mean and median for their data. The following are the computed statistics for the two plants: Plant A Plant B x 25 units M d 25 units x 25 units M d 25 units The division vice president looked at these statistics and concluded the following: 1. Average production is 25 units per week at both plants. 2. The median production is 25 units per week at both plants. 3. Because the mean and median are equal the distribution of production output at the two plants is symmetrical. 4. Based on these statistics there is no reason to believe that the two plants are different in terms of their production output. However if he had taken a closer look at the raw data he would have seen there is a very big difference between the two plants. The difference is the production variation from week to week. Plant B is very stable producing almost the same number of units every week. Plant A varies considerably with some high-output weeks and some low-output weeks. Thus looking at only measures of the data’s central location can be misleading. To fully describe a set of data we need a measure of variation or spread. There is variation in everything that is made by humans or that occurs in nature. The vari- ation may be small but it is there. Given a fine enough measuring instrument we can detect the variation. Variation is either a natural part of a process or inherent to a product or can be attributed to a special cause that is not considered random. Several different measures of variation are used in business decision making. In this section we introduce four of these measures: range interquartile range variance and standard deviation. Variation A set of data exhibits variation if all the data are not the same value. TABLE 3 | Manufacturing Output for Fleetwood Mobile Homes Plant A Plant B 15 units 23 units 25 units 26 units 35 units 25 units 20 units 24 units 30 units 27 units END EXERCISES 3-1 www.downloadslide.com

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Describing Data Using Numerical Measures Range The simplest measure of variation is the range. It is both easy to compute and easy to understand. The range is computed using Equation 7. Range The range is a measure of variation that is computed by finding the difference between the maximum and minimum values in a data set. Range R Maximum Value - MinimumValue 7 BUSINESS APPLICATION CALCULATING THE RANGE FLEETWOOD MOBILE HOMES CONTINUED Table 3 showed the production-volume data for the two Fleetwood Mobile Home plants. The range in production for each plant is determined using Equation 7 as follows: Plant A Plant B R Maximum - Minimum R Maximum - Minimum R 35 - 15 R 27 - 23 R 20 R 4 We see Plant A has a range that is fve times as great as Plant B. Although the range is quick and easy to compute it does have some limitations. First because we use only the high and low values to compute the range it is very sensitive to extreme values in the data. Second regardless of how many values are in the sample or popu- lation the range is computed from only two of these values. For these reasons it is considered a weak measure of variation. Interquartile Range A measure of variation that tends to overcome the range’s susceptibility to extreme values is called the interquartile range. Equation 8 is used to compute the interquartile range. Interquartile Range The interquartile range is a measure of variation that is determined by computing the difference between the third and first quartiles. Interquartile Range Interquartile Range Third Quartile - First Quartile 8 EXAMPLE 10 COMPUTING THE INTERQUARTILE RANGE Verizon Wireless A systems capacity manager for Verizon Wireless is interested in better understanding Verizon customer text messaging use. To do this she has collected a random sample of 100 customers under the age of 25 and recorded the number of text messages sent in a one-week period. She wishes to analyze the variation in these data by computing the range and the interquartile range. She could use the following steps to do so: Step 1 Sort the data into a data array from lowest to highest. The 100 sorted values are as follows: 33 164 173 184 190 197 207 216 224 237 53 164 175 186 191 197 207 217 225 240 150 164 175 186 191 198 208 217 225 240 152 166 175 186 192 200 208 217 229 240 157 166 178 187 193 200 208 219 231 250 160 168 178 188 193 201 210 222 231 251 161 169 179 188 194 202 211 223 234 259 162 171 180 188 194 204 212 223 234 270 162 171 182 190 196 205 213 223 235 379 163 172 183 190 196 205 216 224 236 479 Chapter Outcome 3. Chapter Outcome 3. www.downloadslide.com

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Describing Data Using Numerical Measures Step 2 Compute the range using Equation 7. R Maximum value - Minimum value R 479 - 33 446 Note the range is sensitive to extreme values. The small value of 33 and the high value of 479 cause the range value to be very large. Step 3 Compute the first and third quartiles. Equation 6 can be used to find the location of the third quartile 75th percentile and the first quartile 25th percentile. For Q 3 the location i 75 100 100 75. Thus Q 3 is halfway between the 75th and 76th data values which is found as follows: Q 3 1219 + 22222 220.50 For Q 1 the location is i 25 100 100 25. Then Q 1 is halfway between the 25th and 26th data values. Q 1 1178 + 17822 178 Step 4 Compute the interquartile range. The interquartile range overcomes the range’s problem of sensitivity to extreme values. It is computed using Equation 8: Interquartile range Q 3 - Q 1 220.50 - 178 42.50 Note the interquartile range would be unchanged even if the values on the high or low end of the distribution were even more extreme than those shown in these sample data. END EXAMPLE TRY PROBLEM 30 Population Variance and Standard Deviation Although the range is easy to compute and understand and the interquartile range is designed to overcome the range’s sensitivity to extreme values neither measure uses all the available data in its computation. Thus both measures ignore potentially valuable information in data. Two measures of variation that incorporate all the values in a data set are the variance and the standard deviation. These two measures are closely related. The standard deviation is the positive square root of the variance. The standard deviation is in the original units dollars pounds etc. whereas the units of measure in the variance are squared. Because dealing with original units is easier than dealing with the square of the units we usually use the standard deviation to measure variation in a population or sample. BUSINESS APPLICATION CALCULATING THE VARIANCE AND STANDARD DEVIATION FLEETWOOD MOBILE HOMES CONTINUED Recall the Fleetwood Mobile Home application in which we compared the weekly production output for two of the company’s plants. Table 3 showed the data which are considered a population for our purposes here. Previously we examined the variability in the output from these two plants by com- puting the ranges. Although those results gave us some sense of how much more variable Plant A is than Plant B we also pointed out some of the deficiencies of the range. The variance and standard deviation offer alternatives to the range for measuring variation in data. Equation 9 is the formula for the population variance. Like the population mean the population variance and standard deviation are assigned Greek symbols. Variance The population variance is the average of the squared distances of the data values from the mean. Standard Deviation The standard deviation is the positive square root of the variance. Chapter Outcome 3. www.downloadslide.com

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Describing Data Using Numerical Measures We begin by computing the variance for the output data from Plant A. The frst step in manu- ally calculating the variance is to fnd the mean using Equation 1. ∑ x N 15 25 35 20 30 5 125 5 25 Next subtract the mean from each value as shown in Table 4. Notice the sum of the deviations from the mean is 0. Recall from Section 1 that this will be true for any set of data. The positive differences are cancelled out by the negative differences. To overcome this fact when computing the variance we square each of the differences and then sum the squared differences. These calculations are also shown in Table 4. The final step in computing the population variance is to divide the sum of the squared differences by the population size N 5. 2 2 250 5 50 ∑− x N The population variance is 50 mobile homes squared. Manual calculations for the population variance may be easier if you use an alternative formula for s 2 that is the algebraic equivalent. This is shown as Equation 10. Population Variance 2 x N i i N 2 1 ∑ 9 where: m Population mean N Population size s 2 Population variance 1sigma squared2 TABLE 4 | Computing the Population Variance: Squaring the Deviations x i 1x i -m2 1x i -m2 2 15 15 - 25 -10 100 25 25 - 25 0 0 35 35 - 25 10 100 20 20 - 25-5 25 30 30 - 25 5 25 Σ1x i -m2 0 Σ1x i -m2 2 250 Population Variance Shortcut 2 2 2 ∑− ∑ x x N N 5 s 10 Example 11 will illustrate using Equation 10 to fnd a population variance. Because we squared the deviations to keep the positive values and negative values from canceling the units of measure were also squared but the term mobile homes squared doesn’t have a meaning. To get back to the original units of measure take the square root of the www.downloadslide.com

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Describing Data Using Numerical Measures variance. The result is the standard deviation. Equation 11 shows the formula for the popula- tion standard deviation. Population Standard Deviation 2 2 1 x N i i N − ∑ 11 Therefore the population standard deviation of Plant A’s production output is 50 707 . mobile homes The population standard deviation is a parameter and will not change unless the population values change. We could repeat this process using the data for Plant B which also had a mean output of 25 mobile homes. You should verify that the population variance is 2 2 10 5 2 ∑− x N mobile homes squared s m 555 The standard deviation is found by taking the square root of the variance. 2 1 414 . mobile homes 5 s 5 s Thus Plant A has an output standard deviation that is five times larger than Plant B’s. The fact that Plant A’s range was also five times larger than the range for Plant B is merely a coincidence. EXAMPLE 11 COMPUTING A POPULATION VARIANCE AND STANDARD DEVIATION Boydson Shipping Company Boydson Shipping Company owns and operates a fleet of tanker ships that carry commodities between the countries of the world. In the past six months the company has had seven contracts that called for shipments between Vancouver Canada and London England. For many reasons the travel time varies between these two locations. The scheduling manager is interested in knowing the variance and standard deviation in shipping times for these seven shipments. To find these values he can follow these steps: Step 1 Collect the data for the population. The shipping times are shown as follows: x shipping weeks 55 7 5 9 7 4 66 Step 2 Select Equation 10 to find the population variance. 2 2 2 ∑− ∑ x x N N Step 3 Add the x values and square the sum. Σx 5 + 7 + 5 + 9 + 7 + 4 + 6 43 1Σx 2 2 1432 2 1849 How to do it Example 11 Computing the Population Variance and Standard Deviation The population variance and stand- ard deviation are computed using the following steps: 1. Collect quantitative data for the variable of interest for the entire population. 2. Use either Equation 9 or Equa- tion 10 to compute the variance. 3. If Equation 10 is used find the sum of the x-values 1Σx 2 2 and then square this sum 1Σx2 2 . 4. Square each x value and sum these squared values 1Σx 2 2. 5. Compute the variance using 2 2 2 ∑− ∑ x x N N 6. Compute the standard deviation by taking the positive square root of the variance: 2 Svitlana Kataieva/Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures Step 4 Square each of the x values and sum these squares. Σx 2 5 2 + 7 2 + 5 2 + 9 2 + 7 2 + 4 2 + 6 2 281 Step 5 Compute the population variance. 2 2 2 281 1 849 7 7 2 4082 ∑− ∑ − x x N N . The variance is in units squared so in this example the population variance is 2.4082 weeks squared. Step 6 Calculate the standard deviation as the square root of the variance. 2 2 4082 1 5518 . . weeks Thus the standard deviation for the number of shipping weeks between Vancouver and London for the seven shipments is 1.5518 weeks. END EXAMPLE TRY PROBLEM 27 Sample Variance and Standard Deviation Equations 9 10 and 11 are the equations for the population variance and standard deviation. Any time you are working with a population these are the equations that are used. However in most instances you will be describing sample data that have been selected from the population. In addition to using different notations for the sample variance and sample standard deviation the equations are also slightly different. Equations 12 and 13 can be used to find the sample variance. Note that Equation 13 is considered the shortcut formula for manual computations. Sample Variance s xx n i i n 2 2 1 1 5 5 – – ∑ 12 The sample standard deviation is found by taking the square root of the sample variance as shown in Equation 14. Sample Variance Shortcut s x x n n 2 2 2 1 5 ∑ ∑ – – 13 where: n Sample size x Sample mean s 2 Sample variance Sample Standard Deviation ss xx n i i n 55 5 2 2 1 1 – – ∑ 14 www.downloadslide.com

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Describing Data Using Numerical Measures Take note in Equations 12 13 and 14 that the denominator is n - 1 sample size minus 1. This may seem strange given that the denominator for the population vari- ance and the standard deviation is simply N the population size. The mathematical justification for the n - 1 divisor is outside the scope of this text. However the gen- eral reason for this is that we want the average sample variance to equal the population variance. If we were to select all possible samples of size n from a given population and for each sample we computed the sample variance using Equation 12 or Equation 13 the average of all the sample variances would equal s 2 the population variance provided we used n - 1 as the divisor. Using n instead of n - 1 in the denominator would produce an average sample variance that would be smaller than s 2 the popula- tion variance. Because we want an estimator on average to equal the population vari- ance we use n - 1 in the denominator of s 2 . EXAMPLE 12 COMPUTING A SAMPLE VARIANCE AND STANDARD DEVIATION Balco Industries The internal auditor for Balco Industries a manufacturer of equip- ment used in the food processing business recently examined the transaction records related to 10 of the company’s customers. For each client the auditor counted the number of incorrectly recorded entries i.e “defects”. The ten customers’ accounts can be consid- ered to be samples of all possible Balco customers that could be analyzed. To fully ana- lyze the data the auditor can calculate the sample variance and sample standard deviation using the following steps: Step 1 Select the sample and record the data for the variable of interest. Client Defects x Client Defects x 14 6 0 27 7 3 31 8 2 40 9 6 5 5 10 2 Step 2 Select either Equation 12 or Equation 13 to compute the sample variance. If we use Equation 12 s xx n 2 2 1 5 ∑ – – Step 3 Compute x. The sample mean number of defectives is x x n 55 5 ∑ 30 10 30 . Step 4 Determine the sum of the squared deviations of each x value from x. Client Defectives x 1x - x2 1x - x2 2 14 1 1 27 4 16 31 -2 4 40 -3 9 55 2 4 60 -3 9 73 0 0 82 -1 1 www.downloadslide.com

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Describing Data Using Numerical Measures Client Defectives x 1x - x2 1x - x2 2 96 3 9 10 2 -1 1 Σ 30 Σ 0 Σ 54 Step 5 Compute the sample variance using Equation 12. s xx n 2 2 1 54 9 6 555 ∑− − The sample variance is measured in squared units. Thus the variance in this example is 6 defectives squared. Step 6 Compute the sample standard deviation by taking the square root of the variance see Equation 14. s xx n s 555 5 ∑− − . 2 1 54 9 6 2 4495 defects This sample standard deviation measures the variation in the sample data for the number of incorrectly recorded entries. END EXAMPLE TRY PROBLEM 25 BUSINESS APPLICATION CALCULATING MEASURES OF VARIATION USING EXCEL COLLEGES AND UNIVERSITIES CONTINUED In Section 1 the guidance counselor was interested in describing the data representing the cost of out-of-state tuition for a large number of colleges and universities in the United States. The data for 718 schools are in the file called Colleges and Universities. Previously we determined the following descriptive measures of the center for the variable out-of-state tuition: Mean +9933.38 Median +9433.00 Mode +6550 Next the analyst will turn her attention to measures of variability. The range maximum – minimum is one measure of variability. Excel can be used to compute the range and the standard deviation of tuition which is a more powerful measure of variation than the range. Figure 7 shows the Excel descriptive statistics results. We find the following measures of vari- ation: Range +22340.00 Standard Deviation +3920.07 These values are measures of the spread in the data. You should know that outlier values in a data set will increase both the range and standard deviation. One guideline for identifying outliers is the 3 standard deviation rule. That is if a value falls outside 3 standard devia- tions from the mean it is considered an outlier. Also as shown in Section 1 outliers can be identified using box and whisker plots. Excel Tutorial Excel tutorials www.downloadslide.com

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Describing Data Using Numerical Measures 3-2: Exercises FIGURE 7 |  Excel 2010 Descriptive Statistics for Colleges and Universities Data Excel 2010 Instructions: 1. Open fle: Colleges and Universities.xlsx. 2. Select the Data tab. 3. Click on Data Analysis Descriptive Statistics. 4. Defne data range for the desired variables. 5. Check Summary Statistics. 6. Name new Output Sheet. 7. On Home tab adjust decimal places as desired. Standard Deviation Variance Range Minitab Instructions for similar results: 1. Open fle: Colleges and Universities.MTW. 2. Choose Stat Basic Statistics Display Descriptive Statistics. 3. In Variables enter column out-of-state tuition. 4. Click Statistics. 5. Check required statistics. 6. Click OK. OK. MyStatLab a. Compute the range for these data. b. Compute the variance and standard deviation. c. Assuming that these data represent a sample rather than a population compute the variance and standard deviation. Discuss the difference between the values computed here and in part b. 3-27. The following data are the population of ages of students who have recently purchased a sports video game: 16 15 17 15 15 15 14 916151310 81820171717 18 23 7152010 14 14 12 12 24 21 a. Compute the population variance. b. Compute the population standard deviation. Skill Development 3-25. Google is noted for its generous employee benefits. The following data reflect the number of vacation days that a sample of employees at Google have left to take before the end of the year: 30201352 51300133 43184240 a. Compute the range for these sample data. b. Compute the variance for these sample data. c. Compute the standard deviation for these sample data. 3-26. The following data reflect the number of times a population of business executives flew on business during the previous month: 469457 www.downloadslide.com

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Describing Data Using Numerical Measures 3-28. A county library in Minnesota reported the following number of books checked out in 15 randomly selected months: 5176 6005 5052 5310 4188 4132 5736 5381 4983 4423 5002 4573 4209 5611 4568 Determine the range variance and standard deviation for the sample data. 3-29. The following data show the number of hours spent watching television for 12 randomly selected freshmen attending a liberal arts college in the Midwest: Hours of Television Viewed Weekly 7.5 11.5 14.4 7.8 13.0 10.3 5.4 12.0 12.2 8.9 8.5 6.6 Calculate the range variance standard deviation and interquartile range for the sample data. 3-30. Consider the following two separate samples: 27 27 25 12 15 10 20 37 31 35 and 1 3 216181616 4 16 118 a. Calculate the range variance standard deviation and interquartile range for each data set. b. Which data set is most spread out based on these statistics c. Now remove the largest number from each data set and repeat the calculations called for in part a. d. Compare the results of parts a and c. Which statistic seems to be most affected by outliers 3-31. The following set of data shows the number of alcoholic drinks that students at a Kansas university reported they had consumed in the past month: 24 16 23 26 30 21 15 9 18 27 14 6 14 10 12 a. Assume the data set is a sample. Calculate the range variance standard deviation and interquartile range for the data set. b. Assume the data set is a population. Calculate the range variance standard deviation and interquartile range for the data set. c. Indicate the relationship between the statistics and the respective parameters calculated in parts a and b. Business Applications 3-32. Easy Connect Inc. provides access to computers for business uses. The manager monitors computer use to make sure that the number of computers is sufficient to meet the needs of the customers. Recently the manager collected data on a sample of customers and tracked the time the customers started working at a computer until they were finished. The elapsed times in minutes are shown as follows: 40 42 18 32 43 35 11 39 36 37 8343450203931753317 Compute appropriate measures of the center and variation to describe the time customers spend on the computer. 3-33. A random sample of 20 pledges to a public radio fund- raiser revealed the following dollar pledges: 90 85 100 150 125 75 50 100 75 60 35 90 100 125 75 85 50 100 50 80 a. Compute the range variance standard deviation and interquartile range for these sample data. b. Briefly explain the difference between the range and the interquartile range as a measure of dispersion. 3-34. Gold’s Gym selected a random sample of 10 customers and monitored the number of times each customer used the workout facility in a one-month period. The following data were collected: 10 19 17 19 12 20 20 15 16 13 Gold’s managers are considering a promotion in which they reward frequent users with a small gift. They have decided that they will only give gifts to those customers whose number of visits in a one-month period is 1 standard deviation above the mean. Find the minimum number of visits required to receive a gift. 3-35. The registrar at Whitworth College has been asked to prepare a report about the graduate students. Among other things she wants to analyze the ages of the students. She has taken a sample of 10 graduate students and has found the following ages: 32 22 24 27 27 33 28 23 24 21 a. Compute the range interquartile range and standard deviation for these data. b. An earlier study showed that the mean age of graduate students in U.S. colleges and universities is 37.8 years. Based on your calculations in part a what might you conclude about the age of students in Whitworth’s programs 3-36. The branch manager for the D. L. Evens Bank has been asked to prepare a presentation for next week’s board meeting. At the presentation she will discuss the status of her branch’s loans issued for recreation vehicles RVs. In particular she will analyze the loan balances for a sample of 10 RV loans. The following data were collected: 11509 8088 13415 17028 16754 18626 4917 11740 16393 8757 www.downloadslide.com

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Describing Data Using Numerical Measures a. Compute the mean loan balance. b. Compute the loan balance standard deviation. c. Write a one-paragraph statement that uses the statistics computed in parts a and b to describe the RV loan data at the branch. 3-37. A parking garage in Memphis monitors the time it takes customers to exit the parking structure from the time they get in their car until they are on the streets. A sample of 28 exits was recently taken and is shown here. Garage Exit time in seconds 83 138 145 147 130 79 156 156 90 85 68 93 178 76 73 119 92 146 88 103 116 134 162 71 181 110 105 74 a. Calculate the range interquartile range variance and standard deviation for these sample data. b. If the minimum time and the maximum time in the sample data are both increased by 10 seconds would this affect the value for the interquartile range that you calculated in part a Why or why not c. Suppose the clock that electronically recorded the times was not working properly when the sample was taken and each of the sampled times needs to be increased by 10 seconds. How would adding 10 seconds to each of the sampled speed of service times change the sample variance of the data 3-38. Nielsen Monitor-Plus a service of Nielsen Media Research is one of the leaders in advertising information services in the United States providing advertising activity for 16 media including television tracking in all 210 Designated Market Areas DMAs. One of the issues it has researched is the increasing amount of “clutter”—nonprogramming minutes in an hour of prime time—including network and local commercials and advertisements for other shows. Recently it found the average nonprogramming minutes in an hour of prime- time broadcasting for network television was 15:48 minutes. For cable television the average was 14:55 minutes. a. Calculate the difference in the average clutter between network and cable television. b. Suppose the standard deviation in the amount of clutter for both the network and cable television was either 5 minutes or 15 seconds. Which standard deviation would lead you to conclude that there was a major difference in the two clutter averages Comment. 3-39. The Bureau of Labor Statistics in its Monthly Labor Review published the “over-the-month” percent change in the price index for imports from December 2010 to December 2011. These data are reproduced next. Month Change in Index Jan 0.1 Feb 0.2 Mar 1.3 Apr -0.4 May -2.5 Jun -0.7 Jul 0.7 Aug -0.5 Sep 0.3 Oct -0.3 Nov 1.2 Dec -0.9 a. Calculate the mean standard deviation and interquartile range for these data. b. Consider the mean calculated in part a. What does this value indicate about the price index c. What does the standard deviation indicate about the price index 3-40. The U.S. Government Accountability Office recently indicated the price of college textbooks has been rising an average of 6 annually since the late 1980s. The report estimated that the average cost of books and supplies for first-time full-time students at four-year public universities for the academic year had reached 898. A data set that would produce this average follows: 537.51 1032.52 1119.17 877.27 856.87 739.91 963.79 847.92 1393.81 524.68 1012.91 1176.46 944.60 708.26 1074.35 778.87 967.91 562.55 789.50 1051.65 a. Calculate the mean and standard deviation. b. Determine the number of standard deviations the most extreme cost is away from the mean. If you were to advise a prospective student concerning the money the student should save to afford the cost of books and supplies for at least 90 of the colleges determine the amount you would suggest. Hint: Don’t forget the yearly inflation of the cost of books and supplies. Computer Database Exercises 3-41. The manager of a phone kiosk in the Valley Mall recently collected data on a sample of 50 customers who purchased a cell phone and a monthly call plan. The data she recorded are in the data file called Phone Survey. a. The manager is interested in describing the difference between male and female customers with respect to the price of the phone purchased. She wants to compute mean and standard deviation of phone purchase price for each group of customers. www.downloadslide.com

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Describing Data Using Numerical Measures and their 2005 out-of-pocket medical expenses for prescription drugs were collected. The data are in the file Drug Expenses. a. Calculate the mean and median for the sample data. b. Calculate the range variance standard deviation and interquartile range for the sample data. c. Construct a box and whisker plot for the sample data. d. Write a short report that describes out-of-pocket drug expenses for privately insured adults whose incomes are greater than 200 of the poverty level. 3-45. Executive MBA programs have become increasingly popular. In an article titled “The Best Executive MBAs” Business Week provided data concerning the top 25 executive MBA programs for one specific year. The tuition for each of the schools selected was given. A file titled EMBA contains this data. a. Calculate the 20th 40th 60th and 80th percentile among the ranks. b. Calculate the mean and standard deviation of the tuition for the five subgroups defined by the rank percentiles in part a. Hint: For this purpose are the data subgroups samples or populations c. Do the various subgroups’ descriptive statistics echo their standing among the listed programs Comment. 3-46. When PricewaterhouseCoopers Saratoga released its Human Capital Index Report it indicated that the average hiring cost for an American company to fill a job vacancy was 3270. Sample data for recent job hires is in a file titled Hired. a. Calculate the variance and standard deviation for the sample data. b. Construct a box and whisker plot. Does this plot indicate that extreme values outliers may be inflating the measures of spread calculated in part a c. Suggest and calculate a measure of spread that is not affected by outliers. b. The manager is also interested in an analysis of the phone purchase price based on whether the use will be for home or business. Again she wants to compute mean and standard deviation of phone purchase price for each group of customers. 3-42. Each year Business Week publishes information and rankings of MBA programs. The data file MBA Analysis contains data on several variables for eight reputable MBA programs. The variables include pre– and post–MBA salary percentage salary increase undergraduate GPA average GMAT score annual tuition and expected annual student cost. Compute the mean median range variance and standard deviation for each of the variables in the database and write a short report that summarizes the data using these measures. Include any appropriate charts or graphs to assist in your report. 3-43. The First City Real Estate Company lists and sells residential real estate property in and around Yuma Arizona. At a recent company meeting the managing partner asked the office administrator to provide a descriptive analysis of the asking prices of the homes the company currently has listed. This list includes 319 homes the price data along with other home characteristics are included in the data file called First City Real Estate. These data constitute a population. a. Compute the mean listing price. b. Compute the median listing price. c. Compute the range in listing prices. d. Compute the standard deviation in listing prices. e. Write a short report using the statistics computed in parts a–d to describe the prices of the homes currently listed by First City Real Estate. 3-44. Suppose there is an investigation to determine whether the increased availability of generic drugs Internet drug purchases and cost controls have reduced out-of- pocket drug expenses. As a part of the investigation a random sample of 196 privately insured adults with incomes above 200 of the poverty level was taken Chapter Outcome 4. END EXERCISES 3-2 3 Using the Mean and Standard Deviation Together In the previous sections we introduced several important descriptive measures that are useful for transforming data into meaningful information. Two of the most important of these meas- ures are the mean and the standard deviation. In this section we discuss several statistical tools that combine these two. Coefficient of Variation The standard deviation measures the variation in a set of data. For decision makers the stan- dard deviation indicates how spread out a distribution is. For distributions having the same mean the distribution with the largest standard deviation has the greatest relative spread. When two or more distributions have different means the relative spread cannot be deter- mined by merely comparing standard deviations. The coefficient of variation CV is used to measure the relative variation for distribu- tions with different means. Coefficient of Variation The ratio of the standard deviation to the mean expressed as a percentage. The coefficient of variation is used to measure variation relative to the mean. www.downloadslide.com

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Describing Data Using Numerical Measures The coefficient of variation for a population is computed using Equation 15 whereas Equation 16 is used for sample data. Population Coefficient of Variation CV 100 15 Sample Coefficient of Variation CV s x 5 100 16 When the coefficients of variation for two or more distributions are compared the distribution with the largest CV is said to have the greatest relative spread. In finance the CV measures the relative risk of a stock portfolio. Assume portfolio A has a collection of stocks that average a 12 return with a standard deviation of 3 and portfolio B has an average return of 6 with a standard deviation of 2. We can compute the CV values for each as follows: CV A55 3 12 100 25 and CV B55 2 6 100 33 Even though portfolio B has a lower standard deviation it would be considered more risky than portfolio A because B’s CV is 33 and A’s CV is 25. EXAMPLE 13 COMPUTING THE COEFFICIENT OF VARIATION Agra-Tech Industries Agra-Tech Industries has recently introduced feed supplements for both cattle and hogs that will increase the rate at which the animals gain weight. Three years of feedlot tests indicate that steers fed the supplement will weigh an average of 125 pounds more than those not fed the supplement. However not every steer on the supplement has the same weight gain results vary. The standard deviation in weight-gain advantage for the steers in the three-year study has been 10 pounds. Similar tests with hogs indicate those fed the supplement average 40 additional pounds compared with hogs not given the supplement. The standard deviation for the hogs was also 10 pounds. Even though the standard deviation is the same for both cattle and hogs the mean weight gains differ. Therefore the coefficient of variation is needed to compare rela- tive variability. The coefficient of variation for each is computed using the following steps: Step 1 Collect the sample or population data for the variable of interest. In this case we have two samples: weight gain for cattle and weight gain for hogs. Step 2 Compute the mean and the standard deviation. For the two samples in this example we get Cattle: lb and lb Hogs: lb and xs x 55 5 125 10 40 lb s 510 Step 3 Compute the coefficient of variation using Equation 15 for populations or Equation 16 for samples. Because the data in this example are from samples the CV is computed using CV s x 5 100 www.downloadslide.com

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Describing Data Using Numerical Measures For each data set we get CV CV cattle hogs 55 5 10 125 100 8 10 40 100 5 25 These results indicate that hogs exhibit much greater relative variability in weight gain com- pared with cattle. END EXAMPLE TRY PROBLEM 50 The Empirical Rule A tool that is helpful in describing data in certain circumstances is called the Empirical Rule. For the Empirical Rule to be used the frequency distribution must be bell-shaped such as the one shown in Figure 8. BUSINESS APPLICATION EMPIRICAL RULE BURGER N’ BREW The standard deviation can be thought of as a measure of distance from the mean. Consider the Phoenix Burger n’ Brew restaurant chain which records the number of each hamburger option it sells each day at each location. The numbers of chili burgers sold each day for the past 365 days are in the file called BurgerNBrew. Figure 9 shows the frequency histogram for those data. The distribution is nearly symmetrical and is approximately bell-shaped. The mean number of chili burgers sold was 15.1 with a standard deviation of 3.1. The Empirical Rule is a very useful statistical concept for helping us understand the data in a bell-shaped distribution. In the Burger N’ Brew example with x 15.1 and s 3.1 if we move 1 standard deviation in each direction from the mean approximately 68 of the data should lie within the following range: 15.1 113.12 12.0 18.2 Empirical Rule If the data distribution is bell shaped then the interval m 1s contains approximately 68 of the values m 2s contains approximately 95 of the values m 3s contains virtually all of the data values FIGURE 8 |  Illustrating the Empirical Rule for the Bell-Shaped Distribution x ± 1 ± 2 68 95 Excel tutorials Excel Tutorial Chapter Outcome 5. Dan Peretz/Shutterstock www.downloadslide.com

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Describing Data Using Numerical Measures FIGURE 9 |  Excel 2010 Histogram for Burger n’ Brew Data Minitab Instructions for similar results: 1. Open fle: BurgerNBrew.MTW. 2. Choose Graph Histogram. 3. Click Simple. 4. Click OK. 5. In Graph variables enter data column Chili-Burgers Sold. 6. Click OK. Excel 2010 Instructions: 1. Open fle BurgerNBrew.xlsx. 2. Set up an area in the worksheet for the bins upper limit of each class as 7 9 etc. Be sure to label the column of upper limits as “Bins.” 3. On the Data tab click Data Analysis Histogram. 4. Input Range specifes the actual data values. 5. Put on a new worksheet and include Chart Output. 6. Right click on the bars and use the Format Data Series Options to set gap width to zero and add lines to bars. 7. Convert the bins in column A of the histogram output sheet to actual class labels. Note the bin labeled 7 is changed to “6 - 7”. 8. Click on Layout and set titles as desired. Mean 15.1 Standard Deviation 3.1 The actual number of days Burger n’ Brew sold between 12 and 18 chili burgers is 263. Thus out of 365 days on 72 of the days Burger n’ Brew sold between 12 and 18 chili burgers. The rea- son that we didn’t get exactly 68 is that the distribution in Figure 9 is not perfectly bell-shaped. If we look at the interval 2 standard deviations from either side of the mean we would expect approximately 95 of the data. The interval is 15.1 213.12 15.1 6.2 8.9 21.30 Counting the values between these limits we find 353 of the 365 values or 97. Again this is close to what the Empirical Rule predicted. Finally according to the Empirical Rule we would expect almost all of the data to fall within 3 standard deviations. The interval is 15.1 313.12 15.1 9.3 5.80 24.40 Looking at the data in Figure 9 we fnd that in fact all the data do fall within this interval. Therefore if we know only the mean and the standard deviation for a set of data the Empiri- cal Rule gives us a tool for describing how the data are distributed if the distribution is bell-shaped. Tchebysheff’s Theorem The Empirical Rule applies when a distribution is bell-shaped. But what about the many sit- uations in which a distribution is skewed and not bell-shaped In these cases we can use Tchebysheff’s theorem. Tchebysheff’s Theorem Regardless of how data are distributed at least 11 - 1k 2 2 of the values will fall within k standard deviations of the mean. For example: At least a1 - 1 1 2 b 0 0 of the values will fall within k 1 standard deviation of the mean. At least a1 - 1 2 2 b 3 4 75 of the values will lie within k 2 standard deviations of the mean. At least a1 - 1 3 2 b 8 9 89 of the values will lie within k 3 standard deviations of the mean. Chapter Outcome 5. www.downloadslide.com

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Describing Data Using Numerical Measures Tchebysheff’s theorem is conservative. It tells us nothing about the data within 1 stan- dard deviation of the mean. Tchebysheff indicates that at least 75 of the data will fall within 2 standard deviations—it could be more. If we applied Tchebysheff’s theorem to bell-shaped distributions the percentage estimates are very low. The thing to remember is that Tcheby- sheff’s theorem applies to any distribution. This gives it great flexibility. Standardized Data Values When you are dealing with quantitative data you will sometimes want to convert the meas- ures to a form called standardized data values. This is especially useful when we wish to compare data from two or more distributions when the data scales for the two distributions are substantially different. BUSINESS APPLICATION STANDARDIZING DATA HUMAN RESOURCES Consider a company that uses placement exams as part of its hiring process. The company currently will accept scores from either of two tests: AIMS Hiring and BHS-Screen. The problem is that the AIMS Hiring test has an average score of 2000 and a standard deviation of 200 whereas the BHS-Screen test has an average score of 80 with a standard deviation of 12. These means and standard deviations were developed from a large number of people who have taken the two tests. How can the company compare applicants when the average scores and measures of spread are so different for the two tests One approach is to standardize the test scores. Suppose the company is considering two applicants John and Mary. John took the AIMS Hiring test and scored 2344 whereas Mary took the BHS-Screen and scored 95. Their scores can be standardized using Equation 17. Standardized Data Values The number of standard deviations a value is from the mean. Standardized data values are sometimes referred to as z scores. Standardized Population Data z x − 17 where: x Original data value m Population mean s Population standard deviation z Standard score 1number of standard deviation x is from m2 Standardized Sample Data z xx s 5 − 18 where: x Original data value x Sample mean s Sample standard deviation z The standard score If you are working with sample data rather than a population Equation 18 can be used to standardize the values. We can standardize the test scores for John and Mary using z x − For the AIMS Hiring test the mean m is 2000 and the standard deviation s equals 200. John’s score of 2344 converts to Chapter Outcome 4. www.downloadslide.com

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Describing Data Using Numerical Measures z z 5 5 2 344 2 000 200 172 . − The BHS-Screen’s m 80 and s 12. Mary’s score of 95 converts to z z 5 5 95 80 12 125 − . Compared to the average score on the AIMS Hiring test John’s score is 1.72 standard devia- tions higher. Mary’s score is only 1.25 standard deviations higher than the average score on the BHS-Screen test. Therefore even though the two tests used different scales standardizing the data allows us to conclude John scored relatively better on his test than Mary did on her test. EXAMPLE 14 CONVERTING DATA TO STANDARDIZED VALUES SAT and ACT Exams Many colleges and universities require students to submit either SAT or ACT scores or both. One eastern university requires both exam scores. However in assess- ing whether to admit a student the university uses whichever exam score favors the student among all the applicants. Suppose the school receives 4000 applications for admission. To determine which exam will be used for each student the school will standardize the exam scores from both tests. To do this it can use the following steps: Step 1 Collect data. The university will collect the data for the 4000 SAT scores and the 4000 ACT scores for those students who applied for admission. Step 2 Compute the mean and standard deviation. Assuming that these data reflect the population of interest for the university the population mean is computed using SAT: ACT: 5 ∑ ∑ x N x N 1 255 28 3 . The standard deviation is computed using SAT: ACT: ∑− ∑− . x N x N 22 72 2 4 Step 3 Standardize the data. Convert the x values to z values using z x − Suppose a particular applicant has an SAT score of 1228 and an ACT score of 27. These test scores can be converted to standardized scores. SAT: ACT: z x z x − − − − 1 228 1 255 72 0 375 2 . 77283 24 0 542 − − . . . The negative z values indicate that this student is below the mean on both the SAT and ACT exams. Because the university wishes to use the score that most favors the student it will use the SAT score. The student is only 0.375 standard deviations below the SAT mean compared with 0.542 standard deviations below the ACT mean. END EXAMPLE TRY PROBLEM 52 How to do it Example 14 Converting Data to Standardized Values For a set of quantitative data each data value can be converted to a corresponding standardized value by determining how many standard deviations the value is from the mean. Here are the steps to do this. 1. Collect the population or sam- ple values for the quantitative variable of interest. 2. Compute the population mean and standard deviation or the sample mean and standard deviation. 3. Convert the values to standard- ized z-values using Equation 17 or Equation 18. For populations z x − For samples z xx s − www.downloadslide.com

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Describing Data Using Numerical Measures MyStatLab 3-52. Given two distributions with the following characteristics: Distribution A Distribution B m 45600 m 33.40 s 6333 s 4.05 If a value from distribution A is 50000 and a value from distribution B is 40.0 convert each value to a standardized z value and indicate which one is relatively closer to its respective mean. 3-53. If a sample mean is 1000 and the sample standard deviation is 250 determine the standardized value for a. x 800 b. x 1200 c. x 1000 3-54. The following data represent random samples taken from two different populations A and B: A 31 10 69 25 62 61 46 74 57 B 1030 1111 1155 978 943 983 932 1067 1013 a. Compute the mean and standard deviation for the sample data randomly selected from population A. b. Compute the mean and standard deviation for the sample data randomly selected from population B. c. Which sample has the greater spread when measured by the standard deviation d. Compute the coefficient of variation for the sample data selected from population A and from population B. Which sample exhibits the greater relative variation 3-55. Consider the following sample: 22 46 25 37 35 84 33 54 80 37 76 34 48 86 41 13 49 45 62 47 72 70 91 51 91 43 56 25 12 65 a. Calculate the mean and standard deviation for this data. b. Determine the percentage of data values that fall in each of the following intervals: x s x 2s x 3s. c. Compare these with the percentages that should be expected from a bell-shaped distribution. Does it seem plausible that these data came from a bell- shaped population Explain. 3-56. Consider the following population: 71 89 65 97 46 52 99 41 62 88 73 50 91 71 52 86 92 60 70 91 73 98 56 80 70 63 55 61 40 95 Skill Development 3-47. A population of unknown shape has a mean of 3000 and a standard deviation of 200. a. Find the minimum proportion of observations in the population that are in the range 2600 to 3400. b. Determine the maximum proportion of the observations that are above 3600. c. What statement could you make concerning the proportion of observations that are smaller than 2400 3-48. The mean time that a certain model of light bulb will last is 400 hours with a standard deviation equal to 50 hours. a. Calculate the standardized value for a light bulb that lasts 500 hours. b. Assuming that the distribution of hours that light bulbs last is bell-shaped what percentage of bulbs could be expected to last longer than 500 hours 3-49. Consider the following set of sample data: 78 121 143 88 110 107 62 122 130 95 78 139 89 125 a. Compute the mean and standard deviation for these sample data. b. Calculate the coefficient of variation for these sample data and interpret its meaning. c. Using Tchebysheff’s theorem determine the range of values that should include at least 89 of the data. Count the number of data values that fall into this range and comment on whether your interval range was conservative. 3-50. You are given the following parameters for two populations: Population 1 Population 2 m 700 m 29000 s 50 s 5000 a. Compute the coefficient of variation for each population. b. Based on the answers to part a which population has data values that are more variable relative to the size of the population mean 3-51. Two distributions of data are being analyzed. Distribution A has a mean of 500 and a standard deviation equal to 100. Distribution B has a mean of 10 and a standard deviation equal to 4.0. Based on this information use the coefficient of variation to determine which distribution has greater relative variation. 3-3: Exercises www.downloadslide.com

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Describing Data Using Numerical Measures 3-59. Lockheed Martin is a supplier for the aerospace industry. Recently the company was considering switching to Cirus Systems Inc. a new supplier for one of the component parts it needs for an assembly. At issue is the variability of the components supplied by Cirus Systems Inc. compared to that of the existing supplier. The existing supplier makes the desired part with a mean diameter of 3.75 inches and a standard deviation of 0.078 inches. Unfortunately Lockheed Martin does not have any of the exact same parts from the new supplier. Instead the new supplier has sent a sample of 20 parts of a different size that it claims are representative of the type of work it can do. These sample data are shown here and in the data file called Cirus. Diameters in inches 18.018 17.856 18.095 17.992 18.086 17.812 17.988 17.996 18.129 18.003 18.214 18.313 17.983 18.153 17.996 17.908 17.948 18.219 18.079 17.799 Prepare a short letter to Lockheed-Martin indicating which supplier you would recommend based on relative variability. 3-60. A recent article in The Washington Post Weekly Edition indicated that about 80 of the estimated 200 billion of federal housing subsidies consists of tax breaks mainly deductions for mortgage interest payments and preferential treatment for profits on home sales. Federal housing benefits average 8268 for those with incomes between 50000 and 200000 and 365 for those with income of 40000 to 50000. Suppose the standard deviations of the housing benefits in these two categories were equal to 2750 and 120 respectively. a. Examine the two standard deviations. What do these indicate about the range of benefits enjoyed by the two groups b. Repeat part a using the coefficient of variation as the measure of relative variation. 3-61. Anaheim Human Resources Inc. performs employment screening for large companies in southern California. It usually follows a two-step process. First potential applicants are given a test that covers basic knowledge and intelligence. If applicants score within a certain range they are called in for an interview. If they score below a certain point they are sent a rejection letter. If applicants score above a certain point they are sent directly to the client’s human resources office without the interview. Recently Anaheim Human Resources began working with a new client and formulated a new test just for this company. Thirty people were given the test which is supposed to produce scores that are distributed according to a bell-shaped distribution. The following data reflect the scores of those 30 people: 76 75 74 56 61 76 62 96 68 62 78 76 a. Determine the mean and variance. b. Determine the percentage of data values that fall in each of the following intervals: x 2s x 3s x 4s. c. Compare these with the percentages specified by Tchebysheff’s theorem. Business Applications 3-57. Pfizer Inc. a major U.S. pharmaceutical company is developing a new drug aimed at reducing the pain associated with migraine headaches. Two drugs are currently under development. One consideration in the evaluation of the medication is how long the pain- killing effects of the drugs last. A random sample of 12 tests for each drug revealed the following times in minutes until the effects of the drug were neutralized. The random samples are as follows: Drug A 258 214 243 227 235 222 240 245 245 234 243 211 Drug B 219 283 291 277 258 273 289 260 286 265 284 266 a. Calculate the mean and standard deviation for each of the two drugs. b. Based on the sample means calculated in part a which drug appears to be effective longer c. Based on the sample standard deviations calculated in part a which drug appears to have the greater variability in effect time d. Calculate the sample coefficient of variation for the two drugs. Based on the coefficient of variation which drug has the greater variability in its time until the effect is neutralized 3-58. Wells Fargo Bank’s call center has representatives that speak both English and Spanish. A random sample of 11 calls to English-speaking service representatives and a random sample of 14 calls to Spanish-speaking service representatives was taken and the time to complete the calls was measured. The results in seconds are as follows: Time to Complete the Call in seconds English-Speaking 131 80 140 118 79 94 103 145 113 100 122 Spanish-Speaking 170 177 150 208 151 127 147 140 109 184 119 149 129 152 a. Compute the mean and standard deviation for the time to complete calls to English-speaking service representatives. b. Compute the mean and standard deviation for the time to complete calls to Spanish-speaking service representatives. c. Compute the coefficient of variation for the time to complete calls to English-speaking and Spanish- speaking service representatives. Which group has the greater relative variability in the time to complete calls d. Construct box and whisker plots for the time required to complete the two types of calls and briefly discuss. www.downloadslide.com

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Describing Data Using Numerical Measures Include in your article such descriptive statistics as the mean median and standard deviation. You might consider using percentiles the coefficient of variation and Tchebysheff’s theorem to help describe the data. 3-64. Nike ONE Black is one of the golf balls Nike Inc. produces. It must meet the specifications of the United States Golf Association USGA. The USGA mandates that the diameter of the ball shall not be less than 1.682 inches 42.67 mm. To verify that this specification is met sample golf balls are taken from the production line and measured. These data are found in the file titled Diameter. a. Calculate the mean and standard deviation of this sample. b. Examine the specification for the diameter of the golf ball again. Does it seem that the data could possibly be bell shaped Explain. c. Determine the proportion of diameters in the following intervals: x 2s x 3s x 4s. Compare these with the percentages specified by Tchebysheff’s theorem. 3-65. The Centers for Disease Control and Prevention CDC started the Vessel Sanitation Program VSP in the early 1970s because of several disease outbreaks on cruise ships. The VSP was established to protect the health of passengers and crew by minimizing the risk of gastrointestinal illness on cruise ships. Inspections are scored on a point system of maximum 100 and cruise ships earn a score based on the criteria. Ships that score an 86 or higher have a satisfactory sanitation level. Data from a recent inspection are contained in a file titled Cruiscore. a. Calculate the mean standard deviation median and interquartile range. Which of these measures would seem most appropriate to characterize this data set b. Produce a box and whisker plot of the data. Would the Empirical Rule or Tchebysheff’s theorem be appropriate for describing this data set Explain. c. If you wished to travel only on those ships that are at the 90th percentile or above in terms of sanitation what would be the lowest sanitation score you would find acceptable 3-66. Airfare prices were collected for a round trip from Los Angeles LAX to Salt Lake City SLC. Airfare prices were also collected for a round trip from Los Angeles LAX to Barcelona Spain BCN. Airfares were obtained for the designated and nearby airports during high travel months. The passenger was to fly coach class round-trip staying seven days. The data are contained in a file titled Airfare. a. Calculate the mean and standard deviation for each of the flights. b. Calculate an appropriate measure of the relative variability of these two flights. c. A British friend of yours is currently in Barcelona and wishes to fly to Los Angeles. If the flight 84 67 60 96 77 59 67 81 66 71 69 65 58 77 82 75 76 67 Anaheim Human Resources has in the past issued a rejection letter with no interview to the lower 16 taking the test. They also send the upper 2.5 directly to the company without an interview. Everyone else is interviewed. Based on the data and the assumption of a bell-shaped distribution what score should be used for the two cutoffs 3-62. The College Board’s Annual Survey of Colleges provides up-to-date information on tuition and other expenses associated with attending public and private nonprofit institutions of postsecondary education in the United States. Each fall the College Board releases the survey results on how much colleges and universities are charging undergraduate students in the new academic year. The survey indicated that the average published tuition and fees for 2005–2006 were 8244 at public four-year colleges and universities and 28500 at private nonprofit four-year colleges and universities. The standard deviation was approximately 4500 at public four-year colleges and universities and approximately 12000 for private colleges and universities. a. Do the private nonprofit four-year colleges and universities have the larger relative variability Provide statistical evidence to support your answer. b. If the data on published tuition and fees were bell shaped determine the largest and smallest amount paid at the four-year private nonprofit colleges and universities. c. Based on your answer to part b do you believe that the data are bell shaped Support your answer using statistical reasoning. Computer Database Exercises 3-63. April 15 of every year is a day that most adults in the United States can relate to—the day that federal and state income taxes are due. Although there have been several attempts by Congress and the Internal Revenue Service over the past few years to simplify the income tax process many people still have a difficult time completing their tax returns properly. To draw attention to this problem a West Coast newspaper has asked 50 certified public accountant CPA firms to complete the same tax return for a hypothetical head of household. The CPA firms have their tax experts complete the return with the objective of determining the total federal income tax liability. The data in the file Taxes show the taxes owed as figured by each of the 50 CPA firms. Theoretically they should all come up with the same taxes owed. Based on these data write a short article for the paper that describes the results of this experiment. www.downloadslide.com

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Describing Data Using Numerical Measures the world’s countries. The data file Country Growth contains the most recent United Nations data on the population and the growth rate for the last decade for 231 countries throughout the world. Based on these data which countries had growth rates that exceeded 2 standard deviations higher than the mean growth rate Which countries had growth rates more than 2 standard deviations below the mean growth rate fares are the same but priced in English pounds determine his mean standard deviation and measure of relative dispersion for that data. Note: +1 0.566 GBP. 3-67. Doing business internationally is no longer something reserved for the largest companies. In fact medium- size and in some cases even small companies are finding themselves with the opportunity to do business internationally. One factor that will be important for world trade is the growth rate of the population of END EXERCISES 3-3 www.downloadslide.com

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Describing Data Using Numerical Measures Visual Summary 1 Measures of Center and Location 2 Measures of Variation Outcome 3. Compute the range variance and standard deviation and know what these values mean. 3 Using the Mean and Standard Deviation Together Outcome 4. Compute a z score and the coeffcient of variation and understand how they are applied in decision-making situations. Outcome 5. Understand the Empirical Rule and Tchebysheff’s Theorem Summary The real power of statistical measures of the center and variation come when they are used together to fully describe the data. One particular measure that is used a great deal in business especially in fnancial analysis is the coeffcient of variation . When comparing two or more data sets the larger the coeffcient of variation the greater the relative variation of the data. Another very important way in which the mean and standard deviation are used together is evident in the empirical rule which allows decision makers to better understand the data from a bell-shaped distribution. In cases where the data are not bell-shaped the data can be described using Tchebysheff’s Theorem. The fnal way discussed in this chapter in which the mean and standard deviation are used together is the z-value. Z-values for each individual data point measure the number of standard deviations a data value is from the mean. Conclusion A very important part of the descriptive tools in statistics is the collection of numerical measures that can be computed. When these measures of the center and variation in the data are combined with charts and graphs you can fully describe the data. Figure 10 presents a summary of the key numerical measures that are discussed in this chapter. Remember measures computed from a population are called parameters while measures computed from a sample are called statistics. To fully describe your data not only do you need to use the graphs charts and tables you need to provide the measures of the center and measures of variations in the data that are presented in this chapter. Together the numeric measures and the graphs and charts can paint a complete picture of the data that transform it from just data to useful information for decision-making purposes. Summary The three numerical measures of the center for a set of data are the mean median and the mode. The mean is the arithmetic average and is the most frequently used measure. However if the data are skewed or are ordinal level the median is suggested. Unlike the mean which is sensitive to extreme values in the data the median is unaffected by extremes. The mode is less frequently used as a measure of the center since it is simply the value in the data that occurs most frequently. When one of these measures is computed from a population the measure is said to be a parameter but if the measure is computed from sample data the measure is called a statistic. Other measures of location that are commonly used are percentiles and quartiles. Finally many decision makers prefer to construct a box and whisker plot which uses a box to display the range of the middle 50 percent of the data. The limits of whiskers are calculated based on the numerical distance between the frst and third quartiles. Outcome 1. Compute the mean median mode and weighted average for a set of data and understand what these values represent. Outcome 2. Construct a box and whisker graph and interpret it. Summary One of the major issues that business decision makers face every day is the variation that exists in their operations processes and people. Because virtually all data exhibit variation it is important to measure it. The simplest measure of variation is the range which is the difference between the highest value and the lowest value in the data. An alternative to the range that ignores the extremes in the data is the interquartile range which measures the numerical distance between the 3rd and 1st quartiles. But the two most frequently used measures of variation are the variance and the standard deviation. The equations for these two measures differ slightly depending on whether you are working with a population or a sample. The standard deviation is measured in the same units as the variable of interest and is a measure of the average deviation of the individual data items around the mean. www.downloadslide.com

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Describing Data Using Numerical Measures FIGURE 10 |  Summary of Numerical Statistical Measures Mode Median Mean Mode Median Mode Percentiles/ Quartiles Range Interquartile Range Variance and Standard Deviation Percentiles/ Quartiles Box and Whisker Coefcient of Variation Standardized z-values Ordinal Nominal Ratio/Interval Location Variation Location Location Descriptive Analysis Comparisons Data Level Type of Measures Equations 1 Population Mean ∑ x N i i N 1 2 Sample Mean x x n i i n 5 51 ∑ 3 Median Index in 5 1 2 4 Weighted Mean for a Population w ii i wx w ∑ ∑ 5 Weighted Mean for a Sample x wx w w ii i 5 ∑ ∑ 6 Percentile Location Index in 5 p 100 7 Range R Maximum value - Minimum value 8 Interquartile Range Interquartile range Third quartile - First quartile 9 Population Variance 2 x N i i N 2 1 ∑ 10 Population Variance Shortcut 2 2 2 ∑− ∑ x x N N 5 s 11 Population Standard Deviation 2 2 1 x N i i N − ∑ www.downloadslide.com

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Describing Data Using Numerical Measures 15 Population Coefficient of Variation CV 100 16 Sample Coefficient of Variation CV s x 5 100 17 Standardized Population Data z x − 18 Standardized Sample Data z xx s 5 − 12 Sample Variance s xx n i i n 2 2 1 1 5 5 – – ∑ 13 Sample Variance Shortcut s x x n n 2 2 2 1 5 ∑ ∑ – – 14 Sample Standard Deviation ss xx n i i n 55 5 2 2 1 1 – – ∑ Key Terms Box and whisker plot Coefficient of variation Data array Empirical Rule Interquartile range Left-skewed data Mean Median Mode Parameter Percentiles Population mean Quartiles Range Right-skewed data Sample mean Skewed data Standard deviation Standardized data values Statistic Symmetric data Tchebysheff’s theorem Variance Variation Weighted mean Chapter Exercises Conceptual Questions 3-68. Consider the following questions concerning the sample variance: a. Is it possible for a variance to be negative Explain. b. What is the smallest value a variance can be Under what conditions does the variance equal this smallest value c. Under what conditions is the sample variance smaller than the corresponding sample standard deviation 3-69. For a continuous variable that has a bell-shaped distribution determine the percentiles associated with the endpoints of the intervals specified in the Empirical Rule. 3-70. Consider that the Empirical Rule stipulates that virtually all of the data values are within the interval m 3s. Use this stipulation to determine an approximation for the standard deviation involving the range. 3-71. At almost every university in the United States the university computes student grade point averages GPAs. The following scale is typically used by universities: A 4 points B 3 points C 2 points D 1 point F 0 points Discuss what if any problems might exist when GPAs for two students are compared. What about comparing GPAs for students from two different universities 3-72. Since the standard deviation of a set of data requires more effort to compute than the range does what advantages does the standard deviation have when discussing the spread in a set of data 3-73. The mode seems like a very simple measure of the location of a distribution. When would the mode be preferred over the median or the mean Business Applications 3-74. Home Pros sells supplies to “do-it-yourselfers.” One of the things the company prides itself on is fast service. It uses a number system and takes customers in the order they arrive at the store. Recently the assistant manager tracked the time customers spent in the store from the time they took a number until they left. A sample of 16 customers was selected and the following data measured in minutes were recorded: 15 14 16 14 14 14 13 8 12 9 7 17 10 15 16 16 MyStatLab www.downloadslide.com

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Describing Data Using Numerical Measures 229.00 345.00 599.00 229.00 429.00 605.00 339.00 339.00 229.00 279.00 344.00 407.00 a. Compute the mean quoted airfare. b. Compute the variance and standard deviation in airfares quoted. Treat the data as a sample. 3-78. The manager of the Cottonwood Grille recently selected a random sample of 18 customers and kept track of how long the customers were required to wait from the time they arrived at the restaurant until they were actually served dinner. This study resulted from several complaints the manager had received from customers saying that their wait time was unduly long and that it appeared that the objective was to keep people waiting in the lounge for as long as possible to increase the lounge business. The following data were recorded with time measured in minutes: 34 24 43 56 74 20 19 33 55 43 54 34 27 34 36 24 54 39 a. Compute the mean waiting time for this sample of customers. b. Compute the median waiting time for this sample of customers. c. Compute the variance and standard deviation of waiting time for this sample of customers. d. Develop a frequency distribution using six classes each with a class width of 10. Make the lower limit of the first class 15. e. Develop a frequency histogram for the frequency distribution. f. Construct a box and whisker plot of these data. g. The manager is considering giving a complementary drink to customers whose waiting time is longer than the third quartile. Determine the minimum number of minutes a customer would have to wait in order to receive a complementary drink. 3-79. Simplot Agri-Chemical has decided to implement a new incentive system for the managers of its three plants. The plan calls for a bonus to be paid next month to the manager whose plant has the greatest relative improvement over the average monthly production volume. The following data reflect the historical production volumes at the three plants: Plant 1 Plant 2 Plant 3 m 700 m 2300 m 1200 s 200 s 350 s 30 At the close of next month the monthly output for the three plants was Plant 1 810 Plant 2 2600 Plant 3 1320 a. Compute the mean median mode range interquartile range and standard deviation. b. Develop a box and whisker plot for these data. 3-75. More than 272 million computer and video games were sold in 2010—more than two games for every U.S. household. Gamers spend an average of 3 to 4 hours playing games online every day. The average age of players is 28. Video games and gamers have even created a new form of marketing—called “advergaming.” “Advergaming is taking games— something that people do for recreation—and inserting a message” said Julie Roehm director of marketing communications for the Chrysler Group which sells Chrysler Jeep and Dodge brand vehicles. “It’s important we go to all the places our consumers are.” Suppose it is possible to assume the standard deviation of the ages of video game users is 9 years and that the distribution is bell shaped. To assist the marketing department in obtaining demographics to increase sales determine the proportion of players who are a. between 19 and 28 b. between 28 and 37 c. older than 37 3-76. Travelers are facing increased costs for both driving and flying to chosen destinations. With rising costs for both modes of transportation what really weighs on the decision to drive or to fly To gain a better understanding of the “fly or drive” decision a recent study compared the costs for trips between Los Angeles and Denver 1016 miles one way. Los Angeles to Denver round-trip costs 218 by car and 159.00 by plane. Cost flexibility is greater with the flying trips because of greater airfare choices. The driving trip costs except for the on-road lunches are pretty much set in place. Assume the standard deviation for the cost of flying trips is approximately 54. a. If a flight to Denver from Los Angeles was chosen at random determine the proportion of the time that the cost would be smaller than 164. Assume the flight costs are bell-shaped. b. Determine a flight cost that would qualify as the 25th percentile. c. If nothing can be assumed about the distribution of the flight costs determine the largest percentile that could be attributed to an airfare of 128. 3-77. With the ups and downs in the economy since 2008 many discount airline fares are available if a customer knows how to obtain the discount. Many travelers complain that they get a different price every time they call. The American Consumer Institute recently priced tickets between Spokane Washington and St. Louis Missouri. The passenger was to fly coach class round- trip staying seven days. Calls were made directly to airlines and to travel agents with the following results. Note that the data reflect round-trip airfare. www.downloadslide.com

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Describing Data Using Numerical Measures reason for his interest in this data is that the club is thinking of applying a discount to members who golf more than a specified number of rounds per year. The sample of eight people produced the following number of rounds played: 13 32 12 9 16 17 16 12 a. Compute the mean for these sample data. b. Compute the median for these sample data. c. Compute the mode for these sample data. d. Calculate the variance and standard deviation for these sample data. e. Note that one person in the sample played 32 rounds. What effect if any does this large value have on each of the three measures of location Discuss. f. For these sample data which measure of location provides the best measure of the center of the data Discuss. g. Given this sample data suppose the manager wishes to give discounts to golfers in the top quartile. What should the minimum number of rounds played be to receive a discount 3-83. Stock investors often look to beat the performance of the SP 500 Index which generally serves as a yardstick for the market as a whole. The following table shows a comparison of the five-year cumulative total shareholder returns for IDACORP common stock the SP 500 Index and the Edison Electric Institute EEI Electric Utilities Index. The data assume that 100 was invested on December 31 2002 with beginning-of-period weighting of the peer group indices based on market capitalization and monthly compounding of returns Source: IDACORP 2007 Annual Report. Year IDACORP SP 500 EEI Electric Utilities Inde x 2002 100.00 100.00 100.00 2003 128.86 128.67 123.48 2004 137.11 142.65 151.68 2005 136.92 149.66 176.02 2006 186.71 173.27 212.56 2007 176.26 182.78 247.76 Using the information provided construct appropriate statistical measures that illustrate the performance of the three investments. How well has IDACORP performed over the time periods compared to the SP 500 How well has it performed relative to its industry as measured by the returns of the EEI Electric Utilities Index 3-84. The Zagat Survey ® a leading provider of leisure-based survey results released its San Francisco Restaurants Survey involving participants who dined out an average of 3.2 times per week. The report showed the average Suppose the division manager has awarded the bonus to the manager of Plant 2 since her plant increased its production by 300 units over the mean more than that for any of the other managers. Do you agree with the award of the bonus for this month Explain using the appropriate statistical measures to support your position. 3-80. According to the annual report issued by Wilson Associates an investment firm in Bowling Green the stocks in its Growth Fund have generated an average return of 8 with a standard deviation of 2. The stocks in the Specialized Fund have generated an average return of 18 with a standard deviation of 6. a. Based on the data provided which of these funds has exhibited greater relative variability Use the proper statistical measure to make your determination. b. Suppose an investor who is very risk averse is interested in one of these two funds. Based strictly on relative variability which fund would you recommend Discuss. c. Suppose the distributions for the two stock funds had a bell-shaped distribution with the means and standard deviations previously indicated. Which fund appears to be the best investment assuming future returns will mimic past returns Explain. 3-81. The Dakota Farm Cooperative owns and leases prime farmland in the upper Midwest. Most of its 34000 acres are planted in grain. The cooperative performs a substantial amount of testing to determine what seed types produce the greatest yields. Recently the cooperative tested three types of corn seed on test plots. The following values were observed after the first test year: Seed Type A Seed Type B Seed Type C Mean Bushels/Acre 88 56 100 Standard Deviation 25 15 16 a. Based on the results of this testing which seed seems to produce the greatest average yield per acre Comment on the type of testing controls that should have been used to make this study valid. b. Suppose the company is interested in consistency. Which seed type shows the least relative variability c. Assuming the Empirical Rule applies describe the production distribution for each of the three seed types. d. Suppose you were a farmer and had to obtain at least 135 bushels per acre to escape bankruptcy. Which seed type would you plant Explain your choice. e. Rework your answer to part d assuming the farmer needed 115 bushels per acre instead. 3-82. The Hillcrest Golf and Country Club manager selected a random sample of the members and recorded the number of rounds of golf they played last season. The www.downloadslide.com

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Describing Data Using Numerical Measures at least as good mileage in city driving conditions as the mean mileage for highway driving for all cars 3-87. American Express estimates current Halloween spending to be about 53 per person. Much of the spending was expected to come from young adults. A file titled Halloween contains sample data on Halloween spending. a. Calculate the mean and standard deviation of these data. b. Determine the following intervals for this data set: x 1s x 2s x 3s. c. Suppose your responsibility as an assistant manager was to determine the price of costumes to be sold. The manager has informed you to set the price of one costume so that it was beyond the budget of only 2.5 of the customers. Assume that the data set has a bell-shaped distribution. 3-88. PayScale is a source of online compensation information providing access to accurate compensation data for both employees and employers. PayScale allows users to obtain compensation information providing a snapshot of the job market. Recently it published statistics for the salaries of MBA graduates. The file titled Payscale contains data with the same characteristics as those obtained by PayScale for California and Florida. a. Calculate the standard deviations of the salaries for both states’ MBA graduates. Which state seems to have the widest spectrum of salaries for MBA graduates b. Calculate the average and median salary for each state’s MBA graduates. c. Examining the averages calculated in part b determine which state’s MBA graduates have the largest relative dispersion. 3-89. Yahoo Finance makes available historical stock prices. It lists the opening high and low stock prices for each stock available on NYSE and NASDAQ. A file titled GEstock gives this data for General Electric GE for a recent 99-day period. a. Calculate the difference between the opening and closing stock prices for GE over this time period. Then calculate the mean median and standard deviation of these differences. b. Indicate what the mean in part a indicates about the relative prices of the opening and closing stock prices for GE. c. Compare the dispersion of the opening stock prices with the difference between the opening and closing stock prices. 3-90. Zepolle’s Bakery makes a variety of bread types that it sells to supermarket chains in the area. One of Zepolle’s problems is that the number of loaves of each type of bread sold each day by the chain stores varies considerably making it difficult to know how price per meal falling from the previous year from 34.07 to 33.75. a. If the standard deviation of the price of meals in San Francisco was 10 determine the largest proportion of meal prices that could be larger than 50. b. If the checks were paid in Chinese currency 1 USD 8.0916 Chinese Yuan determine the mean and standard deviation of meal prices in San Francisco. How would this change of currency affect your answer to part a Computer Database Exercises 3-85. The data in the file named Fast100 were collected by D. L. Green Associates a regional investment management company that specializes in working with clients who wish to invest in smaller companies with high growth potential. To aid the investment firm in locating appropriate investments for its clients Sandra Williams an assistant client manager put together a database on 100 fast-growing companies. The database consists of data on eight variables for each of the 100 companies. Note that in some cases data are not available. A code of –99 has been used to signify missing data. These data will have to be omitted from any calculations. a. Select the variable Sales. Develop a frequency distribution and histogram for Sales. b. Compute the mean median and standard deviation for the Sales variable. c. Determine the interquartile range for the Sales variable. d. Construct a box and whisker plot for the Sales variable. Identify any outliers. Discard the outliers and recalculate the measures in part b. e. Each year a goal is set for sales. Next year’s goal will be to have average sales that are at this year’s 65th percentile. Identify next year’s sales goal. 3-86. The Environmental Protection Agency EPA tests all new cars and provides a mileage rating for both city and highway driving conditions. Thirty cars were tested and are contained in the data file Automobiles. The file contains data on several variables. In this problem focus on the city and highway mileage data. a. Calculate the sample mean miles per gallon mpg for both city and highway driving for the 30 cars. Also calculate the sample standard deviation for the two mileage variables. Do the data tend to support the premise that cars get better mileage on the highway than around town Discuss. b. Referring to part a what can the EPA conclude about the relative variability between car models for highway versus city driving Hint: Compute the appropriate measure to compare relative variability. c. Assume that mileage ratings are approximately bell shaped. Approximately what proportion of cars gets www.downloadslide.com

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Describing Data Using Numerical Measures b. For this new variable computed in part a develop a frequency distribution. c. For the new variable computed in part a determine the mean median and standard deviation. d. Determine the percentile that would correspond to the “correct” tax figure if the IRS figure were one of the CPA firms’ estimated tax figures. Describe what this implies about the agreement between the IRS and consultants’ calculated tax. 3-92. The Cozine Corporation operates a garbage hauling business. Up to this point the company has been charged a flat fee for each of the garbage trucks that enters the county landfill. The flat fee is based on the assumed truck weight of 45000 pounds. In two weeks the company is required to appear before the county commissioners to discuss a rate adjustment. In preparation for this meeting Cozine has hired an independent company to weigh a sample of Cozine’s garbage trucks just prior to their entering the landfill. The data file Cozine contains the data the company has collected. a. Based on the sample data what percentile does the 45000-pound weight fall closest to b. Compute appropriate measures of central location for the data. c. Construct a frequency histogram based on the sample data. Use the 2 k Ú n guideline to determine the number of classes. Also construct a box and whisker plot for these data. Discuss the relative advantages of histograms and box and whisker plots for presenting these data. d. Use the information determined in parts a–c to develop a presentation to the county commissioners. Make sure the presentation attempts to answer the question of whether Cozine deserves a rate reduction. many loaves to bake. A sample of daily demand data is contained in the file Bakery. a. Which bread type has the highest average daily demand b. Develop a frequency distribution for each bread type. c. Which bread type has the highest standard deviation in demand d. Which bread type has the greatest relative variability Which type has the lowest relative variability e. Assuming that these sample data are representative of demand during the year determine how many loaves of each type of bread should be made such that demand would be met on at least 75 of the days during the year. f. Create a new variable called Total Loaves Sold. On which day of the week is the average for total loaves sold the highest 3-91. The Internal Revenue Service IRS has come under a great deal of criticism in recent years for various actions it is purported to have taken against U.S. citizens related to collecting federal income taxes. The IRS is also criticized for the complexity of the tax code although the tax laws are actually written by congressional staff and passed by Congress. For the past few years one of the country’s biggest tax-preparing companies has sponsored an event in which 50 certified public accountants from all sizes of CPA firms are asked to determine the tax owed for a fictitious citizen. The IRS is also asked to determine the “correct” tax owed. Last year the “correct” figure stated by the IRS was 11560. The file Taxes contains the data for the 50 accountants. a. Compute a new variable that is the difference between the IRS number and the number determined by each accountant. Video Case 3 video Drive-Thru Service Times  McDonald’s When you’re on the go and looking for a quick meal where do you go If you’re like millions of people every day you make a stop at McDonald’s. Known as “quick service restaurants” in the industry not “fast food” companies such as McDonald’s invest heavily to determine the most efficient and effective ways to provide fast high-quality service in all phases of their business. Drive-thru operations play a vital role. It’s not surprising that attention is focused on the drive-thru process. After all more than 60 of individual restaurant revenues in the United States come from the drive-thru experience. Yet understanding the process is more complex than just counting cars. Marla King professor at the company’s international training center Hamburger University got her start 25 years ago working at a McDonald’s drive-thru. She now coaches new restaurant owners and managers. “Our stated drive-thru service time is 90 seconds or less. We train every man- ager and team member to understand that a quality customer expe- rience at the drive-thru depends on them” says Marla. Some of the factors that affect customers’ ability to complete their purchases within 90 seconds include restaurant staffing equipment layout in the restaurant training efficiency of the grill team and frequency of customer arrivals to name a few. Also customer order patterns play a role. Some customers will just order drinks whereas others seem to need enough food to feed an entire soccer team. And then there are the special orders. Obviously there is plenty of room for variability here. www.downloadslide.com

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Describing Data Using Numerical Measures drive the decisions behind McDonald’s drive-thru service opera- tions. Discussion Questions: 1. After returning from the training session at Hamburger University a McDonald’s store owner selected a random sample of 362 drive-thru customers and carefully measured the time it took from when a customer entered the McDonald’s property until the customer had received the order at the drive-thru window. These data are in the file called McDonald’s Drive- Thru Waiting Times. Note the owner selected some customers during the breakfast period others during lunch and others during dinner. For the overall sample compute the key measures of central tendency. Based on these measures what conclusion might the owner reach with respect to how well his store is doing in meeting the 90-second customer service goal Discuss. 2. Referring to question 1 compute the key measures of central tendency for drive-thru times broken down by breakfast lunch and dinner time periods. Based on these calculations does it appear that the store is doing better at one of these time periods than the others in providing shorter drive-thru waiting times Discuss. 3. Referring to questions 1 and 2 compute the range and standard deviation for drive-thru times for the overall sample and for the three different times of the day. Also calculate the appropriate measure of relative variability for each time period. Discuss these measures of variability and what they might imply about what customers can expect at this McDonald’s drive-thru. 4. Determine the 1st and 3rd quartiles for drive-thru times and develop a box and whisker diagram for the overall sample data. Are there any outliers identified in these sample data Discuss. Yet that doesn’t stop the company from using statistical tech- niques to better understand the drive-thru action. In particular McDonald’s utilizes numerical measures of the center and spread in the data to help transform the data into useful information. For res- taurant managers to achieve the goal in their own restaurants they need training in proper restaurant and drive-thru operations. Ham- burger University McDonald’s training center located near Chi- cago Illinois satisfies that need. In the mock-up restaurant service lab managers go through a “before and after” training scenario. In the “before” scenario they run the restaurant for 30 minutes as if they were back in their home restaurants. Managers in the training class are assigned to be crew customers drive-thru cars special- needs guests such as hearing impaired indecisive clumsy or observers. Statistical data about the operations revenues and ser- vice times are collected and analyzed. Without the right training the restaurant’s operations usually start breaking down after 10 to 15 minutes. After debriefing and analyzing the data collected the managers make suggestions for adjustments and head back to the service lab to try again. This time the results usually come in well within standards. “When presented with the quantitative results managers are pretty quick to make the connections between better operations higher revenues and happier customers” Marla states. When managers return to their respective restaurants the training results and techniques are shared with staff charged with implementing the ideas locally. The results of the training eventu- ally are measured when McDonald’s conducts a restaurant opera- tions improvement process study or ROIP. The goal is simple: improved operations. When the ROIP review is completed statis- tical analyses are performed and managers are given their results. Depending on the results decisions might be made that require additional financial resources building construction staff train- ing or layout reconfiguration. Yet one thing is clear: Statistics Case 1 WGI—Human Resources WGI is a large international construction company with opera- tions in 43 countries. The company has been a major player in the reconstruction efforts in Iraq with a number of subcontracts under the major contractor Haliburton Inc. However the com- pany is also involved in many small projects both in the United States and around the world. One of these is a sewer line installa- tion project in Madison Wisconsin. The contract is what is called a “cost plus” contract meaning that the city of Madison will pay for all direct costs including materials and labor of the project plus an additional fee to WGI. Roberta Bernhart is the human resources HR manager for the Madison project and is respon- sible for overseeing all aspects of employee compensation and HR issues. WGI is required to produce a variety of reports to the Madison city council on an annual basis. Recently the council asked WGI to prepare a report showing the current hourly rates for the nonsalaried work crew on the project. Specifically the council is interested in any proposed pay increases to the work crew that will ultimately be passed along to the city of Madison. In response to the city’s request Ro berta put together a data file for all 19 nonsalaried work crew mem- bers called WGI which shows their current hourly pay rate and the proposed increase to take place the first of next month. These data are as follows: Name Current Rate New Rate Jody 20.55 22.55 Tim 22.15 23.81 Thomas 14.18 15.60 Shari 14.18 15.60 John 18.80 20.20 Jared 18.98 20.20 Loren 25.24 26.42 Mike 18.36 19.28 Patrick 17.20 18.06 Sharon 16.99 17.84 Sam 16.45 17.27 Susan 18.90 19.66 Chris 18.30 19.02 Steve F 27.45 28.12 Kevin 16.00 16.64 Larry 17.47 18.00 MaryAnn 23.99 24.47 Mark 22.62 23.08 Aaron 15.00 15.40 www.downloadslide.com

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Describing Data Using Numerical Measures 3. Compute a new variable called Pay Increase that reflects the difference between the proposed new pay rate and the current rate. Develop a histogram for this variable and then compute key measures of the center and variation for the new variable. 4. Compute a new variable that is the percentage increase in hourly pay rate. Prepare a graphical and numerical description of this new variable. 5. Prepare a report to the city council that contains the results from tasks 1–4. The city council expects the report to contain both graphic and numerical descriptive analyses. Roberta has outlined the following tasks and has asked you to help her. Required Tasks: 1. Develop and interpret histograms showing the distributions of current hourly rates and proposed new hourly rates for the crew members. 2. Compute and interpret key measures of central tendency and variation for the current and new hourly rates. Determine the coefficient of variation for each. Case 2 National Call Center Candice Worthy and Philip Hanson are day shift supervisors at National Call Center’s Austin Texas facility. National provides contract call center services for a number of companies includ- ing banks and major retail companies. Candice and Philip have both been with the company for slightly more than five years having joined National right after graduating with bachelor’s degrees from the University of Texas. As they walked down the hall together after the weekly staff meeting the two friends were discussing the assignment they were just handed by Mark Gonzales the division manager. The assignment came out of a discussion at the meeting in which one of National’s clients wanted a report describing the calls being handled for them by National. Mark had asked Candice and Philip to describe the data in a file called National Call Center and produce a report that would both graphically and numerically analyze the data. The data are for a sample of 57 calls and for the following v ariables: Account Number Caller Gender Account Holder Gender Past Due Amount Current Account Balance Nature of Call Billing Question or Other By the time they reached their office Candice and Philip had out- lined some of the key tasks that they needed to do. Required Tasks: 1. Develop bar charts showing the mean and median current account balance by gender of the caller. 2. Develop bar charts showing the mean and median current account balance by gender of the account holder. 3. Construct a scatter diagram showing current balance on the horizontal axis and past due amount on the vertical axis. 4. Compute the key descriptive statistics for the center and for the variation in current account balance broken down by gender of the caller gender of the account holder and by the nature of the call. 5. Repeat task 4 but compute the statistics for the past due balances. 6. Compute the coefficient of variation for current account balances for male and female account holders. 7. Develop frequency and relative frequency distributions for the gender of callers gender of account holders and nature of the calls. 8. Develop joint frequency and joint relative frequency distributions for the account holder gender by whether or not the account has a past due balance. 9. Write a report to National’s client that contains the results for tasks 1–8 along with a discussion of these statistics and graphs. Case 3 Welco Lumber Company—Part B Welco Lumber makes cedar fencing materials at its Naples Idaho facility employing about 160 people. The head rig is a large saw that breaks down the logs into slabs and cants. Gene recently conducted a study in which he videotaped 365 logs being broken down by the head rig. All three operators April Sid and Jim were involved. Each log was marked as to its true diameter. Then Gene observed the way the log was broken down and the degree to which the cants were properly centered. He then determined the projected value of the finished product from each log given the way it was actually cut. In addition he also determined what the value would have been had the log been cut in the optimal way. Data for this study are in a file called Welco Lumber. In addition to the graphical analysis that you helped Gene per- form you have been asked to assist Gene by analyzing these data using appropriate measures of the center and variation. He wishes to focus on the lost profit to the company and whether there are differences among the operators. Also do the operators tend to perform better on small logs than on large logs In general he is hoping to learn as much as possible from this study and needs your help with the analysis. www.downloadslide.com

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Describing Data Using Numerical Measures Case 4 AJ’s Fitness Center When A. J. Reeser signed papers to take ownership of the fitness center previously known as the Park Center Club he realized that he had just taken the biggest financial step in his life. Every asset he could pull together had been pledged against the mortgage. If the new AJ’s Fitness Center didn’t succeed he would be in really bad shape financially. But A. J. didn’t plan on failing. After all he had never failed at anything. As a high school football All-American A. J. had been heav- ily recruited by major colleges around the country. Although he loved football he and his family had always put academics ahead of sports. Thus he surprised almost everyone other than those who knew him best when he chose to attend an Ivy League university not particularly noted for its football success. Although he excelled at football and was a member of two winning teams he also suc- ceeded in the classroom and graduated in four years. He spent six years working for McKinsey Company a major consulting firm at which he gained significant experience in a broad range of busi- ness situations. He was hired away from McKinsey Company by the Dryden Group a management services company that specializes in running health and fitness operations and recreational resorts throughout the world. After eight years of leading the Fitness Center section at Dryden A. J. found that earning a high salary and the perks associated with corporate life were not satisfying him. Besides the travel was getting old now that he had married and had two young children. When the opportunity to purchase the Park Center Club came he decided that the time was right to control his own destiny. A key aspect of the deal was that AJ’s Fitness Club would keep its existing clientele consisting of 1833 memberships. One of the things A. J. was very concerned about was whether these members would stay with the club after the sale or move on to other fitness clubs in the area. He knew that keeping existing customers is a lot less expensive than attracting new customers. Within days of assuming ownership A. J. developed a survey that was mailed to all 1833 members. The letter that accompa- nied the survey discussed A. J.’s philosophy and asked several key questions regarding the current level of satisfaction. Survey respondents were eligible to win a free lifetime membership in a drawing—an inducement that was no doubt responsible for the 1214 usable responses. To get help with the analysis of the survey data A. J. approached the college of business at a local university with the idea of having a senior student serve as an intern at AJ’s Fitness Center. In addition to an hourly wage the intern would get free use of the fitness facilities for the rest of the academic year. The intern’s first task was to key the data from the survey into a file that could be analyzed using a spreadsheet or a statistical software package. The survey contained eight questions that were keyed into eight columns as follows: Column 1: Satisfaction with the club’s weight- and exer- cise-equipment facilities Column 2: Satisfaction with the club’s staff Column 3: Satisfaction with the club’s exercise programs aerobics etc. Column 4: Satisfaction with the club’s overall service Note columns 1 through 4 were coded on an ordinal scale as follows: 1 2345 Very unsatisfied Unsatisfied Neutral Satisfied Very satisfied Column 5: Number of years that the respondent had been a member at this club Column 6: Gender 11 Male 2 Female2 Column 7: Typical number of visits to the club per week Column 8: Age The data saved in the file AJFitness were clearly too much for anyone to comprehend in raw form. At yesterday’s meeting A. J. asked the intern to “make some sense of the data.” When the intern asked for some direction A. J.’s response was “That’s what I’m paying you the big bucks for. I just want you to develop a descriptive analysis of these data. Use whatever charts graphs and tables that will help us understand our customers. Also use any pertinent numerical measures that will help in the analysis. For right now give me a report that discusses the data. Why don’t we set a time to get together next week to review your report” www.downloadslide.com

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Describing Data Using Numerical Measures Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. x a n i- 1 x i n 26110 26.1 s 2 a n i 1 x - x 2 n - 1 148.910 - 1 16.5444 s 2s 2 216.5444 4.0675 Interquartile range 28 - 23 5 b. Ages are lower at Whitworth than for the U.S. colleges and universities as a group. 37. a. The range is 113.0 the IQR is 62.25 the variance is 1217.14 and the standard deviation is 34.89. b. No c. Adding a constant to all the data values leaves the variance unchanged. 39. a. Mean-.125 Standard Deviation 1.028. IQR 1.1 b. Prices have fallen slightly. c. Variation in prices is greater than the average decrease. 41. a. Men spent an average of 117 whereas women spent an average of 98 for their phones. The standard deviation for men was nearly twice that for women. b. Business users spent an average of 166.67 on their phones whereas home users spent an average of 105.74. The vari- ation in phone costs for the two groups was about equal. 43. a. The population mean is m a x N +178465 b. The population median is m ∼ +173000 c. The range is: R High - Low R +361100 - +54100 +307000 d. The population standard deviation is s Q a x -m 2 N +63172 47. a. at least 75 in the range 2600 to 3400 m2s. b. The range 2400 to 3600 should contain at least 89 of the data values. c. less than 11 49. a. 25.008 b. CV 23.55 c. The range from 31.19 to 181.24 should contain at least 89 of the data values. 51. For Distribution A: CV s m 100 100 500 100 20 For Distribution B: CV s m 100 4.0 10.0 100 40 1. Q 1 4423 Median 5002 Q 3 5381 3. Q 1 13.5 + 13.6 2 13.55 Q 3 15.5 + 15.9 2 15.7 7. a. 31.2 + 32.22 31.7 b. 26.7 + 31.22 28.95 c. 20.8 + 22.82 21.8 9. a. Mean 19 Median 19 + 192 19 Mode 19 b. Symmetrical Mean Median Mode 11. a. 11213.48 b. Use weighted average. 13. a. Mean 114.21 Median 107.50 Mode 156 b. skewed right 15. a. 562.99 b. 551.685 c. 562.90 17. a. 2008 Average Σx i n 6826804000000 8885 768351603.83 2011 Average Σx i n 7966700000000 7436 10713690156 b. Deposits went up and the number of institutions went down. c. If however these data were considered “typical” and were to be used to describe past current averages they would be considered statistics. 19. a. Mean .33 b. Median .31 c. Mode .24 d. The 800th percentile .40 minutes. 21. a. x 1177.1 13 90.55 right-skewed b. 0.392 c. 91.55 d. weighted average 25. a. Range 8 - 0 8 b. 3.99 c. 1.998 27. a. 16.87 b. 4.11 29. Standard deviation 2.8 31. a. Standard deviation 7.21 IQR 24 - 12 12 b. Range Largest - Smallest 30 - 6 24 Standard deviation 6.96 IQR 12 c. s 2 is smaller than s 2 by a factor of N - 12N. s is smaller than s by a factor of 1N - 1N. The range and IQR are not affected. 33. a. The variance is 815.79 and the standard deviation is 28.56. b. Interquartile range overcomes the susceptibility of the range to being highly influenced by extreme values. 35. a. Range 33 - 21 12 www.downloadslide.com

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Describing Data Using Numerical Measures 53. a. z 800 - x s 800 - 1000 250 -0.80 b. z 0.80 c. z 0.00 55. a. x 1530 30 51 Variance 510.76 Standard deviation 22.60 b. 5122.60 51222.60 51322.60 i.e. 28.4 73.6 5.8 96.2 and -16.8 118.8. There are 1930100 63.3 of the data within 28.4 73.6 3030100 100 of the data within 5.8 96.2 3030100 100 of the data within -16.8 118.8. c. bell-shaped population 57. a. b. Based on the sample means of the time each drug is effec- tive Drug B appears to be effective longer than Drug A. c. Based on the standard deviation of effect time Drug B exhibits a higher variability in effect time than Drug A. d. Drug A CV 5.93 Drug B CV 7.35. Drug B has a higher coefficient of variation and the greater relative spread. 59. Existing supplier: CV 0.078 3.75 100 2.08 New supplier: CV 0.135 18.029 100 0.75 61. Anyone scoring below 61.86 rounded to 62 will be rejected without an interview. Anyone scoring higher than 91.98 rounded to 92 will be sent directly to the company. 63. CV 3083.45 11144.48 100 27.67 At least 75 of CPA firms will compute a tax owed between +4977.58+17311.38 65. a. Variable Mean StDev Variance Q1 Median Q3 IQR Scores 94.780 4.130 17.056 93.000 96.000 98.000 5.000 b. Tchebysheff’s Theorem would be preferable. c. 99 Drug A Drug B Mean 234.75 270.92 Standard Deviation 13.92 19.90 Variable Mean StDev Median Close-Open -0.0354 0.2615 -0.0600 73. The mode is a useful measure of location of a set of data if the data set is large and involves nominal or ordinal data. 75. a. 0.34 b. 0.34 c. 0.16 77. a. 364.42 b. Variance 16662.63 Standard deviation 129.08 81. a. Comparing only the mean bushels/acre you would say that Seed Type C produces the greatest average yield per acre. b. CV of Seed Type A 2588 0.2841 or 28.41 CV of Seed Type B 1556 0.2679 or 26.79 CV of Seed Type C 16/100 0.1600 or 16 Seed Type C shows the least relative variability. c. Seed Type A: 68 between 63 and 113 95 between 38 and 138 approximately 100 between 13 and 163 Seed Type B: 68 between 41 and 71 95 between 26 and 86 approximately 100 between 11 and 101 Seed Type C: 68 between 84 and 116 95 between 68 and 132 approximately 100 between 52 and 148 d. Seed Type A e. Seed Type C 87. a. Mean 54.00 Standard Deviation 3.813 b. x1s 543.813 150.187 57.813 x2s 46.374 61.6262 x3s 42.561 65.4392 c. The Empirical Rule indicates that 95 of the data is con- tained within x 2s. This would mean that each tail has 11 - 0.9522 0.025 of the data. Therefore the costume should be priced at 46.37. 89. a. b. It means that the closing price for GE stock is an average of approximately four 0.0354 cents lower than the opening price. c. Variable Mean StDev Median Open 33.947 0.503 33.980 Close-Open -0.0354 0.2615 -0.0600 www.downloadslide.com

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Describing Data Using Numerical Measures Range The range is a measure of variation that is computed by finding the difference between the maximum and minimum values in a data set. Right-Skewed Data A data distribution is right skewed if the mean for the data is larger than the median. Sample Mean The average for all values in the sample com- puted by dividing the sum of all sample values by the sam- ple size. Skewed Data Data sets that are not symmetric. For skewed data the mean will be larger or smaller than the median. Standard Deviation The standard deviation is the positive square root of the variance. Standardized Data Values The number of standard devia- tions a value is from the mean. Standardized data values are sometimes referred to as z scores. Statistic A measure computed from a sample that has been selected from a population. The value of the statistic will depend on which sample is selected. Symmetric Data Data sets whose values are evenly spread around the center. For symmetric data the mean and median are equal. Tchebysheff’s Theorem Regardless of how data are distrib- uted at least 11 - 1k 2 2 of the values will fall within k standard deviations of the mean. For example: At least a1 - 1 1 2 b 0 0 of the values will fall within k 1 standard deviation of the mean. At least a1 - 1 2 2 b 3 4 75 of the values will lie within k 2 standard deviations of the mean. At least a1 - 1 3 2 b 8 9 89 of the values will lie within k 3 standard deviations of the mean. Variance The population variance is the average of the squared distances of the data values from the mean. Variation A set of data exhibits variation if all the data are not the same value. Weighted Mean The mean value of data values that have been weighted according to their relative importance. Box and Whisker Plot A graph that is composed of two parts: a box and the whiskers. The box has a width that ranges from the first quartile 1Q 1 2 to the third quartile 1Q 3 2. A verti- cal line through the box is placed at the median. Limits are located at a value that is 1.5 times the difference between Q 1 and Q 3 below Q 1 and above Q 3 . The whiskers extend to the left to the lowest value within the limits and to the right to the highest value within the limits. Coefficient of Variation The ratio of the standard deviation to the mean expressed as a percentage. The coefficient of varia- tion is used to measure variation relative to the mean. Data Array Data that have been arranged in numerical order. Empirical Rule If the data distribution is bell shaped then the interval m 1s contains approximately 68 of the values m 2s contains approximately 95 of the values m 3s contains virtually all of the data values Interquartile Range The interquartile range is a measure of variation that is determined by computing the difference between the third and first quartiles. Left-Skewed Data A data distribution is left skewed if the mean for the data is smaller than the median. Mean A numerical measure of the center of a set of quantita- tive measures computed by dividing the sum of the values by the number of values in the data. Median The median is a center value that divides a data array into two halves. We use m ∼ to denote the population median and M d to denote the sample median. Mode The mode is the value in a data set that occurs most fre- quently. Parameter A measure computed from the entire population. As long as the population does not change the value of the parameter will not change. Percentiles The pth percentile in a data array is a value that divides the data set into two parts. The lower segment con- tains at least p and the upper segment contains at least 1 100 - p25 of the data. The 50th percentile is the median. Population Mean The average for all values in the population computed by dividing the sum of all values by the popula- tion size. Quartiles Quartiles in a data array are those values that divide the data set into four equal-sized groups. The median cor- responds to the second quartile. Glossary www.downloadslide.com

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Describing Data Using Numerical Measures Berenson Mark L. and David M. Levine Basic Business Sta- tistics: Concepts and Applications 12th ed. Upper Saddle River NJ: Prentice Hall 2012. DeVeaux Richard D. Paul F. Velleman and David E. Bock Stats Data and Models 3rd ed. New York: Addison-Wesley 2012. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. Tukey John W. Exploratory Data Analysis Reading MA: Addison-Wesley 1977. References www.downloadslide.com

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Special Review Section The Where Why and How of Data Collection Graphs Charts and Tables—Describing Your Data Describing Data Using Numerical Measures This text introduces data data collection and statistical tools for describing data. The steps needed to gather “good” statistical data transform it to usable information and present the information in a manner that allows good decisions are outlined in the following figures. Information Statistical Tools Transforming Data into Information Data From Special Review Section 1–3 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Special Review Section A Typical Application Sequence Determine a Need for Data • Research the issue • Analyze business alternatives • Respond to request for information Defne your data requirements Defne the Population • All items of interest—Who What Determine What Data Y ou Will Need • Identify the key variables e.g. age income diameter processing time satisfaction rating • What categorical breakdowns will be needed e.g. analyze by gender race region and class standing Decide How the Data Will Be Collected • Experiment • Observation • Automation • Telephone Survey • Written Survey • Personal Interview Decide on a Census or a Sample • Census: All items in the population Decide on Statistical or Nonstatistical Sampling • Statistical Sampling: • Nonstatistical Sampling: Convenience Sample Judgment Sample Simple Random Sample Stratifed Random Sample Systematic Random Sample Cluster Random Sample Determine Data Types and Measurement Level Te method of descriptive statistical analysis that can be performed depends on the type of data and the level of data measurement for the variables in the data set. Typical studies will involve multiple types of variables and data levels. • Types of Data • Data Timing Quantitative Qualitative Cross-Sectional Time-Series ta Level Nominal Ordinal Interval/Ratio Lowest Level Mid-Level Highest Level Categories—no ordering implied Categories—defned ordering Measurements • Sample: A subset of the population • Determine how to gain access to the population www.downloadslide.com

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Special Review Section Select Graphic Presentation Tools Quantitative Qualitative Data Timing Line Chart Data Type Frequency Distribution Bar Chart Vertical or Horizontal Bar Chart Vertical Relative Frequency Distribution Pie Chart Joint Frequency Histogram Frequency Distribution Cross-Sectional Grouped or Ungrouped Time- Series Relative Frequency Distribution Stem and Leaf Diagram Scatter Diagram Cumulative Relative Frequency Distribution Joint Frequency Distribution Box and Whisker Plot Ogive Discrete or Continuous Interval/Ratio Categorical/ Nominal/Ordinal Data Level Mode Median Mean Mode Median Mode Percentiles/ Quartiles Range Interquartile Range Variance and Standard Deviation Percentiles/ Quartiles Box and Whisker Coefcient of Variation Standardized z-values Ordinal Nominal Ratio/Interval Central Location Variation Central Location Descriptive Analysis Comparisons Compute Numerical Measures Te choice of nu merical descriptive analysis depends on the level of data measurement. If the data are ratio or interval you have the widest range of numerical tools available. Type of Measures www.downloadslide.com

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Special Review Section Write the Statistical Report Tere is no one set format for writing a statistical report. However there are a few suggestions you may fnd useful. Describe the data collection methodology : Explain how the data were gathered and the sampling techniques were used. Lay the foundation : Provide background and motivation for the analysis. Use a logical sequence : Follow a systematic plan for presenting your fndings and analysis. Label fgures and tables by number : Employ a consistent numbering and labeling format. Exercises Integrative Application Exercises This text introduced you to the basics of descriptive statistics. Many of the business application problems advanced business application problems and cases will give you practice in performing descriptive statistical analysis. However too often you are told which procedure you should use or you can surmise which to use by the location of the exercise. It is important that you learn to identify the appropriate procedure on your own in order to solve problems for test purposes. But more important this ability is essential throughout your career when you are required to select procedures for the tasks you will undertake. The following exercises will provide you with identification practice. SR.1. Go to your university library and obtain the Statistical Abstract of the United States. a. Construct a frequency distribution for unemployment rate by state for the most current year available. b. Justify your choice of class limits and number of classes. c. Locate the unemployment rate for the state in which you are attending college. 1 What proportion of the unemployment rates are below that of your state 2 Describe the distribution’s shape with respect to symmetry. 3 If you were planning to build a new manufacturing plant what state would you choose in which to build Justify your answer. 4 Are there any unusual features of this distribution Describe them. SR.2. The State Industrial Development Council is currently working on a financial services brochure to send to out-of-state companies. It is hoped that the brochure will be helpful in attracting companies to relocate to your state. You are given the following frequency distribution on banks in your state: Deposit Size in millions Number of Banks Total Deposits in millions Less than 5 2 7.2 5 to less than 10 7 52.1 10 to less than 25 6 111.5 25 to less than 50 3 95.4 50 to less than 100 2 166.6 100 to less than 500 2 529.8 Over 500 2 1663.0 a. Does this frequency distribution violate any of the rules of construction for frequency distributions If so reconstruct the frequency distribution to remedy this violation. b. The Council wishes to target companies that would require financial support from banks that have at least 25 million in deposits. Reconstruct the frequency distribution to attract such companies to relocate to your state. Do this by considering different classes that would accomplish such a goal. c. Reconstruct the frequency distribution to attract companies that require financial support from banks that have between 5 million and 25 million in deposits. d. Present an eye-catching two-paragraph summary of what the data would mean to a company that is considering moving to the state. Your boss has said you need to include relative frequencies in this presentation. MyStatLab www.downloadslide.com

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Special Review Section in part a to prepare a written report that describes the results of the test. Be sure to include in your report a conclusion regarding whether the scanner outperforms the manual process. c. Which process scanner or manual generated the most values that were more than 2 standard deviations from the mean d. Which of the two processes has the least relative variability SR.5. Excel tutorials The commercial banking industry is undergoing rapid changes due to advances in technology and competitive pressures in the financial services sector. The data file Banks contains selected information tabulated by Fortune concerning the revenues profitability and number of employees for the 51 largest U.S. commercial banks in terms of revenues. Use the information in this file to complete the following: a. Compute the mean median and standard deviation for the three variables revenues profits and number of employees. b. Convert the data for each variable to a z value. Consider Mellon Bank Corporation headquartered in Pittsburgh. How does it compare to the average bank in the study on the three variables Discuss. c. As you can see by examining the data and by looking at the statistics computed in part a not all banks had the same revenue same profit or the same number of employees. Which variable had the greatest relative variation among the banks in the study d. Calculate a new variable: profits per employee. Develop a frequency distribution and a histogram for this new variable. Also compute the mean median and standard deviation for the new variable. Write a short report that describes the profits per employee for the banks. e. Referring to part d how many banks had a profit- per-employee ratio that exceeded 2 standard deviations from the mean SR.3. Excel tutorials As an intern for Intel Corporation suppose you have been asked to help the vice president prepare a newsletter to the shareholders. Y ou have been given access to the data in a file called Intel that contains Intel Corporation financial data for the years 1987 through 1996. Go to the Internet or to Intel’s annual report and update the file to include the same variables for the years 1997 to the present. Then use graphs to effectively present the data in a format that would be usable for the vice president’s newsletter. Write a short article that discusses the information shown in your graphs. SR.4. Excel tutorials The Woodmill Company makes windows and door trim products. The first step in the process is to rip dimension 2 8 2 10 etc. lumber into narrower pieces. Currently the company uses a manual process in which an experienced operator quickly looks at a board and determines what rip widths to use. The decision is based on the knots and defects in the wood. A company in Oregon has developed an optical scanner that can be used to determine the rip widths. The scanner is programmed to recognize defects and to determine rip widths that will optimize the value of the board. A test run of 100 boards was put through the scanner and the rip widths were identified. However the boards were not actually ripped. A lumber grader determined the resulting values for each of the 100 boards assuming that the rips determined by the scanner had been made. Next the same 100 boards were manually ripped using the normal process. The grader then determined the value for each board after the manual rip process was completed. The resulting data in the file Woodmill consist of manual rip values and scanner rip values for each of the 100 boards. a. Develop a frequency distribution for the board values for the scanner and the manual process. b. Compute appropriate descriptive statistics for both manual and scanner values. Use these data along with the frequency distribution developed END SRS 1-3 www.downloadslide.com

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Special Review Section were selected in each county and the state police set up roadblocks on a randomly selected day. Vehicles with in-state license plates were stopped at random until approximately 100 vehicles had been stopped at each location. The target total was about 2400 vehicles statewide. The issue of primary interest was whether the vehicle was insured. This was determined by observing whether the vehicle was carrying the required certificate of insurance. If so the officer took down the insurance company name and address and the policy number. If the certificate was not in the car but the owner stated that insurance was carried the owner was given a postcard to return within five days supplying the required information. A vehi- cle was determined to be uninsured if no postcard was returned or if subsequently the insurance company reported that the policy was not valid on the day of the survey. In addition to the issue of insurance coverage Herb Kriner wanted to collect other information about the vehicle and the owner. This was done using a personal interview during which the police officer asked a series of questions and observed cer- tain things such as seat belt usage and driver’s and vehicle license expiration status. Also the owners’ driving records were obtained through the Transportation Department’s computer division and added to the information gathered by the state police. The Data The data are contained in the file Liabins. The sheet titled “Description” contains an explanation of the data set and the vari- ables. Issues to Address Herb Kriner has two weeks before making a presentation to the legislative subcommittee that has been dealing with the liability insurance issue. As Herb’s chief analyst your job is to perform a comprehensive analysis of the data and to prepare the report that Herb will deliver to the legislature. Remember this report will go a long way in determining whether the state should spend the 1.5 million to implement a full liability insurance audit system. State Department of Insurance Excel tutorials This case study describes the efforts undertaken by the director of the Insurance Division to assess the magnitude of the uninsured motorist problem in a western state. The objective of the case study is to introduce you to a data collection application and show how one organization developed a database. The database Liabins con- tains a subset of the data actually collected by the state department. The impetus for the case came from the legislative transpor- tation committee which heard much testimony during the recent legislative session about the problems that occur when an unin- sured motorist is involved in a traffic accident in which damages to individuals and property occur. The state’s law enforcement offic- ers also testified that a large number of vehicles are not covered by liability insurance. Because of both political pressure and a sense of duty to do what is right the legislative committee spent many hours wres- tling with what to do about drivers who do not carry the mandatory liability insurance. Because the actual magnitude of the problem was unknown the committee finally arrived at a compromise plan which required the state Insurance Division to perform random audits of vehicles to determine whether the vehicle was covered by liability insurance. The audits are to be performed on approxi- mately 1 of the state’s 1 million registered vehicles each month. If a vehicle is found not to have liability insurance the vehicle license and the owner’s driver’s license will be revoked for three months and a 250 fine will be imposed. However before actually implementing the audit process which is projected to cost 1.5 million per year Herb Kriner director of the Insurance Division was told to conduct a prelimi- nary study of the uninsured motorists’ problem in the state and to report back to the legislative committee in six months. The Study A random sample of 12 counties in the state was selected in a man- ner that gave the counties with higher numbers of registered vehi- cles proportionally higher chances of being selected. Two locations Here is an integrative case study designed to give you more experience. In addition we have included several term project assignments that require you to collect and analyze data. Review Case 1 Term Project Assignments For the project selected you are to devise a sampling plan collect appropriate data and carry out a full descriptive analysis aimed at shedding light on the key issues for the project. The finished project will include a written report of a length and format specified by your professor. Project A Issue: Your College of Business and Economics seeks input from business majors regarding class scheduling. Some potential issues are ● Day or evening ● Morning or afternoon ● One-day two-day or three-day schedules ● Weekend ● Location on or off campus Project B Issue: Intercollegiate athletics is a part of most major universi- ties. Revenue from attendance at major sporting events is one key to financing the athletic program. Investigate the drivers of www.downloadslide.com

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Special Review Section many companies are listed in both files but some are just in one or the other. The two files have many of the same variables but the 2003 file has a larger range of financial variables than the 2005 file. For some companies the data for certain variables are not avail- able and a code of NA is used to so indicate. The 2003 file has a special worksheet that contains the description of each variable. These descriptions apply to the 2005 data file as well. You have been given access to these two data files for use in preparing your reports. Your role will be to perform certain statis- tical analyses that can be used to help convert these data into use- ful information in order to respond to the clients’ questions. This morning one of the partners of your company received a call from a client who asked for a report that would compare companies in the financial services industry SIC codes in the 6000s to companies in production-oriented businesses SIC codes in the 2000s and 3000s. There are no firm guidelines on what the report should entail but the partner has suggested the following: ● Start with the 2005 data file. Pull the data for all companies with the desired SIC codes into a new worksheet. ● Prepare a complete descriptive analysis of key financial vari- ables using appropriate charts and graphs to help compare the two types of businesses. ● Determine whether there are differences between the two classes of companies in terms of key financial measures. ● Using data from the 2003 file for companies that have these SIC codes and that are also in the 2005 file develop a compari- son that shows the changes over the time span both within SIC code grouping and between SIC code groupings. Project Deliverables To successfully complete this capstone project you are required to deliver a management report that addresses the partner’s requests listed above and also contains at least one other sub- stantial type of analysis not mentioned by the partner. This latter work should be set off in a special section of the report. The final report should be presented in a professional format using the style or format suggested by your instructor. attendance at your university’s men’s basketball and football games. Some potential issues: ● Game times ● Game days basketball ● Ticket prices ● Athletic booster club memberships ● Competition for entertainment dollars Project C Issue: The department of your major is interested in surveying department alumni. Some potential issues are ● Satisfaction with degree ● Employment status ● Job satisfaction ● Suggestions for improving course content Capstone Project Project Objective The objective of this business statistics capstone project is to pro- vide you with an opportunity to integrate the statistical tools and concepts that you have learned thus far in your business statistics course. Like all real-world applications completing this project will not require you to utilize every statistical technique. Rather an objective of the assignment is for you to determine which of the statistical tools and techniques are appropriate for the situation you have selected. Project Description Assume that you are working as an intern for a financial manage- ment company. Your employer has a large number of clients who trust the company managers to invest their funds. In your posi- tion you are responsible for producing reports for clients when they request information. Your company has two large data files with financial information for a large number of U.S. companies. The first is called US Companies 2003 which contains financial information for the companies’ 2001 or 2002 fiscal year-end. The second file is called US Companies 2005 which has data for the fiscal 2003 or 2004 year-end. The 2003 file has data for 7098 companies. The 2005 file has data for 6992 companies. Thus www.downloadslide.com

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Introduction to Probability From Chapter 4 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Introduction to Probability Quick Prep Links Review the discussion of statistical sampling in. Examine recent business periodicals and newspapers looking for examples in which probability concepts are discussed. Think about how you determine what decision to make in situations in which you are uncertain about your choices. 1 The Basics of Probability 2 The Rules of Probability Outcome 1. Understand the three approaches to assessing probabilities. Outcome 2. Be able to apply the Addition Rule. Outcome 3. Know how to use the Multiplication Rule. Outcome 4. Know how to use Bayes’ Theorem for applications involving conditional probabilities. Why you need to know Recently the Powerball lottery raised the cost of buying a ticket from 1 to 2. With the higher ticket prices lottery offi- cials expect the jackpot prize value to increase more rapidly and thereby entice even greater ticket sales. Most people recognize when buying a lottery ticket that there is a very small probability of winning and that whether they win or lose is based on chance alone. In case you are not familiar with the Powerball lottery system a drum contains 59 balls numbered 1 to 59. The player must choose or have a computer choose five numbers between 1 and 59. The player also chooses a 6th number called the power ball. On the night of the drawing five balls are randomly selected and then placed in numerical order. Lastly a sixth ball the power ball is randomly selected. To win the jackpot the player must match all five numbers plus the power ball. The odds of winning are shown on the Powerball website to be 1 in 175223510 or about 0.00000000571. Later in the chapter you will learn the method for computing probabilities like this. One analogy might put this in perspective. Suppose we take a college football field and cover it with 175223510 tiny red ants. One of these ants has a yellow dot on it. If you were blindfolded your chances of picking the one ant with the yellow dot from the millions of ants on the football field would be the same as winning the Powerball jackpot We suggest you come up with a different retirement strategy. In business decision making in many instances chance is involved in determining the outcome of a decision. For instance when a TV manufacturer establishes a warranty on its television sets there is a certain probability that any given TV will last less than the warranty life and customers will have to be compensated. Accountants perform audits on the finan- cial statements of a client and sign off on the statements as accurate while realizing there is a chance that problems exist that were not uncovered by the audit. A food processor manu- facturer recognizes that there is a chance that one or more of its products will be substandard and dissatisfy the customer. Airlines overbook flights to make sure that the seats on the plane are as full as possible because they know there is a certain probability that customers will not show for their flight. Professional poker players base their decisions to fold or play a hand based on their assessment of the chances that their hand beats those of their opponents. Photo_Ma/Fotolia www.downloadslide.com

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Introduction to Probability If we always knew what the result of our decisions would be our life as decision makers would be a lot less stressful. However in most instances uncertainty exists. To deal with this uncertainty we need to know how to incor- porate probability concepts into the decision process. This text takes the first step in teaching you how to do this by introducing the basic concepts and rules of probability. You need to have a solid understanding of these basics before moving on to the more practical probability applications that you will encounter in business. 1 The Basics of Probability The mathematical study of probability originated more than 300 years ago. The Chevalier de Méré a French nobleman who today would probably own a gaming house in Monte Carlo began ask- ing questions about games of chance. He was mostly interested in the probability of observing various outcomes when dice were repeatedly rolled. The French mathematician Blaise Pascal you may remember studying Pascal’s triangle in a mathematics class with the help of his friend Pierre de Fermat was able to answer de Méré’s questions. Of course Pascal began asking more and more complicated questions of himself and his colleagues and the formal study of probability began. Important Probability Terms Several explanations of what probability is have come out of this mathematical study. How- ever the definition of probability is quite basic. For instance if we look out the window and see rain we can say the probability of rain today is 1 since we know for sure that it will rain. If an airplane has a top speed of 450 mph and the distance between city A and city B is 900 miles we can say the probability the plane will make the trip in 1.5 hours is zero—it can’t happen. These examples involve situations in which we are certain of the outcome and our 1 and 0 probabilities reflect this. However in most business situations we do not have certainty but instead are uncertain. For instance if a real estate investor has the option to purchase a small shopping mall deter- mining rate of return on this investment involves uncertainty. The investor does not know with certainty whether she will make a profit break even or lose money. After looking closely at the situation she might say the chance of making a profit is 0.30. This value between 0 and 1 reflects her uncertainty about whether she will make a profit from purchasing the shopping mall. Events and Sample Space Data come in many forms and are gathered in many ways. In probability language the process that produces the outcomes is an experiment. For instance a very simple experiment might involve flipping a coin one time. When this experiment is performed two possible experimental outcomes can occur: head and tail. If the coin- tossing experiment is expanded to involve two flips of the coin the experimental outcomes are Head on first flip and head on second flip denoted by HH Head on first flip and tail on second flip denoted by HT Tail on first flip and head on second flip denoted by TH Tail on first flip and tail on second flip denoted by TT In business situations the experiment can be things like an investment decision a person- nel decision or a choice of warehouse location. The collection of possible experimental outcomes is called the sample space. EXAMPLE 1 DEFINING THE SAMPLE SPACE Five Guys Hamburgers The sales manager for the Five Guys Hamburger chain is interested in analyzing the sales of its three best-selling hamburgers. As part of this analysis he might be interested in determining the sample space possible outcomes for two randomly selected customers. To do this he can use the following steps. Step 1 Define the experiment. The experiment is the sale. The item of interest is the product sold. Probability The chance that a particular event will occur. The probability value will be in the range 0 to 1. A value of 0 means the event will not occur. A probability of 1 means the event will occur. Anything between 0 and 1 reflects the uncertainty of the event occurring. The definition given is for a countable number of events. Experiment A process that produces a single outcome whose result cannot be predicted with certainty. Sample Space The collection of all outcomes that can result from a selection decision or experiment. ExQuisine/Fotolia www.downloadslide.com

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Introduction to Probability Step 2 Define the outcomes for one trial of the experiment. The manager can define the outcomes to be e 1 Hamburger e 2 Cheeseburger e 3 Bacon Burger Step 3 Define the sample space. The sample space SS for an experiment involving a single sale is SS 5e 1 e 2 e 3 6 If the experiment is expanded to include two sales the sample space is SS 5e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 6 where the outcomes include what happens on both sales and are defined as Outcome Sale 1 Sale 2 e 1 Hamburger Hamburger e 2 Hamburger Cheeseburger e 3 Hamburger Bacon Burger e 4 Cheeseburger Hamburger e 5 Cheeseburger Cheeseburger e 6 Cheeseburger Bacon Burger e 7 Bacon Burger Hamburger e 8 Bacon Burger Cheeseburger e 9 Bacon Burger Bacon Burger END EXAMPLE TRY PROBLEM 3 Using Tree Diagrams A tree diagram is often a useful way to define the sample space for an experiment that helps ensure no outcomes are omitted or repeated. Example 2 illustrates how a tree diagram is used. EXAMPLE 2 USING A TREE DIAGRAM TO DEFINE THE SAMPLE SPACE Clearwater Marketing Research Clearwater Marketing Research is involved in a project in which television viewers were asked whether they objected to hard-liquor advertisements being shown on television. The analyst is interested in listing the sample space using a tree diagram as an aid when three viewers are interviewed. The following steps can be used: Step 1 Define the experiment. Three people are interviewed and asked “Would you object to hard-liquor advertisements on television” Thus the experiment consists of three trials. Step 2 Define the outcomes for a single trial of the experiment. The possible outcomes when one person is interviewed are no yes Step 3 Define the sample space for three trials using a tree diagram. Begin by determining the outcomes for a single trial. Illustrate these with tree branches beginning on the left side of the page: No Yes www.downloadslide.com

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Introduction to Probability For each of these branches add branches depicting the outcomes for a second trial. Continue until the tree has the number of sets of branches corresponding to the number of trials. No Yes Yes Yes Yes Yes Yes Yes No No No No No No No No No No No Y es No Y es No No Y es Y es Y es No No Y es No Y es Y es Y es No Y es Y es Y es Trial 1 Trial 2 Trial 3 Experimental Outcomes END EXAMPLE TRY PROBLEM 4 A collection of possible outcomes is called an event. An example will help clarify these terms. EXAMPLE 3 DEFINING AN EVENT OF INTEREST KPMG Accounting The KPMG Accounting firm is interested in the sample space for an audit experiment in which the outcome of interest is the audit’s completion status. The sample space is the list of all possible outcomes from the experiment. The accounting firm is also interested in specifying the outcomes that make up an event of interest. This can be done using the following steps: Step 1 Define the experiment. The experiment consists of two randomly chosen audits. Step 2 List the outcomes associated with one trial of the experiment. For a single audit the following completion-status possibilities exist: Audit done early Audit done on time Audit done late Step 3 Define the sample space. For two audits two trials we define the sample space as follows: Experimental Outcome Audit 1 Audit 2 e 1 Early Early e 2 Early On time Event A collection of experimental outcomes. www.downloadslide.com

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Introduction to Probability Experimental Outcome Audit 1 Audit 2 e 3 Early Late e 4 On time Early e 5 On time On time e 6 On time Late e 7 Late Early e 8 Late On time e 9 Late Late Step 4 Define the event of interest. The event of interest at least one audit is completed late is composed of all the outcomes in which one or more audits are late. This event E is E 5e 3 e 6 e 7 e 8 e 9 6 There are five ways in which one or more audits are completed late. END EXAMPLE TRY PROBLEM 11 Mutually Exclusive Events Keeping in mind the definitions for experiment sample space and events we introduce two additional concepts. The first is mutually exclusive events. BUSINESS APPLICATION MUTUALLY EXCLUSIVE EVENTS KPMG ACCOUNTING Consider again the KPMG Accounting firm example. The possible outcomes for two audits are Experimental Outcomes Audit 1 Audit 2 e 1 Early Early e 2 Early On time e 3 Early Late e 4 On time Early e 5 On time On time e 6 On time Late e 7 Late Early e 8 Late On time e 9 Late Late Suppose we define one event as consisting of the outcomes in which at least one of the two audits is late. E 1 5e 3 e 6 e 7 e 8 e 9 6 Further suppose we defne a second event as follows: E 2 Neither audit is late 5e 1 e 2 e 4 e 5 6 Events E 1 and E 2 are mutually exclusive: If E 1 occurs E 2 cannot occur if E 2 occurs E 1 cannot occur. That is if at least one audit is late then it is not possible for neither audit to be late. We can verify this fact by observing that no outcomes in E 1 appear in E 2 . This observation pro- vides another way of defning mutually exclusive events: Two events are mutually exclusive if they have no common outcomes. EXAMPLE 4 MUTUALLY EXCLUSIVE EVENTS Tech-Works Inc. Tech-Works Inc. located in Dublin Ireland does contract assembly work for companies such as Hewlett-Packard. Each item produced on the assembly line can be thought of as an experimental trial. The managers at this facility can analyze their process to determine whether the events of interest are mutually exclusive using the following steps: Step 1 Define the experiment. The experiment is producing a part on an assembly line. Mutually Exclusive Events Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other event. www.downloadslide.com

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Introduction to Probability Step 2 Define the outcomes for a single trial of the experiment. On each trial the outcome is either a good or a defective item. Step 3 Define the sample space. If two products are produced two trials the following sample space is defined: Experimental Outcomes Product 1 Product 2 e 1 Good Good e 2 Good Defective e 3 Defective Good e 4 Defective Defective Step 4 Determine whether the events are mutually exclusive. Let event E 1 be defined as both products produced are good and let event E 2 be defined as at least one product is defective: E 1 Both good 5e 1 6 E 2 At least one defective 5e 2 e 3 e 4 6 Then events E 1 and E 2 are determined to be mutually exclusive because the two events have no outcomes in common. Having two good items and at the same time having at least one defective item is not possible. END EXAMPLE TRY PROBLEM 9 Independent and Dependent Events A second probability concept is that of independent versus dependent events. BUSINESS APPLICATION INDEPENDENT AND DEPENDENT EVENTS PETROGL YPH Petroglyph is a subsidiary of the Intermountain Gas Company and is responsible for natural gas exploration in the western U.S. and Canada. During the exploration phase seismic surveys are conducted that provide information about the Earth’s underground formations. Based on history the company knows that if the seismic readings are favorable gas will more likely be discovered than if the seismic readings are not favorable. However the readings are not perfect indicators. Suppose the company currently is exploring in eastern Colorado. The possible outcomes for the seismic survey are defined as e 1 Favorable e 2 Unfavorable If the company decides to drill the outcomes are defned as e 3 Strike gas e 4 Dry hole If we let the event E 1 be that the seismic survey is favorable and event E 2 be that the hole is dry we can say that the events A and B are not mutually exclusive because one event’s occur- rence does not preclude the other event from occurring. We can also say that the two events are dependent because the probability of a dry hole depends on whether the seismic survey is favorable or unfavorable. If the result of drilling was not related to the seismic survey the events would be independent. Methods of Assigning Probability Part of the confusion surrounding probability may be due to the fact that probability can be assigned to outcomes in more than one way. There are three common ways to assign prob- ability to outcomes: classical probability assessment relative frequency assessment and Independent Events Two events are independent if the occurrence of one event in no way influences the probability of the occurrence of the other event. Dependent Events Two events are dependent if the occurrence of one event impacts the probability of the other event occurring. Chapter Outcome 1. www.downloadslide.com

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Introduction to Probability subjective probability assessment. The following notation is used when we refer to the prob- ability of an event: P1E i 2 Probability of event E i occurring Classical Probability Assessment The first method of probability assessment involves classical probability. You are probably already familiar with classical probability. It had its beginning with games of chance and is still most often discussed in those terms. Consider again the experiment of flipping a coin one time. There are two possible out- comes: head and tail. Each of these is equally likely. Thus using the classical assessment method the probability of a head is the ratio of the number of ways a head can occur 1 way to the total number of ways any outcome can occur 2 ways. Thus we get P. Head way ways 1 2 1 2 050 The chance of a head occurring is 1 out of 2 or 0.50. In those situations in which all possible outcomes are equally likely the classical prob- ability measurement is defined in Equation 1. Classical Probability Assessment The method of determining probability based on the ratio of the number of ways an outcome or event of interest can occur to the number of ways any outcome or event can occur when the individual outcomes are equally likely. Classical Probability Assessment PE E i i Number of ways can occur Total numbe er of possible outcomes 1 EXAMPLE 5 CLASSICAL PROBABILITY ASSESSMENT Active Sporting Goods Inc. The managers at Active Sport- ing Goods plan to hold a special promotion over Labor Day Weekend. Each customer making a purchase exceeding 100 will qualify to select an envelope from a large drum. Inside the enve- lope are coupons for percentage discounts off the purchase total. At the beginning of the weekend there were 500 coupons. Four hundred of these were for a 10 discount 50 were for 20 45 were for 30 and 5 were for 50. The probability of getting a particular discount amount can be determined using classical assessment with the following steps: Step 1 Define the experiment. An envelope is selected from a large drum. Step 2 Determine whether the possible outcomes are equally likely. In this case the envelopes with the different discount amounts are unmarked from the outside and are thoroughly mixed in the drum. Thus any one envelope has the same probability of being selected as any other envelope. The outcomes are equally likely. Step 3 Determine the total number of outcomes. There are 500 envelopes in the drum. Step 4 Define the event of interest. We might be interested in assessing the probability that the first customer will get a 20 discount. Step 5 Determine the number of outcomes associated with the event of interest. There are 50 coupons with a discount of 20 marked on them. Step 6 Compute the classical probability using Equation 1: PE E i i Number of ways can occur Total numbe er of possible outcomes discount Num P 20 b ber of ways can occur Total number of p 20 o ossible outcomes 50 500 0.10 Michael Flippo/Fotolia www.downloadslide.com

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Introduction to Probability Note: After the first customer selects an envelope from the drum the probability that the next customer will get a particular discount will change because the values in the denominator and possibly the numerator will change. END EXAMPLE TRY PROBLEM 10 As you can see the classical approach to probability measurement is fairly straightforward. Many games of chance are based on classical probability assessment. However classical prob- ability assessment does not apply in many business situations. Rarely are the individual out- comes equally likely. For instance you might be thinking of starting a business. The  sample space is SS 5Succeed Fail6 Would it be reasonable to use classical assessment to determine the probability that your busi- ness will succeed If so we would make the following assessment: PSucceed 1 2 If this were true then the chance of any business succeeding would be 0.50. Of course this is not true. Many factors go into determining the success or failure of a business. The possible outcomes Succeed Fail are not equally likely. Instead we need another method of probabil- ity assessment in these situations. Relative Frequency Assessment The relative frequency assessment approach is based on actual observations. Equation 2 shows how the relative frequency assessment method is used to assess probabilities. Relative Frequency Assessment The method that defines probability as the number of times an event occurs divided by the total number of times an experiment is performed in a large number of trials. Relative Frequency Assessment PE E N i i Number of times occurs 2 where: E i The event of interest N Number of trials BUSINESS APPLICATION RELATIVE FREQUENCY ASSESSMENT HATHAWAY HEATING AIR CONDITIONING The sales manager at Hathaway Heating Air Conditioning has recently developed the customer profile shown in Table 1. The profile is based on a random sample of 500 customers. As a promotion for the company the sales manager plans to randomly select a customer once a month and perform a free service on the customer’s system. What is the probability that the first customer selected is a residential customer What is the probability that the first customer has a Hathaway heating system TABLE 1 | Hathaway Heating Air Conditioning Co. Customer Category E 1 E 2 Commercial Residential Total E 3 Heating Systems 55 145 200 E 4 Air-Conditioning Systems 45 255 300 Total 100 400 500 www.downloadslide.com

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Introduction to Probability To determine the probability that the customer selected is residential we determine from Table 1 the number of residential customers and divide by the total number of customers both residential and commercial. We then apply Equation 2: PE P . 2 400 500 080 Residential Thus there is an 80 chance the customer selected will be a residential customer. The probability that the customer selected has a Hathaway heating system is determined by the ratio of the number of customers with heating systems to the number of total customers. PE P . 3 200 500 040 Heating There is a 40 chance the randomly selected customer will have a Hathaway heating system. The sales manager hopes the customer selected is a residential customer with a Hatha- way heating system. Because there are 145 customers in this category the relative frequency method assesses the probability of this event occurring as follows: PE E P 23 145 and Residential with heating 5 500 029 . There is a 29 chance the customer selected will be a residential customer with a Hathaway heating system. EXAMPLE 6 RELATIVE FREQUENCY PROBABILITY ASSESSMENT Starbucks Coffee The international coffee chain Starbucks has a store in a busy mall in Pittsburgh Pennsylvania. Starbucks sells caffeinated and decaffeinated drinks. One of the difficulties in this business is determining how much of a given product to prepare for the day. The manager is interested in determining the probability that a customer will select a decaf versus a caffeinated drink. She has maintained records of customer purchases for the past three weeks. The probability can be assessed using relative frequency with the following steps: Step 1 Define the experiment. A randomly chosen customer will select between decaf and caffeinated. Step 2 Define the events of interest. The manager is interested in the event E 1 customer selects caffeinated. Step 3 Determine the total number of occurrences. In this case the manager has observed 2250 sales of decaf and caffeinated in the past week. Thus N 2250. Step 4 For the event of interest determine the number of occurrences. In the past week 1570 sales were for caffeinated drinks. Step 5 Use Equation 2 to determine the probability assessment. PE E N Number of times occurs 1 1 1 570 2 250 0 0 6978 . Thus based on past history the chance that a customer will purchase a caffeinated drink is just under 0.70. END EXAMPLE TRY PROBLEM 9 POTENTIAL ISSUES WITH THE RELATIVE FREQUENCY ASSESSMENT METHOD There are a couple of concerns that you should be aware of before applying the relative frequency assessment method. First for this method to be useful all of the observed frequencies must be comparable. For instance consider again the case in which www.downloadslide.com

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Introduction to Probability you are interested in starting a small business. Two outcomes can occur: business succeeds or business fails. If we are interested in the probability that the business will succeed we might be tempted to study a sample of say 200 small businesses that have been started in the past and determine the number of those that have succeeded—say 50. Using Equation 2 for the relative frequency method we get P. Succeed 50 200 025 However before we can conclude the chance your small business will succeed is 0.25 you must be sure that the conditions of each of the 200 businesses match your conditions that is location type of business management expertise and experience financial standing and so on. If not then the relative frequency method should not be used. Another issue involves the size of the denominator in Equation 2. If the number of pos- sible occurrences is quite small the probability assessment may be unreliable. For instance suppose a basketball player took five free throws during the first ten games of the season and missed them all. The relative frequency method would determine the probability that he will make the next free throw to be P. Make made 5 shots 00 5 00 But do you think that there is a zero chance that he will make his next free throw No even the notoriously poor free-throw shooter Shaquille O’Neal former National Basketball Association NBA star player made some of his free throws. The problem is that the base of five free throws is too small to provide a reliable probability assessment. Subjective Probability Assessment Unfortunately even though managers may have some experience to guide their decision making new factors will always be affecting each decision making that experience only an approximate guide to the future. In other cases managers may have little or no experience and therefore may not be able to use a relative frequency as even a starting point in assessing the desired probability. When experience is not available decision makers must make a subjective probability assessment. A subjective probability is a measure of a personal conviction that an outcome will occur. Therefore in this instance probability represents a person’s belief that an event will occur. BUSINESS APPLICATION SUBJECTIVE PROBABILITY ASSESSMENT BECHTEL CORPORATION The Bechtel Corporation is preparing a bid for a major infrastructure construction project. The company’s engineers are very good at defining all the elements of the projects labor materials and so on and know the costs of these with a great deal of certainty. In finalizing the bid amount the managers add a profit markup to the projected costs. The problem is how much markup to add. If they add too much they won’t be the low bidder and may lose the contract. If they don’t mark the bid up enough they may get the project and make less profit than they might have made had they used a higher markup. The managers are considering four possible markup values stated as percentages of base costs: 10 12 15 20 To make their decision the managers need to assess the probability of winning the con- tract at each of these markup levels. Because they have never done a project exactly like this one they can’t rely on relative frequency assessment. Instead they must subjectively assess the probability based on whatever information they currently have available such as who the other bidders are the rapport Bechtel has with the potential client and so forth. After considering these values the Bechtel managers make the following assessments: P1Win at 102 0.30 P1Win at 122 0.25 P1Win at 152 0.15 P1Win at 202 0.05 Subjective Probability Assessment The method that defines probability of an event as reflecting a decision maker’s state of mind regarding the chances that the particular event will occur. www.downloadslide.com

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Introduction to Probability These assessments indicate the managers’ state of mind regarding the chances of winning the contract. If new information for example a competitor drops out of the bidding be- comes available before the bid is submitted these assessments could change. Each of the three methods by which probabilities are assessed has specific advantages and specific applications. Regardless of how decision makers arrive at a probability assess- ment the rules by which people use these probabilities in decision making are the same. These rules will be introduced in Section 2. MyStatLab Skill Development 4-1. A special roulette wheel which has an equal number of red and black spots has come up red four times in a row. Assuming that the roulette wheel is fair what concept allows a player to know that the probability the next spin of the wheel will come up black is 0.5 4-2. In a survey respondents were asked to indicate their favorite brand of cereal Post or Kellogg’s. They were allowed only one choice. What is the probability concept that implies it is not possible for a single respondent to state both Post and Kellogg’s to be the favorite cereal 4-3. If two customers are asked to list their choice of ice cream flavor from among vanilla chocolate and strawberry list the sample space showing the possible outcomes. 4-4. Use a tree diagram to list the sample space for the number of movies rented by three customers at a video store where customers are allowed to rent one two or three movies assuming that each customer rents at least one movie. 4-5. In each of the following indicate what method of probability assessment would most likely be used to assess the probability. a. What is the probability that a major earthquake will occur in California in the next three years b. What is the probability that a customer will return a purchase for a refund c. An inventory of appliances contains four white washers and one black washer. If a customer selects one at random what is the probability that the black washer will be selected 4-6. Long-time friends Pat and Tom agree on many things but not the outcome of the American League pennant race and the World Series. Pat is originally from Boston and Tom is from New York. They have a steak dinner bet on next year’s race with Pat betting on the Red Sox and Tom on the Yankees. Both are convinced they will win. a. What probability assessment technique is being used by the two friends b. Why would the relative frequency technique not be appropriate in this situation 4-7. Students who live on campus and purchase a meal plan are randomly assigned to one of three dining halls: the Commons Northeast and Frazier. What is the probability that the next student to purchase a meal plan will be assigned to the Commons 4-8. The results of a census of 2500 employees of a mid- sized company with 401k retirement accounts are as follows: Account Balance to nearest Male Female 6+25000 635 495 +25000 - +49999 185 210 +50000 - +99999 515 260 7+100000 155 45 Suppose researchers are going to sample employees from the company for further study. a. Based on the relative frequency assessment method what is the probability that a randomly selected employee will be a female b. Based on the relative frequency assessment method what is the probability that a randomly selected employee will have a 401k account balance of between 25000 and 49999 c. Compute the probability that a randomly selected employee will be a female with an account balance between 50000 and 99999. 4-9. Cross County Bicycles makes two mountain bike models the XB-50 and the YZ-99 in three distinct colors. The following table shows the production volumes for last week: Color Model Blue Brown White XB-50 302 105 200 YZ-99 40 205 130 a. Based on the relative frequency assessment method what is the probability that a mountain bike is brown b. What is the probability that the mountain bike is a YZ-99 4-1: Exercises www.downloadslide.com

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Introduction to Probability CEO stating that the probability the company will earn a profit in excess of 20 million next year is 80. Comment on this probability assessment. 4-13. Five doctors work at the Evergreen Medical Clinic. The plan is to staff Saturdays with three doctors. The office manager has decided to make up Saturday schedules in such a way that no set of three doctors will be in the office together more than once. How many weeks can be covered by this schedule Hint: Use a tree diagram to list the sample space. 4-14. Prince Windows Inc. makes high-quality windows for the residential home market. Recently three marketing managers were asked to assess the probability that sales for next year will be more than 15 higher than the current year. One manager stated that the probability of this happening was 0.40. The second manager assessed the probability to be 0.60 and the third manager stated the probability to be 0.90. a. What method of probability assessment are the three managers using b. Which manager is expressing the least uncertainty in the probability assessment c. Why is it that the three managers did not provide the same probability assessment 4-15. The marketing manager for the Charlotte Times newspaper has commissioned a study of the advertisements in the classified section. The results for the Wednesday edition showed that 204 are help-wanted ads 520 are real estate ads and 306 are other ads. a. If the newspaper plans to select an ad at random each week to be published free what is the probability that the ad for a specific week will be a help-wanted ad b. What method of probability assessment is used to determine the probability in part a c. Are the events that a help-wanted ad is chosen and that an ad for other types of products or services is chosen for this promotion on a specific week mutually exclusive Explain. 4-16. Before passing away in 2009 Larry Miller owned the Utah Jazz basketball team of the NBA and several automobile dealerships in Utah and Idaho. One of the dealerships sells Buick Cadillac and Pontiac automobiles. It also sells used cars that it gets as trade- ins on new car purchases. Supposing two cars are sold on Tuesday by the dealership what is the sample space for the type of cars that might be sold 4-17. The Pacific Northwest has a substantial volume of cedar forests and cedar product manufacturing companies. Welco Lumber manufactures cedar fencing material in Marysville Washington. The company’s quality manager inspected 5900 boards and found that 4100 could be rated as a 1 grade. a. If the manager wanted to assess the probability that a board being produced will be a 1 grade what method of assessment would he likely use b. Referring to your answer in part a what would you assess the probability of a 1 grade board to be c. What is the joint probability that a randomly selected mountain bike is a YZ-99 and brown d. Suppose a mountain bike is chosen at random. Consider the following two events: the event that model YZ-99 is chosen and the event that a white product is chosen. Are these two events mutually exclusive Explain. 4-10. Cyber-Plastics Inc. is in search of a CEO and a CFO. The company has a short list of candidates for each position. The CEO candidates graduated from Chicago C and three Ivy League universities: Harvard H Princeton P and Yale Y. The four CFO candidates graduated from MIT M Northwestern N and two Ivy League universities: Dartmouth D and Brown B. One candidate from each of the respective lists will be chosen randomly to fill the positions. The event of interest is that both positions are filled with candidates from the Ivy League. a. Determine whether the outcomes are equally likely. b. Determine the number of equally likely outcomes. c. Define the event of interest. d. Determine the number of outcomes associated with the event of interest. e. Compute the classical probability of the event of interest using Equation 1. 4-11. Three consumers go to a Best Buy to shop for high- definition televisions HDTVs. Let B indicate that one of the consumers buys an HDTV . Let D be that the consumer doesn’t buy an HDTV . Assume these events are equally likely. Consider the following: 1 only two consumers buy an HDTV 2 at most two consumers buy HDTVs and 3 at least two consumers buy HDTVs. a. Determine whether the outcomes 1 2 and 3 are equally likely. b. Determine the total number of equally likely outcomes for the three shoppers. c. Define the events of interest in each of 1 2 and 3. To define the events of interest list the possible outcomes in each of the following events: . only two consumers buy an HDTV 1E 1 2 . at most two consumers buy HDTVs 1E 2 2 . at least two consumers buy HDTVs 1E 3 2 d. Determine the number of outcomes associated with each of the events of interest. Use the classical probability assessment approach to assign probabilities to each of the possible outcomes and calculate the probabilities of the events. e. Compute the classical probabilities of each of the events in part d by using Equation 1. Business Applications 4-12. Cyber Communications Inc. has a new cell phone product under development in the research and development RD lab. It will increase the megapixel capability of cell phone cameras to the 6+ range. The head of RD made a presentation to the company www.downloadslide.com

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Introduction to Probability 4-18. The results of Fortune Personnel Consultants’ survey of 405 workers was reported in USA Today. One of the questions in the survey asked “Do you feel it’s OK for your company to monitor your Internet use” The possible responses were: 1 Only after informing me 2 Does not need to inform me 3 Only when company believes I am misusing 4 Company does not have right and 5 Only if I have previously misused. The following table contains the results for the 405 respondents: Response 12 3 4 5 Number of Respondents 223 130 32 14 6 a. Calculate the probability that a randomly chosen respondent would indicate that there should be some restriction concerning the company’s right to monitor Internet use. b. Indicate the method of probability assessment used to determine the probability in part a. c. Are the events that a randomly selected respondent chose response 1 and that another randomly selected respondent chose response 2 independent mutually exclusive or dependent events Explain. 4-19. Famous Dave’s is a successful barbeque chain and sells its beef pork and chicken items to three kinds of customers: dine-in delivery and carry-out. Last year’s sales showed that 12753 orders were dine-in 5893 were delivery orders and 3122 orders were carry-out. Suppose an audit of last year’s sales is being conducted. a. If a customer order is selected at random what is the probability it will be a carry-out order b. What method of probability assessment was used to determine the probability in part a c. If two customer orders are selected at random list the sample space indicating the type of order for both customers. 4-20. VERCOR provides merger and acquisition consultants to assist corporations when an owner decides to offer the business for sale. One of their news releases “Tax Audit Frequency Is Rising” written by David L. Perkins Jr. a VERCOR partner originally appeared in The Business Owner . Perkins indicated that audits of the largest businesses those corporations with assets of 10 million and over climbed to 9560 in the previous year. That was up from a low of 7125 a year earlier. He indicated one in six large corporations was being audited. a. Designate the type of probability assessment method that Perkins used to assess the probability of large corporations being audited. b. Determine the number of large corporations that filed tax returns for the previous fiscal year. c. Determine the probability that a large corporation was not audited using the relative frequency probability assessment method. Computer Database Exercises 4-21. According to a September 2005 article on the Womensenews.org Web site “Caesarean sections in which a baby is delivered by abdominal surgery have increased fivefold in the past 30 years prompting concern among health advocates . . .” The data in the file called Babies indicate whether the past 50 babies delivered at a local hospital were delivered using the Caesarean method. a. Based on these data what is the probability that a baby born in this hospital will be born using the Caesarean method b. What concerns might you have about using these data to assess the probability of a Caesarean birth Discuss. 4-22. Recently a large state university conducted a survey of undergraduate students regarding their use of computers. The results of the survey are contained in the data file ComputerUse. a. Based on the data from the survey what is the probability that undergraduate students at this university will have a major that requires them to use a computer on a daily basis b. Based on the data from this survey if a student is a business major what is the probability of the student believing that the computer lab facilities are very adequate 4-23. A company produces scooters used by small businesses such as pizza parlors that find them convenient for making short deliveries. The company is notified whenever a scooter breaks down and the problem is classified as being either mechanical or electrical. The company then matches the scooter to the plant where it was assembled. The file Scooters contains a random sample of 200 breakdowns. Use the data in the file and the relative frequency assessment method to find the following probabilities: a. What is the probability a scooter was assembled at the Tyler plant b. What is the probability that a scooter breakdown was due to a mechanical problem c. What is the probability that a scooter was assem bled at the Lincoln plant and had an electrical problem 4-24. A Harris survey on cell phone use asked in part what was the most important reason that people give for not using a wireless phone exclusively. The responses were: 1 Like the safety of traditional phone 2 Need line for Internet access 3 Pricing not attractive enough 4 Weak or unreliable cell signal at home 5 Coverage not good enough and 6 Other. The file titled Wireless contains the responses for the 1088 respondents. a. Calculate the probability that a randomly chosen respondent would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone. b. Calculate the probability that a randomly chosen person would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone and is over the age of 55. www.downloadslide.com

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Introduction to Probability c. Determine the probability that a randomly chosen person would not use a wireless phone exclusively because of a perceived need for Internet access and the safety of a traditional phone. d. Of those respondents under 36 determine the probability that an individual in this age group would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone. 4-25. CNN staff writer Pariia Bhatnagar reported “Coke Pepsi Losing the Fizz” March 8 2005 that Atlanta- based Coke saw its domestic market share drop to 43.1 in 2004. New York-based PepsiCo had used its “Pepsi Challenge” advertising approach to increase its market share which stood at 31.7 in 2004. A selection of soft-drink users is asked to taste the two disguised soft drinks and indicate which they prefer. The file titled Challenge contains the results of a simulated Pepsi Challenge on a college campus. a. Determine the probability that a randomly chosen student prefers Pepsi. b. Determine the probability that one of the students prefers Pepsi and is less than 20 years old. c. Of those students who are less than 20 years old calculate the probability that a randomly chosen student prefers 1 Pepsi and 2 Coke. d. Of those students who are at least 20 years old calculate the probability that a randomly chosen student prefers 1 Pepsi and 2 Coke. END EXERCISES 4-1 2 The Rules of Probability Measuring Probabilities The probability attached to an event represents the likelihood the event will occur on a speci- fied trial of an experiment. This probability also measures the perceived uncertainty about whether the event will occur. Possible Values and the Summation of Possible Values If we are certain about the outcome of an event we will assign the event a probability of 0 or 1 where P1E i 20 indicates the e v ent E i will not occur and P1E i 21 means that E i will definitely occur. 1 If we are uncertain about the result of an experiment we measure this uncertainty by assigning a probability between 0 and 1. Probability Rule 1 shows that the probability of an event occur- ring is always between 0 and 1. 1 These statements are true only if the number of outcomes of an experiment is countable. They do not apply when the number of outcomes is infinitely uncountable. This will be discussed when continuous probability distributions are discussed. Probability Rule 2 Pe i i k 1 1 S 4 where: k Number of outcomes in the sample e i ith outcome Probability Rule 1 For any event E i 0 … P1E i 2 … 1 for all i 3 All possible outcomes associated with an experiment form the sample space. Therefore the sum of the probabilities of all possible outcomes is 1 as shown by Probability Rule 2. Addition Rule for Individual Outcomes If a single event is composed of two or more individual outcomes then the probability of the event is found by summing the probabilities of the individual outcomes. This is illustrated by Probability Rule 3. Chapter Outcome 2. www.downloadslide.com

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Introduction to Probability Probability Rule 3: Addition Rule for Individual Outcomes The probability of an event E i is equal to the sum of the probabilities of the individual outcomes forming E i . For example if E i 5e 1 e 2 e 3 6 then P1E i 2 P1e 1 2 + P1e 2 2 + P1e 3 2 5 BUSINESS APPLICATION ADDITION RULE GOOGLE Google has become synonymous with Web searches and is clearly the market leader in the search engine marketplace. Officials at the northern California headquarters have recently performed a survey of computer users to determine how many Internet searches individuals do daily using Google. Table 2 shows the results of the survey of Internet users. The sample space for the experiment for each respondent is SS 5e 1 e 2 e 3 e 4 6 where the possible outcomes are e 1 at least 10 searches e 2 3 to 9 searches e 3 1 to 2 searches e 4 0 searches Using the relative frequency assessment approach we assign the following probabilities. P1e 1 2 4005000 0.08 P1e 2 2 19005000 0.38 P1e 3 2 15005000 0.30 P1e 4 2 12005000 0.24 g 1.00 Assume we are interested in the event respondent performs 1 to 9 searches per month. E Internet User Performs 1 to 9 searches per day The outcomes that make up E are E 5e 2 e 3 6 We can find the probability PE by using Probability Rule 3 Equation 5 as follows: P1E2 P1e 2 2 + P1e 3 2 0.38 + 0.30 0.68 TABLE 2 | Google’s Survey Results Searches Per Day Frequency Relative Frequency at least 10 400 0.08 3 to 9 1900 0.38 1 to 2 1500 0.30 0 1200 0.24 Total 5000 1.00 www.downloadslide.com

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Introduction to Probability EXAMPLE 7 THE ADDITION RULE FOR INDIVIDUAL OUTCOMES KFI 640 Radio The KFI 640 radio station in Los Angeles is a combination news/talk and “oldies” station. During a 24-hour day a listener can tune in and hear any of the following four programs being broadcast: “Oldies” music News stories Talk programming Commercials Recently the station has been having trouble with its transmitter. Each day the station’s signal goes dead for a few seconds it seems that these outages are equally likely to occur at any time during the 24-hour broadcast day. There seems to be no pattern regarding what is playing at the time the transmitter problem occurs. The station manager is concerned about the probability that these problems will occur during either a news story or a talk program. Step 1 Define the experiment. The station conducts its broadcast starting at 12:00 midnight extending until a transmitter outage is observed. Step 2 Define the possible outcomes. The possible outcomes are the type of programming that is playing when the transmitter outage occurs. There are four possible outcomes: e 1 Oldies e 2 News e 3 Talk programs e 4 Commercials Step 3 Determine the probability of each possible outcome. The station manager has determined that out of the 1440 minutes per day 540 minutes are oldies 240 minutes are news 540 minutes are talk programs and 120 minutes are commercials. Therefore the probability of each type of programming being on at the moment the outage occurs is assessed as follows: Outcome e i P1e i 2 e 1 Oldies Pe . 1 540 1 440 0375 e 2 News Pe . 2 240 1 440 0 167 e 3 Talk programs Pe . 3 540 1 440 0 375 e 4 Commercials Pe . 4 120 1 440 0 083 a 1.000 Note based on Equation 4 Probability Rule 2 the sum of the probabilities of the individual possible outcomes is 1.0. Step 4 Define the event of interest. The event of interest is a transmitter problem occurring during a news or talk program. This is E 5e 2 e 3 6 Step 5 Use Probability Rule 3 Equation 5 to compute the desired probability. P1E2 P1e 2 2 + P1e 3 2 P1E2 0.167 + 0.375 P1E2 0.542 Tsian/Fotolia www.downloadslide.com

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Introduction to Probability Thus there is slightly higher than a 0.5 probability that when a transmitter problem occurs it will happen during either a news or talk program. END EXAMPLE TRY PROBLEM 26 Complement Rule Closely connected with Probability Rules 1 and 2 is the complement of an event. The complement of event E is represented by E. The Complement Rule is a corol- lary to Probability Rules 1 and 2. Complement The complement of an event E is the collection of all possible outcomes not contained in event E. Complement Rule PE PE 1 6 That is the probability of the complement of event E is 1 minus the probability of event E. EXAMPLE 8 THE COMPLEMENT RULE Capital Consulting The managing partner for Capital Consulting is working on a proposal for a consulting project with a client in Sydney Australia. The manager lists four possible net profits from the consulting engagement and his subjectively assessed probabilities related to each profit level. Outcome POutcome 0 0.70 2000 0.20 15000 0.07 50000 0.03 g 1.00 Note that each probability is between 0 and 1 and that the sum of the probabilities is 1 as required by Rules 1 and 2. The manager plans to submit the proposal if the consulting engagement will have a posi- tive profit so he is interested in knowing the probability of an outcome greater than 0. This probability can be found using the Complement Rule with the following steps: Step 1 Determine the probabilities for the outcomes. P1+02 0.70 P1+20002 0.20 P1+150002 0.07 P1+500002 0.03 Step 2 Find the desired probability. Let E be the consulting outcome event +0. The probability of the zero outcome is P1E2 0.70 The complement E is all investment outcomes greater than 0. Using the Complement Rule the probability of profit greater than 0 is P1Profit 7 +02 1 - P1Profit +02 P1Profit 7 +02 1 - 0.70 P1Profit 7 +02 0.30 Based on his subjective probability assessment there is a 30 chance the consulting project will have a positive profit. END EXAMPLE TRY PROBLEM 32 www.downloadslide.com

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Introduction to Probability Addition Rule for Any Two Events BUSINESS APPLICATION ADDITION RULE GOOGLE CONTINUED Suppose the staff who conducted the survey for Google discussed earlier also asked questions about the computer users’ ages. The Google managers consider age important in designing their search engine methodologies. Table 3 shows the breakdown of the sample by age group and by the number of times a user performs a search each day. Table 3 shows that there are seven events defined. For instance E 1 is the event that a com- puter user performs 10 or more searches per day. This event is composed of three individual outcomes associated with the three age categories. These are E 1 5e 1 e 2 e 3 6 In another case event E 5 corresponds to a survey respondent being younger than 30 years of age. It is composed of four individual outcomes associated with the four levels of search activity. These are E 5 5e 1 e 4 e 7 e 10 6 Table 3 illustrates two important concepts in data analysis: joint frequencies and mar- ginal frequencies. Joint frequencies are the values inside the table. They provide information on age group and search activity jointly. Marginal frequencies are the row and column totals. These values give information on only the age group or only Google search activity. For example 2100 people in the survey are in the 30- to 50-year age group. This column total is a marginal frequency for the age group 30 to 50 years which is represented by E 6 . Now notice that 600 respondents are younger than 30 years old and perform three to nine searches a day. The 600 is a joint frequency whose outcome is represented by e 4 . The joint frequencies are the number of times their associated outcomes occur. Table 4 shows the relative frequencies for the data in Table 3. These values are the prob- abilities of the events and outcomes. Suppose we wish to find the probability of E 4 0 searches or E 6 being in the 30-to-50 age group. That is P1E 4 or E 6 2 Chapter Outcome 2. TABLE 3 | Google Search Study Age Group Searches Per Day E 5 Less than 30 E 6 30 to 50 E 7 Over 50 Total E 1 Ú 10 Searches e 1 200 e 2 100 e 3 100 400 E 2 3 to 9 Searches e 4 600 e 5 900 e 6 400 1900 E 3 1 to 2 Searches e 7 400 e 8 600 e 9 500 1500 E 4 0 Searches e 10 700 e 11 500 e 12 0 1200 Total 1900 2100 1000 5000 TABLE 4 | Google—Joint Probability Table Age Group Searches per Day E 5 Less than 30 E 6 30 to 50 E 7 Over 50 Total E 1 Ú 10 Searches e 1 2005000 0.04 e 2 1005000 0.02 e 3 1005000 0.02 4005000 0.08 E 2 3 to 9 Searches e 4 6005000 0.12 e 5 9005000 0.18 e 6 4005000 0.08 19005000 0.38 E 3 1 to 2 Searches e 7 4005000 0.08 e 8 6005000 0.12 e 9 5005000 0.10 15005000 0.30 E 4 0 Searches e 10 7005000 0.14 e 11 5005000 0.10 e 12 05000 0.00 12005000 0.24 Total 19005000 0.38 21005000 0.42 10005000 0.20 50005000 1.00 www.downloadslide.com

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Introduction to Probability FIGURE 1 |  Venn Diagram—Addition Rule for Any Two Events E 1 E 2 E 1 and E 2 PE 1 or E 2 PE 1 + PE 2 – PE 1 and E 2 To fnd this probability we must use Probability Rule 4. Probability Rule 4: Addition Rule for Any Two Events E 1 and E 2 P1E 1 or E 2 2 P1E 1 2 + P1E 2 2 - P1E 1 and E 2 2 7 The key word in knowing when to use Rule 4 is or. The word or indicates addition. You may have covered this concept as a union in a math class. P1E 1 or E 2 2 P1E 1 h E 2 2. Figure 1 is a Venn diagram that illustrates the application of the Addition Rule for Any Two Events. Notice that the probabilities of the outcomes in the overlap between the two events E 1 and E 2 are double-counted when the probabilities of the outcomes in E 1 are added to those of E 2 . Thus the probabilities of the outcomes in the overlap which is E 1 and E 2 need to be subtracted to avoid the double counting. Referring to the Google situation the probability of E 4 0 searches or E 6 being in the 30-to-50 age group is P1E 4 or E 6 2 Table 5 shows the relative frequencies with the events of interest shaded. The overlap corre- sponds to the joint occurrence intersection of conducting 0 searches and being in the 30-to- 50 age group. The probability of the outcomes in the overlap is represented by P1E 4 and E 6 2 and must be subtracted. This is done to avoid double-counting the probabilities of the out- comes that are in both E 4 and E 6 when calculating the P1E 4 or E 6 2. Thus P1E 4 or E 6 2 P1E 4 2 + P1E 6 2-P1E 4 and E 6 2 0.24 + 0.42 - 0.10 0.56 Therefore the probability that a respondent will either be in the 30-to-50 age group or per- form zero searches on a given day is 0.56. TABLE 5 | Google Searches—Addition Rule Example Age Group Searches Per Day E 5 Less than 30 E 6 30 to 50 E 7 Over 50 Total E 1 Ú 10 Searches e 1 2005000 0.04 e 2 1005000 0.02 e 3 1005000 0.02 4005000 0.08 E 2 3 to 9 Searches e 4 6005000 0.12 e 5 9005000 0.18 e 6 4005000 0.08 19005000 0.38 E 3 1 to 2 Searches e 7 4005000 0.08 e 8 6005000 0.12 e 9 5005000 0.10 15005000 0.30 E 4 0 Searches e 10 7005000 0.14 e 11 5005000 0.10 e 12 05000 0.00 12005000 0.24 Total 19005000 0.38 21005000 0.42 10005000 0.20 50005000 1.00 www.downloadslide.com

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Introduction to Probability What is the probability a respondent will perform 1 to 2 searches or be in the over-50 age group Again we can use Rule 4: P1E 3 or E 7 2 P1E 3 2 + P1E 7 2 - P1E 3 and E 7 2 Table 6 shows the relative frequencies for these events. We have P1E 3 or E 7 2 0.30 + 0.20 - 0.10 0.40 Thus there is a 0.40 chance that a respondent will perform 1- 2 searches or be in the “over 50” age group. EXAMPLE 9 ADDITION RULE FOR ANY TWO EVENTS British Columbia Forest Products British Columbia Forest Products manufactures lum- ber for large material supply centers like Home Depot and Lowe’s in the U.S. and Canada. A representative from Home Depot is due to arrive at the BC plant for a meeting to discuss lumber quality. When the Home Depot representative arrives he will ask BC’s managers to randomly select one board from the finished goods inventory for a quality check. Boards of three dimensions and three lengths are in the inventory. The following chart shows the num- ber of boards of each size and length. Dimension Length E 4 2″ 4″ E 5 2″ 6″ E 6 2″ 8″ Total E 1 8 feet 1400 1500 1100 4000 E 2 10 feet 2000 3500 2500 8000 E 3 12 feet 1600 2000 2400 6000 Total 5000 7000 6000 18000 The BC manager will be selecting one board at random from the inventory to show the Home Depot representative. Suppose he is interested in the probability that the board selected will be 8 feet long or a 2″ 6″. To find this probability he can use the following steps: Step 1 Define the experiment. One board is selected from the inventory and its dimensions are obtained. Step 2 Define the events of interest. The manager is interested in boards that are 8 feet long. E 1 8 foot boards He is also interested in the 2″ 6″ dimension so E 5 2″ 6″ boards Step 3 Determine the probability for each event. There are 18000 boards in inventory and 4000 of these are 8 feet long so PE . 1 4 000 18 000 0 2222 TABLE 6 | Google—Addition Rule Example Age Group Searches Per Day E 5 Less than 30 E 6 30 to 50 E 7 Over 50 Total E 1 Ú 10 Searches e 1 2005000 0.04 e 2 1005000 0.02 e 3 1005000 0.02 4005000 0.08 E 2 3 to 9 Searches e 4 6005000 0.12 e 5 1005000 0.18 e 6 4005000 0.08 19005000 0.38 E 3 1 to 2 Searches e 7 4005000 0.08 e 8 600/5000 0.12 e 9 5005000 0.10 15005000 0.30 E 4 0 Searches e 10 7005000 0.14 e 11 5005000 0.10 e 12 05000 0.00 12005000 0.24 Total 19005000 0.38 21005000 0.42 10005000 0.20 50005000 1.00 www.downloadslide.com

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Introduction to Probability Of the 18000 boards 7000 are 2″ 6″ so the probability is PE . 5 7 000 18 000 0 3889 Step 4 Determine whether the two events overlap and if so compute the joint probability. Of the 18000 total boards 1500 are 8 feet long and 2″ 6″. Thus the joint probability is PE E . 15 1 500 18 000 0 0833 and Step 5 Compute the desired probability using Probability Rule 4. P1E 1 or E 5 2 P1E 1 2 + P1E 5 2-P1E 1 and E 5 2 P1E 1 or E 5 2 0.2222 + 0.3889 - 0.0833 0.5278 The chance of selecting an 8-foot board or a 2″ 6″ board is just under 0.53. END EXAMPLE TRY PROBLEM 31 Addition Rule for Mutually Exclusive Events We indicated previously that when two events are mutually exclusive both events cannot occur at the same time. Thus for mutually exclusive events P1E 1 and E 2 2 0 Therefore when you are dealing with mutually exclusive events the Addition Rule assumes a different form shown as Rule 5. Probability Rule 5: Addition Rule for Mutually Exclusive Events For two mutually exclusive events E 1 and E 2 P1E 1 or E 2 2 P1E 1 2 + P1E 2 2 8 Figure 2 is a Venn diagram illustrating the application of the Addition Rule for Mutually Exclusive Events. Conditional Probability In dealing with probabilities you will often need to determine the chances of two or more events occurring either at the same time or in succession. For example a quality control man- ager for a manufacturing company may be interested in the probability of selecting two suc- cessive defective products from an assembly line. If the probability of this event is low the quality control manager will be surprised when it occurs and might readjust the production process. In other instances the decision maker might know that an event has occurred and may then want to know the probability of a second event occurring. For instance suppose that an oil company geologist who believes oil will be found at a certain drilling site makes a favorable report. Because oil is not always found at locations with a favorable report the oil FIGURE 2 |  Venn Diagram—Addition Rule for Two Mutually Exclusive Events E 1 E 2 PE 1 or E 2 PE 1 + PE 2 www.downloadslide.com

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Introduction to Probability company’s exploration vice president might well be interested in the probability of finding oil given the favorable report. Situations such as this refer to a probability concept known as conditional probability. Probability Rule 6 offers a general rule for conditional probability. The notation P1E 1 E 2 2 reads “probability of event E 1 given event E 2 has occurred.” Thus the probability of one event is conditional upon a second event having occurred. Conditional Probability The probability that an event will occur given that some other event has already happened. Probability Rule 6: Conditional Probability for Any Two Events For any two events E 1 E 2 PE E PE E PE | 12 12 2 and 9 where: P1E 2 2 7 0 Rule 6 uses a joint probability P1E 1 and E 2 2 and a marginal probability P1E 2 2 to calculate the conditional probability P1E 1 E 2 2. Note that to find a conditional probability we find the ratio of how frequently E 1 occurs to the total number of observations given that we restrict our observations to only those cases in which E 2 has occurred. BUSINESS APPLICATION CONDITIONAL PROBABILITY SYRINGA NETWORKS Syringa Networks is an Internet service provider to rural areas in the western United States. The company has studied its customers’ Internet habits. Among the information collected are the data shown in Table 7. The company is focusing on high-volume users and one of the factors that will influence Syringa Networks’ marketing strategy is whether time spent using the Internet is related to a customer’s gender. For example suppose the company knows a user is female and wants to know the chances this woman will spend between 20 and 40 hours a month on the Internet. Let E 2 5e 3 e 4 6 Event: Person uses services 20 to 40 hours per month E 4 5e 1 e 3 e 5 6 Event: User is female A marketing analyst needs to know the probability of E 2 given E 4 . One way to find the desired probability is as follows: 1. We know E 4 has occurred customer is female. There are 850 females in the survey. 2. Of the 850 females 300 use Internet services 20 to 40 hours per month. 3. Then PE E | . 24 300 850 035 However we can also apply Rule 6 as follows: PE E PE E PE | 24 24 4 and TABLE 7 | Joint Frequency Distribution for Syringa Network Gender Hours per Month E 4 Female E 5 Male Total E 1 6 20 e 1 450 e 2 500 950 E 2 20 to 40 e 3 300 e 4 800 1100 E 3 7 40 e 5 100 e 6 350 450 Total 850 1650 2500 www.downloadslide.com

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Introduction to Probability Table 8 shows the relative frequencies of interest. From Table 8 we get the joint probability P1E 2 and E 4 2 0.12 and P1E 4 2 0.34 Then applying Rule 6 PE E | . . . 24 012 034 035 EXAMPLE 10 COMPUTING CONDITIONAL PROBABILITIES Retirement Planning Most financial publications suggest that the older the investor is the more conservative his or her investment strategy should be. For example younger investors might hold more in stocks while older investors might hold more in bonds. A recent survey conducted by a major financial publication yielded the following table which shows the num- ber of people in the study by age group and percentage of retirement funds in the stock market. Percentage of Retirement Investments in the Stock Market Age of Investor E 5 6 5 E 6 5 6 10 E 7 10 6 30 E 8 30 6 50 E 9 50 or more Total E 1 6 30 years 70 240 270 80 55 715 E 2 30 6 50 years 90 300 630 1120 1420 3560 E 3 50 6 65 years 110 305 780 530 480 2205 E 4 65+ years 200 170 370 260 65 1065 Total 470 1015 2050 1990 2020 7545 The publication’s editors are interested in knowing the probability that someone 65 or older will have 50 or more of retirement funds invested in the stock market. Assuming the data collected in this study reflect the population of investors the editors can find this conditional probability using the following steps: Step 1 Define the experiment. A randomly selected person age 65 or older has his or her portfolio analyzed for percentage of retirement funds in the stock market. Step 2 Define the events of interest. In this case we are interested in two events: E 4 At least 65 years old E 9 50 or more in stocks Step 3 Define the probability statement of interest. The editors are interested in P1E 9 E 4 2 Probability of 50 or more stocks given at least 65 years Step 4 Convert the data to probabilities using the relative frequency assessment method. We begin with the event that is given to have occurred 1E 4 2. A total of 1065 people in the study were at least 65 years of age. Of the 1065 people 65 had 50 or more of their retirement funds in the stock market. TABLE 8 | Joint Relative Frequency Distribution for Syringa Networks Gender Hours per Month E 4 Female E 5 Male Total E 1 6 20 e 1 4502500 0.18 e 2 5002500 0.20 9502500 0.38 E 2 20 to 40 e 3 3002500 0.12 e 4 8002500 0.14 11002500 0.44 E 3 7 40 e 5 1002500 0.14 e 6 3502500 0.14 4502500 0.18 Total 8502500 0.34 16502500 0.66 25002500 1.00 www.downloadslide.com

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Introduction to Probability PE E | . 94 65 1 065 0 061 Thus the conditional probability that someone at least 65 will have 50 or more of retirement assets in the stock market is 0.061. This value can be found using Step 5 as well. Step 5 Use Probability Rule 6 to find the conditional probability. PE E PE E PE | 94 94 4 and The necessary probabilities are found using the relative frequency assessment method: PE . 4 1 065 7 545 0 1412 and the joint probability is PE E . 94 65 7 545 0 0086 and Then using Probability Rule 6 we get PE E PE E PE | . . . 94 94 4 0 0086 0 1412 00 and 6 61 END EXAMPLE TRY PROBLEM 34 Tree Diagrams Another way of organizing the events of an experiment that aids in the calculation of probabilities is the tree diagram. BUSINESS APPLICATION USING TREE DIAGRAMS SYRINGA NETWORKS CONTINUED Figure 3 illustrates the tree diagram for Syringa Networks the Internet service provider discussed earlier. Note that the branches at each node in the tree diagram represent mutually exclusive events. Moving from left to right the first two branches indicate the two customer types male and female—mutually exclusive events. Three branches grow from each of these original branches representing the three possible categories for Internet use. The probabilities for the events male and female are shown on the first two branches. The probabilities shown on the right of the tree are the joint probabilities for each combination of gender and hours of use. These figures are found using Table 8 which was shown earlier. The probabilities on the branches following the male and female branches showing hours of use are conditional probabilities. For example we can find the probability that a male customer 1E 5 2 will spend more than 40 hours on the Internet 1E 3 2 by PE |E PE E PE . . . 35 35 5 014 066 0 2121 and Conditional Probability for Independent Events We previously discussed that two events are independent if the occurrence of one event has no bearing on the probability that the second event occurs. Therefore when two events are independent the rule for conditional probability takes a different form as indicated in Probability Rule 7. Probability Rule 7: Conditional Probability for Independent Events For independent events E 1 E 2 PE E P E P E | 12 1 2 0 10 and PE E P E P E | 21 2 1 0 www.downloadslide.com

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Introduction to Probability As Rule 7 shows the conditional probability of one event occurring given a second independ- ent event has already occurred is simply the probability of the event occurring. EXAMPLE 11 CHECKING FOR INDEPENDENCE British Columbia Forest Products In Example 9 the manager at the British Columbia Forest Products Company reported the following data on the boards in inventory: Dimension Length E 4 2″ 4″ E 5 2″ 6″ E 6 2″ 8″ Total E 1 8 feet 1400 1500 1100 4000 E 2 10 feet 2000 3500 2500 8000 E 3 12 feet 1600 2000 2400 6000 Total 5000 7000 6000 18000 He will be selecting one board at random from the inventory to show a visiting customer. Of interest is whether the length of the board is independent of the dimension. This can be determined using the following steps: Step 1 Define the experiment. A board is randomly selected and its dimensions determined. Step 2 Define one event for length and one event for dimension. Let E 2 Event that the board is 10 feet long and E 5 Event that the board is a 2″ 6″ dimension. Step 3 Determine the probability for each event. PE PE . 25 8 000 18 000 0 4444 7 000 18 00 and 0 0 0 3889 . Step 4 Assess the joint probability of the two events occurring. PE E . 25 3 500 18 000 0 1944 and Step 5 Compute the conditional probability of one event given the other using Probability Rule 6. PE E PE E PE | . . . 25 25 5 0 1944 0 3889 05 and 0 0 FIGURE 3 | Tree Diagram for Syringa Networks PE 1 and E 5 0.20 PE 1 |E 5 20 hours PE 3 |E 5 40 hours 0.4848 PE 2 |E 5 20 to 40 hours Male PE 5 0.66 Female PE 4 0.34 PE 2 and E 5 0.32 PE 3 and E 5 0.14 PE 1 and E 4 0.18 PE 2 and E 4 0.12 PE 3 and E 4 0.04 0.3529 PE 2 |E 4 20 to 40 hours PE 1 |E 4 20 hours 0.1176 PE 3 |E 4 40 hours 0.3030 0.2121 0.5294 www.downloadslide.com

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Introduction to Probability Step 6 Check for independence using Probability Rule 7. Because P1E 2 E 5 2 0.50 7 P1E 2 2 0.4444 the two events board length and board dimension are not independent. END EXAMPLE TRY PROBLEM 42 Multiplication Rule We needed to find the joint probability of two events in the discussion on addition of two events and in the discussion on conditional probability. We were able to find P1E 1 and E 2 2 simply by examining the joint relative frequency tables. However we often need to find P1E 1 and E 2 2 when we do not know the joint relative frequencies. When this is the case we can use the multiplication rule for two events. Multiplication Rule for Two Events Chapter Outcome 3. Probability Rule 8: Multiplication Rule for Any Two Events For two events E 1 and E 2 P1E 1 and E 2 2 P1E 1 2P1E 2 E 1 2 11 BUSINESS APPLICATION MULTIPLICATION RULE HONG KONG FIREWORKS To illustrate how to find a joint probability consider an example involving the Hong Kong Fireworks company a manufacturer of fireworks used by cities fairs and other commercial establishments for large-scale fireworks displays. The company uses two suppliers of material used in making a particular product. The materials from the two suppliers are intermingled on the manufacturing process. When a case of fireworks is being made the material is pulled randomly from inventory without regard to which company made it. Recently a customer ordered two products. At the time of assembly the material inventory contained 30 units of MATX and 50 units of Quinex. What is the probability that both fireworks products ordered by this customer will have MATX material To answer this question we must recognize that two events are required to form the desired outcome. Therefore let E 1 Event: MATX Material in first product E 2 Event: MATX Material in second product The probability that both fireworks products contain MATX material is written as P1E 1 and E 2 2. The key word here is and as contrasted with the Addition Rule in which the key word is or. The and signifies that we are interested in the joint probability of two events as noted by P1E 1 and E 2 2. To find this probability we employ Probability Rule 8. P1E 1 and E 2 2 P1E 1 2P1E 2 E 1 2 We start by assuming that each unit of material in the inventory has the same chance of being selected for assembly. For the frst freworks product PE 1 5 5 5 Number of MATX units Number of Firework Materials in inventory 30 80 0 375 . Then because we are not replacing the frst frework material we fnd P1E 2 E 1 2 by PE E | 21 Number of remaining MATX units Number of remaining Firework Materials units 29 79 0 3671 . 5 5 5 www.downloadslide.com

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Introduction to Probability Now by Rule 8 P1E 1 and E 2 2 P1E 1 2P1E 2 E 1 2 10.375210.36712 0.1377 Therefore there is a 13.77 chance the two freworks products will contain the MA TX material Using a Tree Diagram BUSINESS APPLICATION MULTIPLICATION RULE HONG KONG FIREWORKS CONTINUED A tree diagram can be used to display the situation facing Hong Kong Fireworks. The company uses material from two suppliers which is intermingled in the inventory. Recently a customer ordered two products and found that both contained the MATX material. Assuming that the inventory contains 30 MATX and 50 Quinex units to determine the probability of both products containing the MATX material you can use a tree diagram. The two branches on the left side of the tree in Figure 4 show the possible material options for the first product. The two branches coming from each of the first branches show the possible material options for the second product. The probabilities at the far right are the joint probabilities for the material options for the two products. As we determined previously the probability that both products will contain a MATX unit is 0.1377 as shown on the top right on the tree diagram. We can use the Multiplication Rule and the Addition Rule in one application when we determine the probability that two products will have different materials. Looking at Figure 4 we see there are two ways this can happen. P31MATX and Quinex2 or 1Quinex and MATX24 If the first product is a MATX and the second one is a Quinex then the first cannot be a Quinex and the second a MATX. These two events are mutually exclusive and therefore Rule 5 can be used to calculate the required probability. The joint probabilities generated from the Multiplication Rule are shown on the right side of the tree. To find the desired prob- ability using Rule 5 we can add the two joint probabilities: P31MATX and Quinex2 or Quinex and MATX4 0.2373 + 0.2373 0.4746 The chance that a customer buying two products will get two different materials is 47.46. Multiplication Rule for Independent Events When we determined the probability that two products would have MATX material we used the general multiplication rule Rule 8. The general multiplication rule requires that conditional probability be used because the FIGURE 4 |  Tree Diagram for the Fireworks Product Example PMATX and MATX 0.375 3 0.3671 0.1377 MATX P 29/79 0.3671 Quinex P 50/79 0.6329 MATX P 30/80 0.375 Quinex P 50/80 0.625 Product 1 Product 2 PMATX and Quinex 0.375 3 0.6329 0.2373 PQuinex and MATX 0.625 3 0.3797 0.2373 PQuinex and Quinex 0.625 3 0.6203 0.3877 MATX P 30/79 0.3797 Quinex P 49/79 0.6203 www.downloadslide.com

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Introduction to Probability probability associated with the second product depends on the material selected for the first product. The chance of obtaining a MATX was lowered from 30/80 to 29/79 given the first material was a MATX. However if the two events of interest are independent the imposed condition does not alter the probability and the Multiplication Rule takes the form shown in Probability Rule 9. Probability Rule 9: Multiplication Rule for Independent Events For independent events E 1 E 2 P1E 1 and E 2 2 P1E 1 2P1E 2 2 12 The joint probability of two independent events is simply the product of the probabilities of the two events. Rule 9 is the primary way that you can determine whether any two events are independent. If the product of the probabilities of the two events equals the joint probability then the events are independent. EXAMPLE 12 USING THE MULTIPLICATION RULE AND THE ADDITION RULE Christiansen Accounting Christiansen Accounting prepares tax returns for individuals and companies. Over the years the firm has tracked its clients and has discovered that 12 of the individual returns have been selected for audit by the Internal Revenue Service. On one particular day the firm signed two new individual tax clients. The firm is interested in the probability that at least one of these clients will be audited. This probability can be found using the following steps: Step 1 Define the experiment. The IRS randomly selects a tax return to audit. Step 2 Define the possible outcomes. For a single client the following outcomes are defined: A Audit N No audit For each of the clients we define the outcomes as Client 1: A 1 N 1 Client 2: A 2 N 2 Step 3 Define the overall event of interest. The event that Christiansen Accounting is interested in is E At least one client is audited Step 4 List the outcomes for the events of interest. The possible outcomes for which at least one client will be audited are as follows: E 1 : A 1 A 2 both are audited E 2 : A 1 N 2 only one client is audited E 3 : N 1 A 2 Step 5 Compute the probabilities for the events of interest. Assuming the chances of the clients being audited are independent of each other probabilities for the events are determined using Probability Rule 9 for independent events: P1E 1 2 P1A 1 and A 2 2 0.12 0.12 0.0144 P1E 2 2 P1A 1 and N 2 2 0.12 0.88 0.1056 P1E 3 2 P1N 1 and A 2 2 0.88 0.12 0.1056 www.downloadslide.com

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Introduction to Probability Step 6 Determine the probability for the overall event of interest. Because events E 1 E 2 and E 3 are mutually exclusive compute the probability of at least one client being audited using Rule 5 the Addition Rule for Mutually Exclusive Events: P1E 1 or E 2 or E 3 2 P1E 1 2 + P1E 2 2 + P1E 3 2 0.0144 + 0.1056 + 0.1056 0.2256 The chance of one or both of the clients being audited is 0.2256. END EXAMPLE TRY PROBLEM 30 Bayes’ Theorem As decision makers you will often encounter situations that require you to assess probabili- ties for events of interest. Your assessment may be based on relative frequency or subjectivity. However you may then come across new information that causes you to revise the prob- ability assessment. For example a human resources manager who has interviewed a person for a sales job might assess a low probability that the person will succeed in sales. However after seeing the person’s very high score on the company’s sales aptitude test the manager might revise her assessment upward. A medical doctor might assign an 80 chance that a patient has a particular disease. However after seeing positive results from a lab test he might increase his assessment to 95. In these situations you will need a way to formally incorporate the new informa- tion. One very useful tool for doing this is called Bayes’ Theorem which is named for the Reverend Thomas Bayes who developed the special application of conditional probability in the 1700s. Letting event B be an event that is given to have occurred the conditional probability of event E i occurring can be computed as shown earlier using Equation 9: PE B PE B PB i i | and The numerator can be reformulated using the Multiplication Rule Equation 11 as P1E i and B2 P1E i 2P1B E i 2 The conditional probability is then PE B PE P B E PB i ii | | The denominator P1B2 can be found by adding the probability of the k possible ways that event B can occur. This is P1B2 P1E 1 2P1B E 1 2 + P1E 2 2P1B E 2 2 + g+ P1E k 2P1B E k 2 Then Bayes’ Theorem is formulated as Equation 13. Chapter Outcome 4. Bayes’ Theorem PE B PE P B E PE P B E P E P B E i ii | | | | 11 2 2 2 | ... PE P B E kk 13 where: E i ith event of interest of the k possible events B Event that has occurred that might impact P1E i 2 Events E 1 to E k are mutually exclusive and collectively exhaustive. www.downloadslide.com

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Introduction to Probability BUSINESS APPLICATION BAYES’ THEOREM TILE PRODUCTION The Glerum Tile and Flooring Company has two production facilities one in Ohio and one in Virginia. The company makes the same type of tile at both facilities. The Ohio plant makes 60 of the company’s total tile output and the Virginia plant 40. All tiles from the two facilities are sent to regional warehouses where they are intermingled. After extensive study the quality assurance manager has determined that 5 of the tiles produced in Ohio and 10 of the tiles produced in Virginia are unusable due to quality problems. When the company sells a defective tile it incurs not only the cost of replacing the item but also the loss of goodwill. The vice president for production would like to allocate these costs fairly between the two plants. To do so he knows he must first determine the probability that a defective tile was produced by a particular production line. Specifically he needs to answer these questions: 1. What is the probability that the tile was produced at the Ohio plant given that the tile is defective 2. What is the probability that the tile was produced at the Virginia plant given that the tile is defective In notation form with D representing the event that an item is defective what the manager wants to know is P1Ohio D2 P1Virginia D2 We can use Bayes’ Theorem Equation 13 to determine these probabilities as follows: PD PPD PD | | Ohio Ohio Ohio We know that event DDefective tile can happen if it is made in either Ohio or Virginia. Thus P1D2 P1Ohio and Defective2 + P1Virginia and Defective2 P1D2 P1Ohio2P1D Ohio2 + P1Virginia2P1D Virginia2 We already know that 60 of the tiles come from Ohio and 40 from Virginia. So P1Ohio2 0.60 and P1Virginia2 0.40. These are called the prior probabilities. Without Bayes’ Theorem we would likely allocate the total cost of defects in a 60/40 split between Ohio and Virginia based on total production. However the new information about the quality from each line is P1D Ohio2 0.05 and P1D Virginia2 0.10 which can be used to properly allocate the cost of defects. This is done using Bayes’ Theorem. PD PPD PPD | | | Ohio Ohio Ohio Ohio OhioPPD | Virginia Virginia then PD | . . . . . Ohio 060 0 05 060 005 040 0 .. . 10 0 4286 and PD PPD P | | Virginia Virginia Virginia Virg ginia Virginia Ohio Ohio Virg | | PD P P D P i inia | . . . . . . D 040 010 040 010 060 0 0 05 0 5714 . These probabilities are revised probabilities. The prior probabilities have been revised given the new quality information. We now see that 42.86 of the cost of defects should be al- located to the Ohio plant and 57.14 should be allocated to the Virginia plant. severija kirilovaite/Fotolia www.downloadslide.com

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Introduction to Probability TABLE 9 | Bayes’ Theorem Calculations for Glerum Tile and Flooring Events Prior Probabilities Conditional Probabilities Joint Probabilities Revised Probabilities Ohio 0.60 0.05 10.60210.052 0.03 0.030.07 0.4286 Virginia 0.40 0.10 10.40210.102 0.04 0.040.07 0.5714 0.07 1.0000 Note the denominator P1D2 is the overall probability of a defective tile. This pro- bability is P1D2 P1Ohio2P1D Ohio2 + P1Virginia2P1D Virginia2 10.60210.052 + 10.40210.102 0.03 + 0.04 0.07 Thus 7 of all the tiles made by Glerum are defective. You might prefer to use a tabular approach like that shown in Table 9 when you apply Bayes’ Theorem. EXAMPLE 13 BAYES’ THEOREM Techtronics Equipment Corporation The Techtronics Equipment Corporation has developed a new electronic device that it would like to sell to the U.S. military for use in fighter aircraft. The sales manager believes there is a 0.60 chance that the military will place an order. However after making an initial sales presentation military officials will often ask for a second presentation to other military decision makers. Historically 70 of successful companies are asked to make a second presentation whereas 50 of unsuccessful companies are asked back a second time. Suppose Techtronics Equipment has just been asked to make a second presentation what is the revised probability that the company will make the sale This probability can be determined using the following steps: Step 1 Define the events. In this case there are two events: S Sale N No sale Step 2 Determine the prior probabilities for the events. The probability of the events prior to knowing whether a second presentation will be requested are P1S2 0.60 P1N2 0.40 Step 3 Define an event that if it occurs could alter the prior probabilities. In this case the altering event is the invitation to make a second presentation. We label this event as SP. Step 4 Determine the conditional probabilities. The conditional probabilities are associated with being invited to make a second presentation: P1SP S2 0.70 P1SP N2 0.50 Step 5 Use the tabular approach for Bayes’ Theorem to determine the revised probabilities. These correspond to P1S SP2 and P1N SP2 www.downloadslide.com

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Introduction to Probability Skill Development 4-26. Based on weather data collected in Racine Wisconsin on Christmas Day the weather had the following distribution: Event Relative Frequency Clear dry 0.20 Cloudy dry 0.30 Rain 0.40 Snow 0.10 a. Based on these data what is the probability that next Christmas will be dry b. Based on the data what is the probability that next Christmas will be rainy or cloudy and dry c. Supposing next Christmas is dry determine the probability that it will also be cloudy. 4-27. The Jack In The Box franchise in Bangor Maine has determined that the chance a customer will order a soft drink is 0.90. The probability that a customer will order a hamburger is 0.60. The probability that a customer will order french fries is 0.50. a. If a customer places an order what is the probability that the order will include a soft drink and no fries if these two events are independent b. The restaurant has also determined that if a customer orders a hamburger the probability the customer will also order fries is 0.80. Determine the probability that the order will include a hamburger and fries. 4-28. Ponderosa Paint and Glass carries three brands of paint. A customer wants to buy another gallon of paint to match paint she purchased at the store previously. She can’t recall the brand name and does not wish to return home to find the old can of paint. So she selects two of the three brands of paint at random and buys them. Event Prior Probabilities Conditional Probabilities Joint Probabilities Revised Probabilities S Sale 0.60 P1SP S2 0.70 P1S2P1SP S210.60210.702 0.42 0.420.62 0.6774 N No sale 0.40 P1SP N2 0.50 P1N2P1SP N210.40210.502 0.20 0.200.62 0.3226 0.62 1.0000 Thus using Bayes’ Theorem if Techtronics Equipment gets a second presentation opportunity the probability of making the sale is revised upward from 0.60 to 0.6774. END EXAMPLE TRY PROBLEM 33 MyStatLab 4-2: Exercises a. What is the probability that she matched the paint brand b. Her husband also goes to the paint store and fails to remember what brand to buy. So he also purchases two of the three brands of paint at random. Determine the probability that both the woman and her husband fail to get the correct brand of paint. Hint: Are the husband’s selections independent of his wife’s selections 4-29. The college basketball team at West Texas State University has 10 players 5 are seniors 2 are juniors and 3 are sophomores. Two players are randomly selected to serve as captains for the next game. What is the probability that both players selected are seniors 4-30. Micron Technology has sales offices located in four cities: Dallas Seattle Boston and Los Angeles. An analysis of the company’s accounts receivables reveals the number of overdue invoices by days as shown here. Days Overdue Dallas Seattle Boston Los Angeles Under 30 days 137 122 198 287 30–60 days 85 46 76 109 61–90 days 33 27 55 48 Over 90 days 18 32 45 66 Assume the invoices are stored and managed from a central database. a. What is the probability that a randomly selected invoice from the database is from the Boston sales office b. What is the probability that a randomly selected invoice from the database is between 30 and 90 days overdue c. What is the probability that a randomly selected invoice from the database is over 90 days old and from the Seattle office www.downloadslide.com

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Introduction to Probability d. If a randomly selected invoice is from the Los Angeles office what is the probability that it is 60 or fewer days overdue 4-31. Three events occur with probabilities P1E 1 2 0.35 P1E 2 2 0.15 P1E 3 2 0.40. If the event B occurs the probability becomes P1E 1 B2 0.25 P1B2 0.30. a. Calculate P1E 1 and B2 b. Compute P1E 1 or B2 c. Assume that E 1 E 2 and E 3 are independent events. Calculate P1E 1 and E 2 and E 3 2. 4-32. The URS construction company has submitted two bids one to build a large hotel in London and the other to build a commercial office building in New York City. The company believes it has a 40 chance of winning the hotel bid and a 25 chance of winning the office building bid. The company also believes that winning the hotel bid is independent of winning the office building bid. a. What is the probability the company will win both contracts b. What is the probability the company will win at least one contract c. What is the probability the company will lose both contracts 4-33. Suppose a quality manager for Dell Computers has collected the following data on the quality status of disk drives by supplier. She inspected a total of 700 disk drives. Drive Status Supplier Working Defective Company A 120 10 Company B 180 15 Company C 50 5 Company D 300 20 a. Based on these inspection data what is the probability of randomly selecting a disk drive from company B b. What is the probability of a defective disk drive being received by the computer company c. What is the probability of a defect given that company B supplied the disk drive 4-34. Three events occur with probabilities of P1E 1 2 0.35 P1E 2 2 0.25 P1E 3 2 0.40. Other probabilities are: P1B E 1 2 0.25 P1B E 2 2 0.15 P1B E 3 2 0.60. a. Compute P1E 1 B2. b. Compute P1E 2 B2. c. Compute P1E 3 B2. 4-35. Men have a reputation for not wanting to ask for directions. A Harris study conducted for Lincoln Mercury indicated that 42 of men and 61 of women would stop and ask for directions. The U.S. Census Bureau’s 2007 population estimate was that for individuals 18 or over 48.2 were men and 51.8 were women. This exercise addresses this age group. a. A randomly chosen driver gets lost on a road trip. Determine the probability that the driver is a woman and stops to ask for directions. b. Calculate the probability that the driver stops to ask for directions. c. Given that a driver stops to ask for directions determine the probability that the driver was a man. Business Applications 4-36. A local FedEx/Kinkos has three black-and-white copy machines and two color copiers. Based on historical data the chance that each black-and-white copier will be down for repairs is 0.10. The color copiers are more of a problem and are down 20 of the time each. a. Based on this information what is the probability that if a customer needs a color copy both color machines will be down for repairs b. If a customer wants both a color copy and a black- and-white copy what is the probability that the necessary machines will be available Assume that the color copier can also be used to make a black- and-white copy if needed. c. If the manager wants to have at least a 99 chance of being able to furnish a black-and-white copy on demand is the present configuration sufficient Assume that the color copier can also be used to make a black-and-white copy if needed. Back up your answer with appropriate probability computations. d. What is the probability that all five copiers will be up and running at the same time Suppose the manager added a fourth black-and-white copier how would the probability of all copiers being ready at any one time be affected 4-37. Suppose the managers at FedEx/Kinkos wish to meet the increasing demand for color photocopies and to have more reliable service. Refer to Problem 36. As a goal they would like to have at least a 99.9 chance of being able to furnish a black-and-white copy or a color copy on demand. They also wish to purchase only four copiers. They have asked for your advice regarding the mix of black-and-white and color copiers. Supply them with your advice. Provide calculations and reasons to support your advice. 4-38. The Snappy Service gas station manager is thinking about a promotion that she hopes will bring in more business to the full-service island. She is considering the option that when a customer requests a fill-up if the pump stops with the dollar amount at 19.99 the customer will get the gasoline free. Previous studies show that 70 of the customers require more than 20.00 when they fill up so would not be eligible for the free gas. What is the probability that a customer will get free gas at this station if the promotion is implemented www.downloadslide.com

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Introduction to Probability 4-39. Suppose the manager in Problem 38 is concerned about alienating customers who buy more than 20.00 since they would not be eligible to win the free gas under the original concept. To overcome this she is thinking about changing the contest. The customer will get free gas if any of the following happens: 21.11 22.22 23.33 24.44 25.55 26.66 27.77 28.88 29.99 Past data show that only 5 of all customers require 30.00 or more. If one of these big-volume customers arrives he will get to blindly draw a ball from a box containing 100 balls 99 red 1 white. If the white ball is picked the customer gets his gas free. Considering this new promotion what is the probability that a customer will get free gas 4-40. Hubble Construction Company has submitted a bid on a state government project that is to be funded by the federal government’s stimulus money in Arizona. The price of the bid was predetermined in the bid specifications. The contract is to be awarded on the basis of a blind drawing from those who have bid. Five other companies have also submitted bids. a. What is the probability of the Hubble Construction Company winning the bid b. Suppose that there are two contracts to be awarded by a blind draw. What is the probability of Hubble winning both contracts Assume sampling with replacement. c. Referring to part b what is the probability of Hubble not winning either contract d. Referring to part b what is the probability of Hubble winning exactly one contract 4-41. Drake Marketing and Promotions has randomly surveyed 200 men who watch professional sports. The men were separated according to their educational level college degree or not and whether they preferred the NBA or the National Football League NFL. The results of the survey are shown: Sports Pr efer ence College Degree No College Degree NBA 40 55 NFL 10 95 a. What is the probability that a randomly selected survey participant prefers the NFL b. What is the probability that a randomly selected survey participant has a college degree and prefers the NBA c. Suppose a survey participant is randomly selected and you are told that he has a college degree. What is the probability that this man prefers the NFL d. Is a survey participant’s preference for the NBA independent of having a college degree 4-42. Until the summer of 2008 the real estate market in Fresno California had been booming with prices skyrocketing. Recently a study showed the sales patterns in Fresno for single-family homes. One chart presented in the commission’s report is reproduced here. It shows the number of homes sold by price range and number of days the home was on the market. Days on the Market Price Range 000 1-78- 30 Over 30 Under 200 125 15 30 200–500 200 150 100 501–1000 400 525 175 Over 1000 125 140 35 a. Using the relative frequency approach to probability assessment what is the probability that a house will be on the market more than 7 days b. Is the event 1-7 days on the market independent of the price 200–500 c. Suppose a home has just sold in Fresno and was on the market less than 8 days what is the most likely price range for that home 4-43. Vegetables from the summer harvest are currently being processed at Skone and Conners Foods Inc. The manager has found a case of cans that have not been properly sealed. There are three lines that processed cans of this type and the manager wants to know which line is most likely to be responsible for this mistake. Provide the manager this information. Line Contribution to Total Proportion Defective 1 0.40 0.05 2 0.35 0.10 3 0.25 0.07 4-44. A corporation has 11 manufacturing plants. Of these seven are domestic and four are outside the United States. Each year a performance evaluation is conducted for four randomly selected plants. What is the probability that a performance evaluation will include at least one plant outside the United States Hint: Begin by finding the probability that only domestic plants are selected. 4-45. Parts and materials for the skis made by the Downhill Adventures Company are supplied by two suppliers. Supplier A’s materials make up 30 of what is used with Supplier B providing the rest. Past records indicate that 15 of Supplier A’s materials are defective and 10 of B’s are defective. Since it is impossible to tell which supplier the materials came from once they are in inventory the manager wants to know which supplier more likely supplied the defective materials the foreman has brought to his attention. Provide the manager this information. www.downloadslide.com

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Introduction to Probability 4-46. A major electronics manufacturer has determined that when one of its televisions is sold there is 0.08 chance that the set will need service before the warranty period expires. It has also assessed a 0.05 chance that a DVD player will need service prior to the expiration of the warranty. a. Suppose a customer purchases one of the company’s televisions and one of the DVD players. What is the probability that at least one of the products will require service prior to the warranty expiring b. Suppose a retailer sells four televisions on a particular Saturday. What is the probability that none of the four will need service prior to the warranty expiring c. Suppose a retailer sells four televisions on a particular Saturday. What is the probability that at least one will need repair 4-47. The Committee for the Study of the American Electorate indicated that 60.7 of the voting-age voters cast ballots in the 2004 presidential election. It also indicated that 85.3 of registered voters voted in the election. The percentage of those who voted for President Bush was 50.8. a. Determine the proportion of voting-age voters who voted for President Bush. b. Determine the proportion of voting-age voters who were registered to vote. 4-48. A distributor of outdoor yard lights has four suppliers. This past season she purchased 40 of the lights from Franklin Lighting 30 from Wilson Sons 20 from Evergreen Supply and the rest from A. L. Scott. In prior years 3 of Franklin’s lights were defective 6 of the Wilson lights were defective 2 of Evergreen’s were defective and 8 of the Scott lights were defective. When the lights arrive at the distributor she puts them in inventory without identifying the supplier. Suppose that a defective light string has been pulled from inventory what is the probability that it was supplied by Franklin Lighting 4-49. USA Today reported “Study Finds Better Survival Rates at ‘High-V olume’ Hospitals” that “high- volume” hospitals performed at least 77 of bladder removal surgeries “low-volume” hospitals performed at most 23. Assume the percentages are 77 and 23. In the first two weeks after surgery 3.1 of patients at low-volume centers died compared to 0.7 at the high-volume hospitals. a. Calculate the probability that a randomly chosen bladder-cancer patient had surgery at a high-volume hospital and survived the first two weeks after surgery. b. Calculate the probability that a randomly chosen bladder-cancer patient survived the first two weeks after surgery. c. If two bladder-cancer patients were chosen randomly determine the probability that only one would survive the first two weeks after surgery. d. If two bladder-cancer patients were chosen randomly determine the probability that at least one would survive the first two weeks after surgery. 4-50. Suppose an auditor has 18 tax returns 12 of which are for physicians. If three of the 18 tax returns are randomly selected then what is the probability that at least one of the three selected will be a physician’s tax return 4-51. A box of 50 remote control devices contains 3 that have a defective power button. If devices are randomly sampled from the box and inspected one at a time determine a. The probability that the first control device is defective. b. The probability that the first control device is good and the second control device is defective. c. The probability that the first three sampled devices are all good. Computer Database Exercises 4-52. The data file Colleges contains data on more than 1300 colleges and universities in the United States. Suppose a company is planning to award a significant grant to a randomly selected college or university. Using the relative frequency method for assessing probabilities and the rules of probability respond to the following questions. If data are missing for a needed variable reduce the number of colleges in the study appropriately. a. What is the probability that the grant will go to a private college or university b. What is the probability that the grant will go to a college or university that has a student/faculty ratio over 20 c. What is the probability that the grant will go to a college or university that is both private and has a student/faculty ratio over 20 d. If the company decides to split the grant into two grants what is the probability that both grants will go to California colleges and universities What might you conclude if this did happen 4-53. A Courtyard Hotel by Marriott conducted a survey of its guests. Sixty-two surveys were completed. Based on the data from the survey found in the file CourtyardSurvey determine the following probabilities using the relative frequency assessment method. a. Of two customers selected what is the probability that both will be on a business trip b. What is the probability that a customer will be on a business trip or will experience a hotel problem during a stay at the Courtyard c. What is the probability that a customer on business has an in-state area code phone number d. Based on the data in the survey can the Courtyard manager conclude that a customer’s rating regarding staff attentiveness is independent of whether he or www.downloadslide.com

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Introduction to Probability she is traveling on business pleasure or both Use the rules of probability to make this determination. 4-54. Continuing with the Marriott survey done by the managers of a Marriott Courtyard Hotel based on the data from the survey found in the file CourtyardSurvey determine the following probabilities using the relative frequency assessment method. a. Of two customers selected what is the probability that neither will be on a business trip b. What is the probability that a customer will be on a business trip or will not experience a hotel problem during a stay at the Courtyard c. What is the probability that a customer on a pleasure trip has an in-state area code phone number 4-55. A Harris survey asked in part what the most important reason was that people give for not using a wireless phone exclusively. The responses were: 1 Like the safety of traditional phone 2 Need line for Internet access 3 Pricing not attractive enough 4 Weak or unreliable cell signal at home 5 Coverage not good enough and 6 Other. The file titled Wireless contains the responses for the 1088 respondents. a. Of those respondents 36 or older determine the probability that an individual in this age group would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone. b. If three respondents were selected at random from those respondents younger than 36 calculate the probability that at least one of the respondents stated the most important reason for not using a wireless exclusively was that they need a line for Internet access. 4-56. A recent news release published by Ars Technica LLD presented the results of a study concerning the world and domestic market share for the major manufacturers of personal computers PCs. The file titled PCMarket contains a sample that would produce the market shares alluded to in the article and the highest academic degrees achieved by the owners of those PCs. a. Determine the probability that the person had achieved at least a bachelor’s degree and owns a Dell PC. b. If a randomly selected person owned a Dell PC determine the probability that the person had achieved at least a bachelor’s degree. c. Consider these two events: 1 At least a bachelor’s degree and 2 Owns a Dell PC. Are these events independent dependent or mutually exclusive Explain. 4-57. PricewaterhouseCoopers Saratoga in its 2005/2006 Human Capital Index Report indicated the average number of days it took for an American company to fill a job vacancy in 2004 was 48 days. Sample data similar to those used in the study are in a file titled Hired. Categories for the days and hire cost are provided under the headings “Time” and “Cost” respectively. a. Calculate the probability that a company vacancy took at most 100 days or cost at most 4000 to fill. b. Of the vacancies that took at most 100 days to fill calculate the probability that the cost was at most 4000. c. If three of the vacancies were chosen at random calculate the probability that two of the vacancies cost at most 4000 to fill. 4-58. A company produces scooters used by small businesses such as pizza parlors that find them convenient for making short deliveries. The company is notified whenever a scooter breaks down and the problem is classified as being either mechanical or electrical. The company then matches the scooter to the plant where it was assembled. The file Scooters contains a random sample of 200 breakdowns. Use the data in the file to find the following probabilities. a. If a scooter was assembled in the Tyler plant what is the probability its breakdown was due to an electrical problem b. Is the probability of a scooter having a mechanical problem independent of the scooter being assembled at the Lincoln plant c. If mechanical problems are assigned a cost of 75 and electrical problems are assigned a cost of 100 how much cost would be budgeted for the Lincoln and Tyler plants next year if a total of 500 scooters were expected to be returned for repair END EXERCISES 4-2 www.downloadslide.com

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Introduction to Probability Visual Summary 1 The Basics of Probability 2 The Rules of Probability Outcome 2. Be able to apply the Addition Rule. Outcome 3. Know how to use the Multiplication Rule. Outcome 4. Know how to use Bayes’ Theorem for applications involving conditional probabilities Conclusion Probability is how we measure our uncertainty about whether an outcome will occur. The closer the probability assessment is to 1.0 or 0.0 the more certain we are that event will or will not occur. Assessing probabilities and then using those probabilities to help make decisions is a central part of what business decision-makers do on a regular basis. This chapter has introduced the fundamentals of probability and the rules that are used when working with probability. Probability is used in our everyday lives and in business decision-making all the time. You might base your decision to call ahead for dinner reservations based on your assessment of the probability of having to wait for seating. A company may decide to switch suppliers based on their assessment of the probability that the new supplier will provide higher quality products or services. Probability is the way we measure our uncertainty about events. However in order to properly use probability you need to know the probability rules and the terms associated with probability. Summary In order to effectively use probability it is important to understand key concepts and terminology. Some of the most important of these are discussed in section 1 including sample space dependent and independent events and mutually exclusive events. Probabilities are assessed in three main ways classical assessment relative frequency assessment and subjective assessment. Outcome 1. Understand the three approaches to assessing probabilities. Summary To effectively work with probability it is important to know the probability rules. Section 2 introduces nine rules including three addition rules and two multiplication rules. Rules for conditional probability and the complement rule are also very useful. Bayes’ Theorem is used to calculate conditional probabilities in situations where the probability of the given event is not provided and must be calculated. www.downloadslide.com

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Introduction to Probability 8 Probability Rule 5 Addition rule for mutually exclusive events E 1 E 2 : P1E 1 or E 2 2 P1E 1 2 + P1E 2 2 9 Probability Rule 6 Conditional probability for any two events E 1 E 2 : PE E PE E PE | 12 12 2 and 10 Probability Rule 7 Conditional probability for independent events E 1 E 2 : PE E P E P E | 12 1 2 0 and PE E P E P E | 21 2 1 0 11 Probability Rule 8 Multiplication rule for any two events E 1 and E 2 : P1E 1 and E 2 2 P1E 1 2P1E 2 E 1 2 12 Probability Rule 9 Multiplication rule for independent events E 1 E 2 : P1E 1 and E 2 2 P1E 1 2P1E 2 2 13 Bayes’ Theorem PE B PE P B E PE P B E P E P B E i ii | | | | 11 2 2 2 | ... PE P B E kk 1 Classical Probability Assessment PE E i i Number of ways can occur Total numbe er of possible outcomes 2 Relative Frequency Assessment PE E N i i Number of times occurs 3 Probability Rule 1 0 … P1E i 2 … 1 for all i 4 Probability Rule 2 Pe i i k 1 1 S 5 Probability Rule 3 Addition rule for individual outcomes: The probability of an event E i is equal to the sum of the proba bilities of the possible outcomes forming E i . For example if E i 5e 1 e 2 e 3 6 then P1E i 2 P1e 1 2 + P1e 2 2 + P1e 3 2 6 Complement Rule PE PE 1 7 Probability Rule 4 Addition rule for any two events E 1 and E 2 : P1E 1 or E 2 2 P1E 1 2 + P1E 2 2-P1E 1 and E 2 2 Equations Key Terms Classical probability assessment Complement Conditional probability Dependent events Event Experiment Independent events Mutually exclusive events Probability Relative frequency assessment Sample space Subjective probability assessment Chapter Exercises Conceptual Questions 4-59. Discuss what is meant by classical probability assessment and indicate why classical assessment is not often used in business applications. 4-60. Discuss what is meant by the relative frequency assessment approach to probability assessment. Provide a business-related example other than the one given in the text in which this method of probability assessment might be used. 4-61. Discuss what is meant by subjective probability. Provide a business-related example in which subjective probability assessment would likely be used. Also provide an example of when you have personally used subjective probability assessment. 4-62. Examine the relationship between independent dependent and mutually exclusive events. Consider two events A and B that are mutually exclusive such that P1A2 ≠ 0. a. Calculate P1A B2. b. What does your answer to part a say about whether two mutually exclusive events are dependent or independent c. Consider two events C and D such that P1C2 0.4 and P1C D2 0.15. 1 Are events MyStatLab www.downloadslide.com

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Introduction to Probability b. Calculate the probability that a randomly chosen U.S. adult does not take a multiple formula. c. If three U.S. adults were chosen at random compute the probability that only two of them take a multiple formula as their primary source. 4-67. USA Today reported the IRS audited 1 in 63 wealthy individuals and families about 1 of every 107 individuals 20 of corporations in general and 44 of the largest corporations with assets of at least 250 million. a. Calculate the probability that at least 1 wealthy individual in a sample of 10 would be audited. b. Compute the probability that at least 1 from a sample of 10 corporations with assets of at least 250 million would be audited. c. Calculate the probability that a randomly chosen wealthy CEO of a corporation with assets of 300 million would be audited or that the corporation would be audited. 4-68. Simmons Furniture Company is considering changing its starting hour from 8:00 a.m. to 7:30 a.m. A census of the company’s 1200 office and production workers shows 370 of its 750 production workers favor the change and a total of 715 workers favor the change. To further assess worker opinion the region manager decides to talk with random workers. a. What is the probability a randomly selected worker will be in favor of the change b. What is the probability a randomly selected worker will be against the change and be an office worker c. Are the events job type and opinion independent Explain. 4-69. A survey released by the National Association of Convenience Stores NACS indicated that 70 of gas purchases paid for at the pump were made with a credit or debit card. a. Indicate the type of probability assessment method that NACS would use to assess this probability. b. In one local store 10 randomly chosen customers were observed. All 10 of these customers used a credit or a debit card. If the NACS statistic applies to this area determine the probability that 10 out of 10 customers would use a credit or debit card. c. If 90 of gas purchases paid for at the pump were made with a credit or debit card determine the probability that 10 out of 10 customers would use a credit or debit card. d. Based on your answers to parts b and c does it appear that a larger percentage of local individuals use credit or debit cards than is true for the nation as a whole Explain. 4-70. Ponderosa Paint and Glass makes paint at three plants. It then ships the unmarked paint cans to a central warehouse. Plant A supplies 50 of the paint and past records indicate that the paint is incorrectly mixed 10 of the time. Plant B contributes 30 with paint mixed C and D mutually exclusive 2 Are events C and D independent or dependent Are dependent events necessarily mutually exclusive events 4-63. Consider the following table: A A Totals B 800 200 1000 B 600 400 1000 Totals 1400 600 2000 Explore the complements of conditional events: a. Calculate the following probabilities: P1A B2 P1A B2 P1A B2 P1A B2. b. Now determine which pair of events are complements of each other. Hint: Use the probabilities calculated in part a and the Complement Rule. 4-64. Examine the following table: A A Totals B 200 800 1000 B 300 700 1000 Totals 500 1500 2000 a. Calculate the following probabilities: P1A2 P1A2 P1A B2 P1A B2 P1A B2 and P1A B2. b. Show that 1 A and B 2 A and B 3 A and B 4 A and B are dependent events. Business Applications 4-65. An accounting professor at a state university in V ermont recently gave a three-question multiple-choice quiz. Each question had four optional answers. a. What is the probability of getting a perfect score if you were forced to guess at each question b. Suppose it takes at least two correct answers out of three to pass the test. What is the probability of passing if you are forced to guess at each question What does this indicate about studying for such an exam c. Suppose through some late-night studying you are able to correctly eliminate two answers on each question. Now answer parts a and b. 4-66. Simmons Market Research conducted a national consumer study of 13787 respondents. A subset of the respondents was asked to indicate the primary source of the vitamins or mineral supplements they consume. Six out of 10 U.S. adults take vitamins or mineral supplements. Of those who do 58 indicated a multiple formula was their choice. a. Calculate the probability that a randomly chosen U.S. adult takes a multiple formula as her or his primary source. www.downloadslide.com

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Introduction to Probability b. What is the probability that a randomly selected executive from this group is a male whose functional background is marketing c. Assume that an executive is selected and you are told that the executive’s functional background was in operations. What is the probability that this executive is a female d. Assume that an executive is selected and you are told that the executive is a female. What is the probability the executive’s functional area is marketing e. Are gender and functional background independent for this set of executives 4-74. A manufacturing firm has two suppliers for an electrical component used in its process: one in Mexico and one in China. The supplier in Mexico ships 82 of all the electrical components used by the firm and has a defect rate of 4. The Chinese supplier ships 18 of the electrical components used by the firm and has a defect rate of 6. a. Calculate the probability that an electrical component is defective. b. Suppose an electrical component is defective. What is the probability that component was shipped from Mexico Hint: Use Bayes’ theorem. Computer Database Exercises 4-75. A survey of 300 CEOs was conducted in which the CEOs were to list their corporation’s geographical location: Northeast NE Southeast SE Midwest MW Southwest SW and West W. They were also requested to indicate their company’s industrial type: Communication C Electronics E Finance F and Manufacturing M. The file titled CEOInfo contains sample data similar to that used in this study. a. Determine the probability that a randomly chosen CEO would have a corporation in the West. b. Compute the probability that a randomly chosen CEO would have a corporation in the West and head an electronics corporation. c. Calculate the probability that a randomly chosen CEO would have a corporation in the East or head a communications corporation. d. Of the corporations located in the East calculate the probability that a randomly selected CEO would head a communications corporation. 4-76. The ECCO company makes backup alarms for machinery like forklifts and commercial trucks. When a customer returns one of the alarms under warranty the quality manager logs data on the product. From the data available in the file named Ecco use relative frequency to find the following probabilities. a. What is the probability the product was made at the Salt Lake City plant b. What is the probability the reason for the return was a wiring problem incorrectly 5 of the time. Plant C supplies 20 with paint mixed incorrectly 20 of the time. If Ponderosa guarantees its product and spent 10000 replacing improperly mixed paint last year how should the cost be distributed among the three plants 4-71. Recently several long-time customers at the Sweet Haven Chocolate Company have complained about the quality of the chocolates. It seems there are several partially covered chocolates being found in boxes. The defective chocolates should have been caught when the boxes were packed. The manager is wondering which of the three packers is not doing the job properly. Clerk 1 packs 40 of the boxes and usually has a 2 defective rate. Clerk 2 packs 30 with a 2.5 defective rate. Clerk 3 boxes 30 of the chocolates and her defective rate is 1.5. Which clerk is most likely responsible for the boxes that raised the complaints 4-72. Tamarack Resorts and Properties is considering opening a skiing area near McCall Idaho. It is trying to decide whether to open an area catering to family skiers or to some other group. To help make its decision it gathers the following information. Let A 1 Family will ski A 2 Family will not ski B 1 Family has children but none in the 8–16 age group B 2 Family has children in the 8–16 age group B 3 Family has no children then for this location P1A 1 2 0.40 P1B 2 2 0.35 P1B 1 2 0.25 P1A 1 B 2 2 0.70 P1A 1 B 1 2 0.30 a. Use the probabilities given to construct a joint probability distribution table. b. What is the probability a family will ski and have children who are not in the 8–16 age group How do you write this probability c. What is the probability a family with children in the 8–16 age group will not ski d. Are the categories skiing and family composition independent 4-73. Fifty chief executive officers of small to medium-sized companies were classified according to their gender and functional background as shown in the table below: Functional Background Male Female Total Marketing 4 10 14 Finance 11 5 16 Operations 17 3 20 Total 32 18 50 a. If a chief executive is randomly selected from this group what is the probability that the executive is a female www.downloadslide.com

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Introduction to Probability b. Determine the probability that an individual who has saved for retirement has saved less than 50000. Use relative frequencies. c. Determine the probability that a randomly chosen individual has saved less than 50000 toward retirement. d. Calculate the probability that at least two of four individuals have saved less than 50000 toward retirement. 4-79. USA Today reported on the impact of Generation Y on the workforce. The workforce is comprised of 1 Silent generation born before 1946 7.5 2 Baby boomers 1946–1964 42 3 Generation X 11965-19762 29.5 and 4 Generation Y 11977-19892 21. Ways of communication are changing. Randstad Holding an international supplier of services to businesses and institutions examined the different methods of communication preferred by the different elements of the workforce. The file titled Communication contains sample data comparable to those found in this study. a. Construct a frequency distribution for each of the generations. Use the communication categories 1 Gp Meeting 2 Face-to-Face 3 E-mail and 4 Other. b. Calculate the probability that a randomly chosen member of the workforce prefers communicating face to face. c. Given that an individual in the workforce prefers to communicate face to face determine the generation of which the individual is most likely a member. c. What is the joint probability the returned item was from the Salt Lake City plant and had a wiring- related problem d. What is the probability that a returned item was made on the day shift at the Salt Lake plant and had a cracked lens problem e. If an item was returned what is the most likely profile for the item including plant location shift and cause of problem 4-77. Continuing with the ECCO company from Problem 76 when a customer returns one of the alarms under warranty the quality manager logs data on the product. From the data available in the Ecco file use relative frequency to find the following probabilities. a. If a part was made in the Salt Lake plant what is the probability the cause of the returned part was wiring b. If the company incurs a 30 cost for each returned alarm what percentage of the cost should be assigned to each plant if it is known that 70 of all production is done in Boise 20 in Salt Lake and the rest in Toronto 4-78. The Employee Benefit Research Institute EBRI issued a news release “Saving in America: Three Key Sets of Figures” on October 25 2005. In 2005 about 69 of workers said they have saved for retirement. The file titled Retirement contains sample data similar to those used in this study. a. Construct a frequency distribution of the total savings and investments using the intervals 1 Less than 25000 2 25000–49999 3 50000– 99999 4 100000–249999 and 5 250000 or more. Case 1 Great Air Commuter Service The Great Air Commuter Service Company started in 1984 to pro- vide efficient and inexpensive commuter travel between Boston and New York City. People in the airline industry know Peter Wil- son the principal owner and operating manager of the company as “a real promoter.” Before founding Great Air Peter operated a small regional airline in the Rocky Mountains with varying suc- cess. When Cascade Airlines offered to buy his company Peter decided to sell and return to the East. Peter arrived at his office near Fenway Park in Boston a little later than usual this morning. He had stopped to have a business breakfast with Aaron Little his long-time friend and sometime partner in various business deals. Peter needed some advice and through the years has learned to rely on Aaron as a ready source no matter what the subject. Peter explained to Aaron that his commuter service needed a promotional gimmick to improve its visibility in the business com- munities in Boston and New York. Peter was thinking of running a contest on each flight and awarding the winner a prize. The idea would be that travelers who commute between Boston and New York might just as well have fun on the way and have a chance to win a nice prize. As Aaron listened to Peter outlining his contest plans his mind raced through contest ideas. Aaron thought that a large variety of contests would be needed because many of the passengers would likely be repeat customers and might tire of the same old thing. In addition some of the contests should be chance-type contests whereas others should be skill based. “Well what do you think” asked Peter. Aaron finished his scrambled eggs before responding. When he did it was completely in character. “I think it will fly” Aaron said and proceeded to offer a variety of suggestions. Peter felt good about the enthusiastic response Aaron had given to the idea and thought that the ideas discussed at breakfast presented a good basis for the promotional effort. Now back at the office Peter does have some concerns with one part of the plan. Aaron thought that in addition to the regular in-flight contests for www.downloadslide.com

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Introduction to Probability match and therefore the chance of giving away a grand prize on any one flight is small. Peter likes the idea but when he asks Mar- garet what the probability is that a match will occur her response does not sound quite right. She believes the probability for a match will be 40/365 for a full plane and less than that when there are fewer than 40 passengers aboard. After Margaret leaves Peter decides that it would be useful to know the probability of one or more birthday matches on flights with 20 30 and 40 passengers. He realizes that he will need some help from someone with knowledge of statistics. Required Tasks: 1. Assume that there are 365 days in a year in other words there is no leap year. Also assume there is an equal probability of a passenger’s birthday falling on any one of the 365 days. Calculate the probability that there will be at least one birthday match for a flight containing exactly 20 passengers. Hint: This calculation is made easier if you will first calculate the probability that there are no birthday matches for a flight containing 20 passengers. 2. Repeat 1 above for a flight containing 30 passengers and a flight containing 40 passengers. Again it will be easier to compute the probabilities of one or more matches if you first compute the probability of no birthday matches. 3. Assuming that each of the six daily flights carries 20 passengers calculate the probability that the airline will have to award two or more major prizes that month. Hint: It will be easier to calculate the probability of interest by first calculating the probability that the airline will award one or fewer prizes in a month. prizes such as free flights dictation equipment and business peri- odical subscriptions each month on a randomly selected day a major prize should be offered on all Great Air flights. This would encourage regular business fliers to fly Great Air all the time. Aaron proposed that the prize could be a trip to the Virgin Islands or somewhere similar or the cash equivalent. Great Air has three flights daily to New York and three flights returning to Boston for a total of six flights. Peter is concerned that the cost of funding six prizes of this size each month plus six daily smaller prizes might be excessive. He also believes that it might be better to increase the size of the large prize to something such as a new car but use a contest that will not guarantee a winner. But what kind of a contest can be used Just as he is about to dial Aaron’s number Margaret Runyon Great Air’s market- ing manager enters Peter’s office. He has been waiting for her to return from a meeting so he can run the contest idea past her and get her input. Margaret’s response is not as upbeat as Aaron’s but she does think the idea is worth exploring. She offers an idea for the large- prize contest that she thinks might be workable. She outlines the contest as follows. On the first of each month she and Peter will randomly select a day for that month on which the major contest will be run. That date will not be disclosed to the public. Then on each flight that day the flight attendant will have passengers write down their birthdays month and day. If any two people on the plane have the same birthday they will place their names in a hat and one name will be selected to receive the grand prize. Margaret explains that because the capacity of each flight is 40 passengers plus the crew there is a very low chance of a birthday Case 2 Let’s Make a Deal Quite a few years ago a popular show called Let’s Make a Deal appeared on network television. Contestants were selected from the audience. Each contestant would bring some silly item that he or she would trade for a cash prize or a prize behind one of three doors. Suppose that you have been selected as a contestant on the show. Y ou are given a choice of three doors. Behind one door is a new sports car. Behind the other doors are a pig and a chicken—booby prizes to be sure Let’s suppose that you pick door number one. Before open- ing that door the host who knows what is behind each door opens door two to show you the chicken. He then asks you “Would you be willing to trade door one for door three” What should you do Required Tasks: 1. Given that there are three doors one of which hides a sports car calculate the probability that your initial choice is the door that hides the sports car. What is the probability that you have not selected the correct door 2. Given that the host knows where the sports car is and has opened door 2 which revealed a booby prize does this affect the probability that your initial choice is the correct one 3. Given that there are now only two doors remaining and that the sports car is behind one of them is it to your advantage to switch your choice to door 3 Hint: Eliminate door 2 from consideration. The probability that door 1 is the correct door has not changed from your initial choice. Calculate the probability that the prize must be behind door 3. This problem was discussed in the movie 21 starring Jim Sturgess Kate Bosworth and Kevin Spacey. www.downloadslide.com

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Introduction to Probability Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. c. For Pepsi Probability 5+6+6 12+12+11 17 35 0.486 For Coke Probability 6+6+6 12+12+11 18 35 0.514 d. For Pepsi Probability 7+6+8+5 19+16+14+16 26 65 0.4 For Coke Probability 12+10+6+11 19+16+14+16 39 65 0.6 27. a. 0.9 1 - 0.5 0.45 b. 0.60.8 0.48 29. P1senior 1 and senior 22 a 5 10 ba 4 9 b 20 90 0.22 31. a. PE 1 and B PE 1 BPB 0.250.30 0.075 b. PE 1 or B PE 1 + PB - PE 1 and B 0.35 + 0.30 - 0.075 0.575 c. PE 1 and E 2 and E 3 PE 1 PE 2 PE 3 0.350.150.40 0.021 33. a. PB Number of drives from B Total drives 195 700 0.2786 b. PDefect Number of defective drives Total drives 50 700 0.0714 c. PDefect B PDefect and B PB 0.0214 0.2786 0.0769 PDefect B Number of defective drives from B Number of drives from B 15 195 0.0769 35. a. 0.61 0.316 b. 0.42 0.202 0.518 c. 0.39 37. They cannot get to 99.9 on color copies. 39. PFree gas 0.00015 + 0.00585 + 0.0005 0.0065 41. a. PNFL 105200 0.5250 b. PCollege degree and NBA 40200 0.20 c. 1050 0.20 d. The two events are not independent. 43. PLine 1 Defective 0.050.40.0725 0.2759 PLine 2 Defective 0.100.350.0725 0.4828 PLine 3 Defective 0.070.250.0725 0.2413 The unsealed cans probably came from Line 2. 45. PSupplier A Defective 0.150.30.115 0.3913 PSupplier B Defective 0.100.70.115 0.6087 Supplier B is the most likely to have supplied the defective parts. 47. a. PE 1 and E 2 PE 1 E 2 PE 2 0.5080.607 0.308 b. PE 1 and E 3 PE 1 E 3 0.6070.853 0.712 1. independent events 3. V V V C V S C V C C C S S V S C S S 5. a. subjective probability based on expert opinion b. relative frequency based on previous customer return history c. 15 0.20 7. 13 0.333333 9. a. PBrown BrownTotal 310982 0.3157 b. PYZ99 YZ99Total 375982 0.3819 c. PYZ99 and Brown 205982 0.2088 d. not mutually exclusive since their joint probability is 0.1324 11. 0.375 15. a. 0.1981 b. relative frequency c. yes 17. a. relative frequency assessment method b. P1 4000 5900 0.69 19. a. 312221768 0.1434 b. relative frequency assessment method 21. a. PCaesarean 22 50 0.44 b. New births may not exactly match the 50 in this study. 23. The following joint frequency table developed using Excel’s pivot table feature summarizes the data. Electrical Mechanical Total Lincoln 28 39 67 Tyler 64 69 133 Total 92 108 200 a. 133200 0.665 b. 108200 0.54 c. 28200 0.14 25. a. 43 100 0.43 b. 5 + 6 + 6 100 0.17 Type of Ad Occurrences Help-Wanted Ad 204 Real Estate Ad 520 Other Ad 306 Total 1030 www.downloadslide.com

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Introduction to Probability 71. Clerk 1 is most likely responsible for the boxes that raised the complaints. 73. a. 1850 0.36 b. 450 0.08 c. 320 0.15 d. 1018 0.556 e. There are a higher proportion of females whose functional background is marketing and a higher proportion of males whose functional background is operations. 75. a. 100 300 0.33 b. 30 300 0.10 c. PEast or C PEast + PC - PEast and C 0.25 + 0.333 - 0.103 0.48 d. PC East PC and EastPEast 0.1030.25 0.41 77. a. 0.3333 b. Boise will get 70.91 of the cost Salt Lake will get 21.82 and Toronto will get 7.27 regardless of production volume. 49. a. 0.76 b. 0.988 c. 0.024 d. 0.9999 51. a. Probability first sampled device is defective 350 0.06 b. 4750349 0.0576 c. 475046494548 0.8273 53. a. 0.1856 b. 0.50 c. 0.0323 d. 0.3653 55. a. 0.119 b. 0.4727 57. a. 0.50 b. 0.755 c. 0.269 63. a. 0.80 0.40 0.20 0.60 b. A B and A B are complements. 65. a. 0.0156 b. 0.1563 c. 0.50 67. a. 0.149 b. 0.997 c. 0.449 69. a. the relative frequency assessment approach b. 0.028 c. 0.349 d. yes Blyth C. R. “Subjective vs. Objective Methods in Statistics” American Statistician 26 June 1972 pp. 20–22. DeV eaux Richard D. Paul F. V elleman and David E. Bock Stats Data and Models 3rd ed. New Y ork: Addison-Wesley 2012. Hogg R. V . and Elliot A. Tanis Probability and Statistical Infer- ence 8th ed. Upper Saddle River NJ: Prentice Hall 2010. Larsen Richard J. and Morris L. Marx An Introduction to Mathematical Statistics and Its Applications 5th ed. Upper Saddle River NJ: Prentice Hall 2012. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Mlodinow Leonard The Drunkard’s Walk: How Randomness Rules Our Lives New York: Pantheon Books 2008. Raiffa H. Decision Analysis: Introductory Lectures on Choices Under Uncertainty Reading MA: Addison-Wesley 1968. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. References www.downloadslide.com

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Introduction to Probability Mutually Exclusive Events Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other event. Probability The chance that a particular event will occur. The probability value will be in the range 0 to 1. A value of 0 means the event will not occur. A probability of 1 means the event will occur. Anything between 0 and 1 reflects the uncertainty of the event occurring. The definition given is for a countable number of events. Relative Frequency Assessment The method that defines probability as the number of times an event occurs divided by the total number of times an experiment is performed in a large number of trials. Sample Space The collection of all outcomes that can result from a selection decision or experiment. Subjective Probability Assessment The method that defines probability of an event as reflecting a decision maker’s state of mind regarding the chances that the particular event will occur. Classical Probability Assessment The method of determin- ing probability based on the ratio of the number of ways an outcome or event of interest can occur to the number of ways any outcome or event can occur when the individual outcomes are equally likely. Complement The complement of an event E is the collection of all possible outcomes not contained in event E. Conditional Probability The probability that an event will occur given that some other event has already happened. Dependent Events Two events are dependent if the occurrence of one event impacts the probability of the other event occur- ring. Event A collection of experimental outcomes. Experiment A process that produces a single outcome whose result cannot be predicted with certainty. Independent Events Two events are independent if the occur- rence of one event in no way influences the probability of the occurrence of the other event. Glossary www.downloadslide.com

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Discrete Probability Distributions From Chapter 5 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Why you need to know Pak-Sense a manufacturer of temperature sensor equipment for the food industry receives component parts for its sensors weekly from suppliers. When a batch of parts arrives the quality-assurance section randomly samples a fixed number of parts and tests them to see if any are defective. Sup- pose in one such test a sample of 50 parts is selected from a supplier whose contract calls for at most 2 defective parts. How many defective parts in the sample of 50 should Pak-Sense expect if the contract is being satisfied What should be concluded if the sample contains three defects Answers to these ques- tions require calculations based on a probability distribution known as the binomial distribution. How many toll stations should be constructed when a new toll bridge is built in California If there are four toll stations will drivers have to wait too long or will there be too many toll stations and excess employees To help answer these questions decision makers use a probability distribution known as the Poisson distribution. A personnel manager has a chance to promote 3 people from 10 equally quali- fied candidates. Suppose none of six women are selected by the manager. Is this evidence of gender bias or would we expect to see this type of result A distribution known as the hypergeometric distribution would be very helpful in addressing this issue. The binomial Poisson and hypergeometric distributions are three discrete probability distributions used in business decision making. This chapter introduces discrete probability distributions and shows how they are used in business settings. Probability is the way decision makers express their uncertainty about outcomes and events. Through the use of well-established discrete probability distributions you will be better prepared for making decisions in an uncertain environment. including the Addition and Multiplication Rules. Review the discussion of weighted averages. Review the basic rules of probability Quick Prep Links Review the concepts of simple random sampling. Outcome 3. Be able to compute probabilities for the Poisson and hypergeometric distributions and apply these distributions to decision-making situations. Outcome 2. Be able to apply the binomial distribution to business decision-making situations. Outcome 1. Be able to calculate and interpret the expected value of a discrete random variable. Introduction to Discrete Probability Distributions The Binomial Probability Distribution Other Discrete Probability Distributions Discrete Probability Distributions Cobalt Creative/Shutterstock www.downloadslide.com

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Discrete Probability Distributions Introduction to Discrete Probability Distributions Random Variables When a random experiment is performed some outcome must occur. When the experiment has a quantitative characteristic we can associate a number with each outcome. For example an inspector who examines three plasma flat-panel televisions can judge each television as “acceptable” or “unacceptable.” The outcome of the experiment defines the specific number of acceptable televisions. The possible outcomes are x 50 1 2 36 The value x is called a random variable since the numerical values it takes on are random and vary from trial to trial. Although the inspector knows these are the possible values for the variable before she samples she does not know which value will occur in any given trial. Further the value of the random variable may be different each time three plasma televisions are inspected. Two classes of random variables exist: discrete random variables and continuous random variables. For instance if a bank auditor randomly examines 15 accounts to verify the accuracy of the balances the number of inaccurate account balances can be represented by a discrete random variable with the following values: x 50 1c 156 In another situation 10 employees were recently hired by a major electronics company. The number of females in that group can be described as a discrete random variable with pos- sible values equal to x 50 1 2 3c 106 Notice that the value for a discrete random variable is often determined by counting. In the bank auditing example the value of variable x is determined by counting the number of accounts with errors. In the hiring example the value of variable x is determined by counting the number of females hired. In other situations the random variable is said to be continuous. For example the exact time it takes a city bus to complete its route may be any value between two points say 30 minutes to 35 minutes. If x is the time required then x is continuous because if measured precisely enough the possible values x can be any value in the interval 30 to 35 minutes. Other examples of continuous variables include measures of distance and measures of weight when measured precisely. A continuous random variable is generally defined by measuring which is contrasted with a discrete random variable whose value is typically determined by counting. Displaying Discrete Probability Distributions Graphically The probability distribu- tion for a discrete random variable is composed of the values the variable can assume and the probabilities for each of the possible values. For example if three parts are tested to determine if they are defective the probability distribution for the number of defectives might be x Number of Defectives Px 0 0.10 1 0.30 2 0.40 3 0.20 g 1.00 Graphically the discrete probability distribution associated with these defectives can be represented by the areas of rectangles in which the base of each rectangle is one unit wide and the height corresponds to the probability. The areas of the rectangles sum to 1. Figure 1 illus- trates two examples of discrete probability distributions. Figure 1a shows a discrete random Random Variable A variable that takes on different numerical values based on chance. Discrete Random Variable A random variable that can only assume a finite number of values or an infinite sequence of values such as 0 1 2.… Continuous Random Variables Random variables that can assume an uncountably infinite number of values. www.downloadslide.com

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Discrete Probability Distributions variable with only three possible outcomes. Figure 1b shows the probability distribution for a discrete variable that has 21 possible outcomes. Note as the number of possible outcomes increases the distribution becomes smoother and the individual probability of any particular value tends to be reduced. In all cases the sum of the probabilities is 1. Discrete probability distributions have many applications in business decision-making situations. In the remainder of this section we discuss several important issues that are of particular importance to discrete probability distributions. Mean and Standard Deviation of Discrete Distributions A probability distribution like a frequency distribution can be only partially described by a graph. To aid in a decision situation you may need to calculate the distribution’s mean and standard deviation. These values measure the central location and spread respectively of the probability distribution. Calculating the Mean The mean of a discrete probability distribution is also called the expected value of the random variable from an experiment. The expected value is actually a weighted average of the random variable values in which the weights are the probabilities assigned to the values. The expected value is given in Equation 1. Expected Value The mean of a probability distribution. The average value when the experiment that generates values for the random variable is repeated over the long run. Probability 10 20 30 Possible Values of x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 a Discrete Probability Distribution 3 possible outcomes x Px Probability 123456789 101112131415161718192021 Possible Values of x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0 b Discrete Probability Distribution 21 possible outcomes x Px FIGURE 1 | Discrete Probability Distributions Chapter Outcome 1. www.downloadslide.com

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Discrete Probability Distributions Calculating the Standard Deviation The standard deviation measures the spread or dispersion in a set of data. The standard deviation also measures the spread in the values of a random variable. To calculate the standard deviation for a discrete probability distribution use Equation 2. Expected Value of a Discrete Probability Distribution E1x2 a xP1x2 1 where: E1x2 Expected value of x x Values of the random variable P1x2 probability of the random variable taking on the value x Standard Deviation of a Discrete Probability Distribution x xEx Px ∑− 2 2 where: x Values of the random variable E1x2 Expected value of x P1x2 Probability of the random variable taking on the value x Equation 2 is different in form than other equations given for standard deviation. This is because we are now dealing with a discrete probability distribution rather than population or sample values. EXAMPLE 1 COMPUTING THE MEAN AND STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE Swenson Security Sales Swenson Security Sales is a com- pany that sells and installs home security systems throughout the eastern United States. Each week the company’s quality manag- ers examine one randomly selected security installation in all 21 states where the company operates to see whether the installers made errors. The discrete random variable x is the number of errors discovered on each installation examined ranging from 0 to 3. The following frequency distribution was developed: x Frequency 0 150 1 110 250 3 90 g 400 Assuming that these data reflect typical performance by the installers the company leadership wishes to develop a discrete probability distribution for the number of install errors and compute the mean and standard deviation for the distribution. This can be done using the following steps: Step 1 Convert the frequency distribution into a probability distribution using the relative frequency assessment method. x Frequency 0 150400 0.375 1 110400 0.275 Scanrail/Fotolia www.downloadslide.com

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Discrete Probability Distributions x Frequency 2 50400 0.125 3 90400 0.225 g 1.000 Step 2 Compute the expected value using Equation 1. Ex xP x Ex . . ∑ 0 0 375 1 0 275 2 0. . . . 125 3 0 225 120 Ex The expected value is 1.20 errors per install. Thus assuming the distribution of the number of errors is representative of that of each individual installation the long-run average number of errors per installation will be 1.20. Step 3 Compute the standard deviation using Equation 2. x xEx Px ∑ 2 xP1x23x - E1x24 3x - E1x24 2 3x - E1x24 2 P1x2 0 0.375 0-1.2-1.20 1.44 0.540 1 0.275 1-1.2- 0.20 0.04 0.011 2 0.125 2-1.2 0.80 0.64 0.080 3 0.225 3-1.2 1.80 3.24 0.729 g 1.360 x 136 117 .. The standard deviation of the discrete probability distribution is 1.17 errors per security system installed. END EXAMPLE TRY PROBLEM 4 BUSINESS APPLICATION EXPECTED VALUES BENSON LIGHT FIXTURES Benson Light Fixtures imports fixtures from Taiwan and other Far East countries for distribution in France and Spain. For one particular product line Benson currently has two suppliers. Both suppliers have poor records when it comes to quality. Benson is planning to purchase 100000 of a particular light fixture and wants to use the least-cost supplier for the entire purchase. Supplier A is less expensive by 1.20 per fixture and has an ongoing record of supplying 10 defects. Supplier B is more expensive but may be a higher-quality supplier. Benson records indicate that the rate of defects from supplier B varies. Table 1 shows the probability distributions for the defect percentages for supplier B. Each defect is thought to cost the company 9.50. Looking first at supplier A at a defect rate of 0.10 out of 100000 units the number of defects is expected to be 10000. The cost of these is +9.50 10000 +95000. For sup- plier B the expected defect rate is found using Equation 1 as follows: E1Defect rate2 a x P1x2 E1Defect rate2 10.01210.32 + 10.05210.42 + 10.10210.22 + 10.15210.12 E1Defect rate2 0.058 Thus supplier B is expected to supply 5.8 defects or 5800 out of the 100000 units ordered for an expected cost of +9.50 5800 +55100. Based on defect cost alone supplier B is less expensive 55100 versus 95000. However recall that supplier B’s TABLE 1 | Probability Distribution—Defect Rate for Supplier B Defect Rate x Probability Px 0.01 0.3 0.05 0.4 0.10 0.2 0.15 0.1 www.downloadslide.com

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Discrete Probability Distributions product sells for 1.20 per unit more. Thus on a 100000-unit order supplier B costs an extra +1.20 100000 +120000 more than supplier A. The relative costs are Supplier A +95000 Supplier B +55100 + +120000 +175100 Therefore based on expected costs supplier A should be selected to supply the 100000 light fixtures. Skill Development 5-1. An economics quiz contains six multiple-choice questions. Let x represent the number of questions a student answers correctly. a. Is x a continuous or discrete random variable b. What are the possible values of x 5-2. Two numbers are randomly drawn without replacement from a list of five. If the five numbers are 2 2 4 6 8 what is the probability distribution of the sum of the two numbers selected Show the probability distribution graphically. 5-3. If the Prudential Insurance Company surveys its customers to determine the number of children under age 22 living in each household a. What is the random variable for this survey b. Is the random variable discrete or continuous 5-4. Given the following discrete probability distribution xPx 50 0.375 65 0.15 70 0.225 75 0.05 90 0.20 a. Calculate the expected value of x. b. Calculate the variance of x. c. Calculate the standard deviation of x. 5-5. Because of bad weather the number of days next week that the captain of a charter fishing boat can leave port is uncertain. Let x number of days that the boat is able to leave port per week. The following probability distribution for the variable x was determined based on historical data when the weather was poor: xPx 0 0.05 1 0.10 2 0.10 3 0.20 4 0.20 5 0.15 6 0.15 7 0.05 Based on the probability distribution what is the expected number of days per week the captain can leave port 5- Exercises MyStatLab 5-6. Consider the following discrete probability distribution: xP x 3 0.13 6 0.12 9 0.15 12 0.60 a. Calculate the variance and standard deviation of the random variable. b. Let y x + 7. Calculate the variance and standard deviation of the random variable y. c. Let z 7x. Calculate the variance and standard deviation of the random variable z. d. From your calculations in part a and part b indicate the effect that adding a constant to a random variable has on its variance and standard deviation. e. From your calculations in part a and part c indicate the effect that multiplying a random variable with a constant has on the variance and the standard deviation of the random variable. 5-7. Given the following discrete probability distribution xP x 100 0.25 125 0.30 150 0.45 a. Calculate the expected value of x. b. Calculate the variance of x. c. Calculate the standard deviation of x. 5-8. The roll of a pair of dice has the following probability distribution where the random variable x is the sum of the values produced by each die: xP x 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 www.downloadslide.com

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Discrete Probability Distributions 5-12. Jennings Assembly in Hartford Connecticut uses a com- ponent supplied by a company in Brazil. The component is expensive to carry in inventory and consequently is not always available in stock when requested. Furthermore shipping schedules are such that the lead time for trans- portation of the component is not a constant. Using his- torical records the manufacturing firm has developed the following probability distribution for the product’s lead time. The distribution is shown here where the random variable x is the number of days between the placement of the replenishment order and the receipt of the item. xP x 2 0.15 3 0.45 4 0.30 5 0.075 6 0.025 a. What is the average lead time for the component b. What is the coefficient of variation for delivery lead time c. How might the manufacturing firm in the United States use this information 5-13. Marque Electronics is a family-owned electronics repair business in Kansas City. The owner has read an advertisement from a local competitor that guarantees all high-definition television HDTV repairs within four days. Based on his company’s experience he wants to know if he can offer a similar guarantee. His past service records are used to determine the following probability distribution: Number of Days Probability 1 0.15 2 0.25 3 0.30 4 0.18 5 0.12 a. Calculate the mean number of days his customers wait for an HDTV repair. b. Also calculate the variance and standard deviation. c. Based on the calculations in parts a and b what conclusion should the manager reach regarding his company’s repair times 5-14. Cramer’s Bar and Grille in Dallas can seat 130 people at a time. The manager has been gathering data on the number of minutes a party of four spends in the restaurant from the moment they are seated to when they pay the check. What is the mean number of minutes for a dinner party of four What is the variance and standard deviation Number of Minutes Probability 60 0.05 70 0.15 a. Calculate the expected value of x. b. Calculate the variance of x. c. Calculate the standard deviation of x. 5-9. Consider the following discrete probability distribution: xP x 5 0.10 10 0.15 15 0.25 20 0.50 a. Calculate the expected value of the random variable. b. Let y x + 5. Calculate the expected value of the new random variable y. c. Let z 5x. Calculate the expected value of the new random variable z. d. From your calculations in part a and part b indicate the effect that adding a constant to a random variable has on the expected value of the random variable. e. From your calculations in part a and part c indicate the effect that multiplying a random variable by a constant has on the expected value of the random variable. 5-10. Examine the following probability distribution: x 5 101520253035404550 Px 0.01 0.05 0.14 0.20 0.30 0.15 0.05 0.04 0.01 0.05 a. Calculate the expected value and standard deviation for this random variable. b. Denote the expected value as m. Calculate m -s and m +s. c. Determine the proportion of the distribution that is contained within the interval m s. d. Repeat part c for 112m 2s and 122m 3s. Business Applications 5-11. The U.S. Census Bureau Annual Social Economic Supplement collects demographics concerning the number of people in families per household. Assume the distribution of the number of people per household is shown in the following table: xP x 2 0.27 3 0.25 4 0.28 5 0.13 6 0.04 7 0.03 a. Calculate the expected number of people in families per household in the United States. b. Compute the variance and standard deviation of the number of people in families per household. www.downloadslide.com

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Discrete Probability Distributions x Weekly Demand Px 0 0.05 1 0.05 2 0.10 3 0.20 4 0.40 5 0.10 6 0.05 7 0.05 a. What is the expected weekly demand for the alarm clock radio b. What is the probability that weekly demand will be greater than the number of available radios c. What is the expected weekly profit from the sale of the alarm clock radio Remember: There are only four clock radios available in any week to meet demand. d. On average how much profit is lost each week because the radio is not available when demanded 5-17. Fiero Products LTD of Bologna Italy makes a variety of footwear including indoor slippers children’s shoes and flip-flops. To keep up with increasing demand it is considering three expansion plans: 1 a small factory with yearly costs of 150000 that will increase the production of flip-flops to 400000 2 a mid-sized factory with yearly costs of 250000 that will increase the production of flip-flops by 600000 and 3 a large factory with yearly costs of 350000 that will increase the production of flip-flops by 900000. The profit per flip-flop is projected to be 0.75. The probability distribution of demand for flip- flops is considered to be Demand 300000 700000 900000 Probability 0.2 0.5 0.3 a. Compute the expected profit for each of the expansion plans. b. Calculate the standard deviation for each of the expansion plans. c. Which expansion plan would you suggest Provide the statistical reasoning behind your selection. 5-18. A large corporation in search of a CEO and a CFO has narrowed the fields for each position to a short list. The CEO candidates graduated from Chicago C and three Ivy League universities: Harvard H Princeton P and Yale Y. The four CFO candidates graduated from MIT M Northwestern N and two Ivy League universities Dartmouth D and Brown B. The personnel director wishes to determine the distribution of the number of Ivy League graduates who could fill these positions. a. Assume the selections were made randomly. Construct the probability distribution of the number of Ivy League graduates who could fill these positions. Number of Minutes Probability 80 0.20 90 0.45 100 0.10 110 0.05 5-15. Rossmore Brothers Inc. sells plumbing supplies for commercial and residential applications. The company currently has only one supplier for a particular type of faucet. Based on historical data that the company has maintained the company has assessed the following probability distribution for the proportion of defective faucets that it receives from this supplier: Proportion Defective Probability 0.01 0.4 0.02 0.3 0.05 0.2 0.10 0.1 This supplier charges Rossmore Brothers Inc. 29.00 per unit for this faucet. Although the supplier will replace any defects free of charge Rossmore managers figure the cost of dealing with the defects is about 5.00 each. a. Assuming that Rossmore Brothers is planning to purchase 2000 of these faucets from the supplier what is the total expected cost to Rossmore Brothers for the deal b. Suppose that Rossmore Brothers has an opportunity to buy the same faucets from another supplier at a cost of 28.50 per unit. However based on its investigations Rossmore Brothers has assessed the following probability distribution for the proportion of defective faucets that will be delivered by the new supplier: Proportion Defective x Probability Px 0.01 0.1 0.02 0.1 0.05 0.7 0.10 0.1 Assuming that the defect cost is still 5.00 each and based on total expected cost for an order of 2000 faucets should Rossmore buy from the new supplier or stick with its original supplier 5-16. Radio Shack stocks four alarm clock radios. If it has fewer than four clock radios available at the end of a week the store restocks the item to bring the in-stock level up to four. If weekly demand is greater than the four units in stock the store loses the sale. The radio sells for 25 and costs the store 15. The Radio Shack manager estimates that the probability distribution of weekly demand for the radio is as follows: www.downloadslide.com

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Discrete Probability Distributions It presents the number of days Revolution could remain effective when applied to mature cats. a. Produce a frequency distribution for these data. Convert the frequency distribution into a probability distribution using the relative frequency assessment method. b. Calculate the expected value and standard deviation for the number of days Revolution could remain effective. c. If the marketing department wished to advertise the number of days that 90 of the cats remain protected while using Revolution what would this number of days be 5-21. Fiber Systems makes boat tops for a number of boat manufacturers. Its fabric has a limited two-year warranty. Periodic testing is done to determine if the warranty policy should be changed. One such study may have examined those covers that became unserviceable while still under warranty. Data that could be produced by such a study are contained in the file titled Covers. The data represent assessment of the number of months a cover was used until it became unserviceable. a. Produce a frequency distribution for these data. Convert the frequency distribution into a probability distribution using the relative frequency assessment method. b. Calculate the expected value and standard deviation for the time until the covers became unserviceable. c. The quality-control department thinks that among those covers that do become unserviceable while still under warranty the majority last longer than 19 months. Produce the relevant statistic to verify this assumption. b. Would it be surprising if both positions were filled with Ivy League graduates c. Calculate the expected value and standard deviation of the number of Ivy League graduates who could fill these positions. Computer Database Exercises 5-19. Starbucks has entered into an agreement with a publisher to begin selling a food and beverage magazine on a trial basis. The magazine retails for 3.95 in other stores. Starbucks bought it for 1.95 and sold it for 3.49. During the trial period Starbucks placed 10 copies of the magazine in each of 150 stores throughout the country. The file titled Sold contains the number of magazines sold in each of the stores. a. Produce a frequency distribution for these data. Convert the frequency distribution into a probability distribution using the relative frequency assessment method. b. Calculate the expected profit from the sale of these 10 magazines. c. Starbucks is negotiating returning all unsold magazines for a salvage price. Determine the salvage price Starbucks will need to obtain to yield a positive expected profit from selling 10 magazines. 5-20. Pfizer Inc. is the manufacturer of Revolution Selamectin a topical parasiticide used for the treatment control and prevention of flea infestation heartworm and ear mites for dogs and cats. One of its selling points is that it provides protection for an entire month. Such claims are made on the basis of research and statistical studies. The file titled Fleafree contains data similar to those obtained in Pfizer’s research. END EXERCISES 5-1 The Binomial Probability Distribution In Section 1 you learned that random variables can be classified as either discrete or continu- ous. In most instances the value of a discrete random variable is determined by counting. For instance the number of customers who arrive at a store each day is a discrete variable. Its value is determined by counting the customers. Several theoretical discrete distributions have extensive application in business decision making. A probability distribution is called theoretical when the mathematical properties of its random variable are used to produce its probabilities. Such distributions are different from the distributions that are obtained subjectively or from observation. Sections 2 and 3 focus on theoretical discrete probability distributions. The Binomial Distribution The first theoretical probability distribution we will consider is the binomial distribution that describes processes whose trials have only two possible outcomes. The physical events described by this type of process are widespread. For instance a quality-control system in a manufacturing plant labels each tested item as either defective or acceptable. A firm bidding for a contract either will or will not get the contract. A marketing research firm may receive responses to a questionnaire in the form of “Yes I will buy” or “No I will not buy.” The personnel manager in an organization is faced with two possible outcomes each time he offers a job—the applicant either accepts the offer or rejects it. Binomial Probability Distribution Characteristics A distribution that gives the probability of x successes in n trials in a process that meets the following conditions: 1. A trial has only two possible outcomes: a success or a failure. 2. There is a fixed number n of identical tri- als. 3. The trials of the experiment are independent of each other. This means that if one out- come is a success this does not influence the chance of another outcome being a success. 4. The process must be consistent in gener- ating successes and failures. That is the probability p associated with a success remains constant from trial to trial. 5. If p represents the probability of a success then11 - p2 q is the probability of a failure. Chapter Outcome 2. www.downloadslide.com

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Discrete Probability Distributions Characteristics of the Binomial Distribution The binomial distribution requires that the experiment’s trials be independent. This can be assured if the sampling is performed with replacement from a finite population. This means that an item is sampled from a population and returned to the population after its characteristics have been recorded before the next item is sampled. However sampling with replacement is the exception rather than the rule in business applications. Most often the sampling is performed without replacement. Strictly speaking when sampling is per- formed without replacement the conditions for the binomial distribution cannot be satisfied. However the conditions are approximately satisfied if the sample selected is quite small relative to the size of the population from which the sample is selected. A commonly used rule of thumb is that the binomial distribution can be applied if the sample size is at most 5 of the population size. BUSINESS APPLICATION USING THE BINOMIAL DISTRIBUTION LAKE CITY AUTOMOTIVE Lake City Automotive performs 300 automobile transmission repairs every week. Transmission repair work is not an exact science and a percentage of the work done must be reworked. Lake City shop managers have determined that even when the mechanics perform the work in a proper manner 10 of the time the car will have to be worked on again to fix a problem. The binomial distribution applies to this situation because the following conditions exist: 1. There are only two possible outcomes when a car is repaired: It either needs rework or it doesn’t. We are interested in the cars that need rework so we will consider a reworked car to be a “success.” A success occurs when we observe the outcome of interest. 2. All repair jobs are considered to be equivalent in the work required. 3. The outcome of a repair rework or no rework is independent of whether the preceding repair required rework or not. 4. The probability of rework being needed p 0.10 remains constant from car to car. 5. The probability of a car not needing rework q 1 - p 0.90 remains constant from car to car. To determine whether the Lake City mechanics are continuing to function at the stand- ard level of performance the shop foreman randomly selects four cars from the week’s list of repaired vehicles and tracks those to see whether they need rework or not note the need for rework is known within a few hours of the work being completed. Because the sample size is small 14300 0.0133 or 1.332 relative to the size of the population 300 units per week the conditions of independence and constant probability will be approximately satis- fied because the sample is less than 5 of the population. We let the number of reworked cars be the random variable of interest. The number of reworked units is limited to discrete values x 0 1 2 3 or 4. We can determine the prob- ability that the random variable will have any of the discrete values. One way is to list the sample space as shown in Table 2. We can find the probability of zero cars needing rework for instance by employing the Multiplication Rule for Independent Events. P1x 0 reworks2 P1G and G and G and G2 where: G Car does not require rework Here P1G2 0.90 and we have assumed the repair jobs are independent. Using the Multiplication Rule for Independent Events P1G and G and G and G2 P1G2P1G2P1G2P1G2 10.90210.90210.90210.902 0.90 4 0.6561 Dmitry V ereshchagin/Fotolia www.downloadslide.com

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Discrete Probability Distributions We can also find the probability of exactly one reworked car in a sample of four. This is accomplished using both the Multiplication Rule for Independent Events and the Addition Rule for Mutually Exclusive Events: D car needs rework P1x 1 rework2 P1G and G and G and D2 + P1G and G and D and G2 + P1G and D and G and G2 + P1D and G and G and G2 where: P1G and G and G and D2 P1G2P1G2P1G2P1D2 10.90210.90210.90210.102 10.90 3 210.102 Likewise: P1G and G and D and G2 10.90 3 210.102 P1G and D and G and G2 10.90 3 210.102 P1D and G and G and G2 10.90 3 210.102 Then: P11 rework2 10.90 3 210.102 + 10.90 3 210.102 + 10.90 3 210.102 + 10.90 3 210.102 14210.90 3 210.102 0.2916 Note that each of the four possible ways of finding one rework car has the same probabil- ity 10.90 3 210.102. We determine the probability of one of the ways to obtain one rework car and multiply this value by the number of ways four of obtaining one reworked car. This produces the overall probability of one reworked car. Combinations In this relatively simple application we can fairly easily list the sample space and from that count the number of ways that each possible number of reworked cars can occur. However for examples with larger sample sizes this approach is inefficient. A more effective method exists for counting the number of ways binomial events can occur. This method is called the counting rule for combinations. This rule is used to find the number of outcomes from an experiment in which x objects are to be selected from a group of n objects. Equation 3 is used to find the combinations. TABLE 2 | Sample Space for Lake City Automotive Results No. of Reworked Cars No. of Ways GGGG 01 GGGD 1 GGDG 14 GDGG 1 DGGG 1 GGDD 2 GDGD 2 DGGD 26 GDDG 2 DGDG 2 DDGG 2 DDDG 3 DDGD 34 DGDD 3 GDDD 3 DDDD 41 www.downloadslide.com

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Discrete Probability Distributions Using Equation 3 we find the number of ways that x 2 reworked cars can occur in a sam- ple of n 4 as C n xn x x n 55 5 −− 4 24 2 43 2 1 21 21 24 4 6 55 ways Refer to Table 2 to see that this is the same value for two reworked cars in a sample of four that was obtained by listing the sample space. Now we can find the probabilities of two reworked cars. P12 reworks2 16210.90 2 210.10 2 2 0.0486 Use this method to verify the following: P13 reworks2 14210.90210.10 3 2 0.0036 P14 reworks2 11210.10 4 2 0.0001 The key to developing the probability distribution for a binomial process is first to determine the probability of any one way the event of interest can occur and then to multi- ply this probability by the number of ways that event can occur. Table 3 shows the binomial probability distribution for the number of reworked cars in a sample size of four when the probability of any individual car requiring rework is 0.10. The probability distribution is graphed in Figure 2. Most samples would contain zero or one reworked car when the mechanics at Lake City are performing the work to standard. Counting Rule for Combinations C n xn x x n 3 where: Cx x n Number of combinations of objects sel lected from objects n nnn n ... 12 2 1 0 1 by defnition TABLE 3 | Binomial Distribution for Lake City Automotive: n 4 p 0.10 x of Reworks P1x2 0 0.6561 1 0.2916 2 0.0486 3 0.0036 4 0.0001 g 1.0000 Probability 04 3 2 1 x 5 Number of Reworks Px 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIGURE 2 | Binomial Distribution for Lake City Automotive www.downloadslide.com

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Discrete Probability Distributions Binomial Formula The steps that we have taken to develop this binomial probability distribution can be summarized through a formula called the binomial formula shown as Equation 4. Note this formula is composed of two parts: the combinations of x items selected from n items and the probability that x items can occur. Binomial Formula Px n xn x xn x pq 4 where: n Random sample size x Number of successes 1when a success is defined as what we are looking for2 n - x Number of failures p Probability of a success q 1 - p Probability of a failure n n1n - 121n - 221n - 32 . . . 122112 0 1 by definition Applying Equation 4 to the Lake City Automotive example for n 4 p 0.10 and x 2 reworked cars we get Px n xn x pq xn x 5 − − P . . . . 2 0 10 0 90 6 0 10 0 90 0 22 22 55 5 4 22 . .0486 This is the same value we calculated earlier when we listed out the sample space above. EXAMPLE 2 USING THE BINOMIAL FORMULA Creative Style and Cut Creative Style and Cut an upscale beauty salon in San Francisco offers a full refund to anyone who is not satis- fied with the way his or her hair looks after it has been cut and styled. The owners believe the hairstyle satisfaction from customer to customer is independent and that the probability a customer will ask for a refund is 0.20. Suppose a random sample of six customers is observed. In four instances the customer has asked for a refund. The owners might be interested in the probability of four refund requests from six customers. If the binomial distribution applies the probability can be found using the following steps: Step 1 Define the characteristics of the binomial distribution. In this case the characteristics are n 6 p 0.20 q 1 - p 0.80 Step 2 Determine the probability of x successes in n trials using the binomial formula Equation 4. In this case n 6 p 0.20 q 0.80 and we are interested in the probability of x 4 successes. Px n xn x pq P xn x . 4 6 46 4 020 4 . . . . 080 4 15 0 20 0 80 4 0 0154 64 42 P P Szasz-Fabian Ilka Erika/Shutterstock www.downloadslide.com

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Discrete Probability Distributions There is only a 0.0154 chance that exactly four customers will want a refund in a sample of six if the chance that any one of the customers will want a refund is 0.20. END EXAMPLE TRY PROBLEM 24 Using the Binomial Distribution Table Using Equation 4 to develop the binomial dis- tribution is not difficult but it can be time consuming. To make binomial probabilities easier to find you can use the binomial table in Cumulative Binomial Distribution Table. This table is constructed to give cumulative probabilities for different sample sizes and probabilities of success. Each column is headed by a probability p which is the probability associated with a success. The column headings correspond to probabilities of success ranging from 0.01 to 1.00. Down the left side of the table are integer values that correspond to the number of suc- cesses x for the specified sample size n. The values in the body of the table are the cumula- tive probabilities of x or fewer successes in a random sample of size n. BUSINESS APPLICATION BINOMIAL DISTRIBUTION TABLE BIG O TIRE COMPANY Big O Tire Company operates tire and repair stores throughout the western United States. Upper management is considering offering a money-back warranty if one of their tires blows out in the first 20000 miles of use. The managers are willing to make this warranty if 2 or fewer of the tires they sell blow out within the 20000-mile limit. The company plans to test 10 randomly selected tires over 20000 miles of use in test conditions. The number of possible blowouts will be x 0 1 2 c10. We can use the bino- mial table in Cumulative Binomial Distribution Table to develop the probability distribution. This table is called a cumulative probability table. Go to the table for n 10 and the column for p 0.02. The values of x are listed down the left side of the table. For example the cumu- lative probability of x … 2 occurrences is 0.9991. This means that it is extremely likely that 2 or fewer tires in a sample of 10 would blow out in the first 20000 miles of use if the overall fraction of tires that will blow out is 0.02. The probability of 3 or more blowouts in the sample of n 10 is P1x Ú 32 1 - P1x … 22 1 - 0.9991 0.0009 There are about 9 chances in 10000 that we would fnd 3 or more tires in a sample of 10 that will blow out if the probability of it happening for any one tire is p 0.02. If the test did show that 3 tires blew out the true rate of tire blowouts likely exceeds 2 and the company should have serious doubts about making the warranty. Note the decision about this will depend on which sample outcome occurs. EXAMPLE 3 USING THE BINOMIAL TABLE Nielsen Television Ratings The Nielsen Media Group is the best-known television ratings company. On Monday Feb- ruary 6 2012 the day after the 2012 Super Bowl between the Giants and the Patriots Nielsen reported that the game was the most-watched program in history with just over 40 of all televisions in the United States tuned to the game on Sunday. Assuming that the 40 rating is correct what is the probabil- ity that in a random sample of 20 television sets 2 or fewer would have been tuned to the Super Bowl This question can be answered assuming that the binomial distribution applies using the following steps: Step 1 Define the characteristics of the binomial distribution. In this case the characteristics are n 20 p 0.40 q 1 - p 0.60 Step 2 Define the event of interest. We are interested in knowing P1x … 22 P102 + P112 + P122 Ronen/Shutterstock www.downloadslide.com

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Discrete Probability Distributions Step 3 Go to the binomial table in Cumulative Binomial Distribution Table to find the desired probability. In this case we locate the section of the table corresponding to sample size equal to n 20 and go to the column headed p 0.40 and the row labeled x 2. The cumulative P1x … 22 listed in the table is 0.0036. Thus there is only a 0.0036 chance that 2 or fewer sets in a random sample of 20 were tuned to the Super Bowl. Thus it is unlikely that 2 or fewer sets in a sample of 20 TV sets would have been tuned to the Super Bowl in 2012. END EXAMPLE TRY PROBLEM 28 EXAMPLE 4 USING THE BINOMIAL DISTRIBUTION Clearwater Research Clearwater Research is a full-service marketing research consulting firm. Recently it was retained to do a project for a major U.S. airline. The airline was considering changing from an assigned-seating reservation system to one in which fliers would be able to take any seat they wished on a first-come first-served basis. The airline believes that 80 of its fliers would like this change if it was accompanied with a reduction in ticket prices. Clearwater Research will survey a large number of customers on this issue but prior to conducting the full research it has selected a random sample of 20 customers and determined that 12 like the proposed change. What is the probability of finding 12 or fewer who like the change if the probability is 0.80 that a customer will like the change If we assume the binomial distribution applies we can use the following steps to answer this question: Step 1 Define the characteristics of the binomial distribution. In this case the characteristics are n 20 p 0.80 q 1 - p 0.20 Step 2 Define the event of interest. We are interested in knowing P1x … 122 Step 3 Go to the binomial table in Cumulative Binomial Distribution Table to find the desired probability. Locate the table for the sample size n. Locate the column for p 0.80. Go to the row corresponding to x 12 and the column for p 0.80 in the section of the table for n 20 to get P1x … 122 0.0321 Thus it is quite unlikely that if 80 of customers like the new seating plan 12 or fewer in a sample of 20 would like it. The airline may want to rethink its plan. END EXAMPLE TRY PROBLEM 29 Mean and Standard Deviation of the Binomial Distribution In Section 1 we stated the mean of a discrete probability distribution is also referred to as the expected value. The expected value of a discrete random variable x is found using Equation 1. m x E1x2 ΣxP1x2 Fernando Blanco Calzada/ Shutterstock www.downloadslide.com

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Discrete Probability Distributions MEAN OF A BINOMIAL DISTRIBUTION This equation for the expected value can be used with any discrete probability distribution including the binomial. However if we are working with a binomial distribution the expected value can be found more easily by using Equation 5. Expected Value of a Binomial Distribution m x E1x2 np 5 where: n Sample size p Probability of a success BUSINESS APPLICATION BINOMIAL DISTRIBUTION CATALOG SALES Catalog sales have been a part of the U.S. economy for many years and companies such as Lands’ End L.L. Bean and Eddie Bauer have enjoyed increased business. One feature that has made mail-order buying so popular is the ease with which customers can return merchandise. Nevertheless one mail-order catalog has the goal of no more than 11 of all purchased items returned. The binomial distribution can describe the number of items returned. For instance in a given hour the company shipped 300 items. If the probability of an item being returned is p 0.11 the expected number of items mean to be returned is m x E1x2 np m x E1x2 1300210.112 33 Thus the average number of returned items for each 300 items shipped is 33. Suppose the company sales manager wants to know if the return rate is stable at 11. To test this she monitors a random sample of 300 items and finds that 44 have been returned. This return rate exceeds the mean of 33 units which concerns her. However before reaching a conclusion she will be interested in the probability of observing 44 or more returns in a sample of 300. P1x Ú 442 1 - P1x … 432 The Cumulative Binomial Distribution Table does not contain sample sizes as large as 300. Instead we can use Excel’s BINOM.DIST function to find the probability. The Excel output in Figure 3 shows the cumulative probability of 43 or fewer is equal to P1x … 432 0.97 Then the probability of 44 or more returns is P1x Ú 442 1 - 0.97 0.03 There is only a 3 chance of 44 or more items being returned if the 11 return rate is still in effect. This low probability suggests that the return rate may have increased above 11 because we would not expect to see 44 returned items. The probability is very small. Excel Tutorial Excel tutorials www.downloadslide.com

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Discrete Probability Distributions Excel 2010 Instructions: 1. Open blank worksheet. 2. Click on f x function wizard. 3. Select Statistical category. 4. Select the BINOM.DIST function. 5. Fill in the requested information in the template. 6. True indicates cumulative probabilities. Minitab Instructions for similar results: 1. Choose Calc Probability Distribution Binomial. 2. Choose Cumulative probability. 3. In Number of trials enter sample size. 4. In Probability of success enter p. 5. In Input constant enter the number of successes: x. 6. Click OK. Cumulative Probability FIGURE 3 |  Excel 2010 Binomial Distribution Output for Catalog Sales EXAMPLE 5 FINDING THE MEAN OF THE BINOMIAL DISTRIBUTION Clearwater Research In Example 4 Clearwater Research had been hired to do a study for a major airline that is planning to change from a designated-seat assignment plan to an open- seating system. The company believes that 80 of its customers approve of the idea. Clearwater Research interviewed a sample of n 20 and found 12 who like the proposed change. If the airline is correct in its assess- ment of the probability what is the expected number of people in a sample of n 20 who will like the change We can find this using the following steps: Step 1 Define the characteristics of the binomial distribution. In this case the characteristics are n 20 p 0.80 q 1 - p 0.20 Step 2 Use Equation 5 to find the expected value. m x E1x2 np E1x2 2010.802 16 The average number who would say they like the proposed change is 16 in a sample of 20. END EXAMPLE TRY PROBLEM 33a STANDARD DEVIATION OF A BINOMIAL DISTRIBUTION The standard deviation for any discrete probability distribution can be calculated using Equation 2. We show this again as x xEx Px ∑ 2 Fernando Blanco Calzada/ Shutterstock www.downloadslide.com

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Discrete Probability Distributions If a discrete probability distribution meets the binomial distribution conditions the stand- ard deviation is more easily computed by Equation 6. Standard Deviation of the Binomial Distribution npq 6 where: n Sample size p Probability of a success q 1 - p Probability of a failure Probability Px 14 18 22 26 30 34 38 Number of Successes x c 0.12 0.10 0.08 0.06 0.04 0.02 0 Probability Px 010 8 6 429 7 5 3 1 Number of Successes x b 0.30 0.25 0.20 0.15 0.10 0.05 0 Probability Px 05 4 3 2 1 Number of Successes x a n 5 p 0.50 n 10 p 0.50 n 50 p 0.50 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 FIGURE 4 |  The Binomial Distribution with Varying Sample Sizes 1p 0.502 EXAMPLE 6 FINDING THE STANDARD DEVIATION OF A BINOMIAL DISTRIBUTION Clearwater Research Refer to Examples 4 and 5 in which Clearwater Research surveyed a sample of n 20 airline custom- ers about changing the way seats are assigned on flights. The airline believes that 80 of its customers approve of the proposed change. Example 5 showed that if the airline is correct in its assessment the expected number in a sample of 20 who would like the change is 16. However there are other possible outcomes if 20 customers are surveyed. What is the standard deviation of the random variable x in this case We can find the standard deviation for the binomial distribution using the following steps: Step 1 Define the characteristics of the binomial distribution. In this case the characteristics are n 20 p 0.80 q 1 - p 0.20 Step 2 Use Equation 6 to calculate the standard deviation. npq 20 0 80 0 20 1 7889 . . . END EXAMPLE TRY PROBLEM 33b Additional Information about the Binomial Distribution At this point several com- ments about the binomial distribution are worth making. If p the probability of a success is 0.50 the binomial distribution is symmetrical and bell-shaped regardless of the sam- ple size. This is illustrated in Figure 4 which shows frequency histograms for samples of n 5 n 10 and n 50. Notice that all three distributions are centered at the expected value E1x2 np. When the value of p differs from 0.50 in either direction the binomial distribution is skewed. The skewness will be most pronounced when n is small and p approaches 0 or 1. However the binomial distribution becomes more bell shaped as n increases. The frequency histograms shown in Figure 5 bear this out. Fernando Blanco Calzada/ Shutterstock www.downloadslide.com

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Discrete Probability Distributions Probability Px 011 8910 6 4 27 5 3 1 Number of Successes x c 0.30 0.25 0.20 0.15 0.10 0.05 0 05 4 3 2 1 Number of Successes x a 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Probability Px Probability Px 08 6 4 27 5 3 1 Number of Successes x b 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 n 10 p 0.05 n 20 p 0.05 n 50 p 0.05 FIGURE 5 |  The Binomial Distribution with Varying Sample Sizes 1p 0.052 Skill Development 5-22. The manager for State Bank and Trust has recently examined the credit card account balances for the customers of her bank and found that 20 have an outstanding balance at the credit card limit. Suppose the manager randomly selects 15 customers and finds 4 that have balances at the limit. Assume that the properties of the binomial distribution apply. a. What is the probability of finding 4 customers in a sample of 15 who have “maxed out” their credit cards b. What is the probability that 4 or fewer customers in the sample will have balances at the limit of the credit card 5-23. For a binomial distribution with a sample size equal to 10 and a probability of a success equal to 0.30 what is the probability that the sample will contain exactly three successes Use the binomial formula to determine the probability. 5-24. Use the binomial formula to calculate the following probabilities for an experiment in which n 5 and p 0.4: a. the probability that x is at most 1 b. the probability that x is at least 4 c. the probability that x is less than 1 5-25. If a binomial distribution applies with a sample size of n 20 find a. the probability of 5 successes if the probability of a success is 0.40 b. the probability of at least 7 successes if the probability of a success is 0.25 c. the expected value n 20 p 0.20 d. the standard deviation n 20 p 0.20 5-26. A report issued by the American Association of Building Contractors indicates that 40 of all home buyers will do some remodeling to their home within the first five years of home ownership. Assuming this is true use the binomial distribution to determine the probability that in a random sample of 20 homeowners 2 or fewer will remodel their homes. Use the binomial table. 5-27. Find the probability of exactly 5 successes in a sample of n 10 when the probability of a success is 0.70. 5-28. Assuming the binomial distribution applies with a sample size of n 15 find a. the probability of 5 or more successes if the probability of a success is 0.30 b. the probability of fewer than 4 successes if the probability of a success is 0.75 c. the expected value of the random variable if the probability of success is 0.40 d. the standard deviation of the random variable if the probability of success is 0.40 5-29. A random variable follows a binomial distribution with a probability of success equal to 0.65. For a sample size of n 7 find a. the probability of exactly 3 successes b. the probability of 4 or more successes c. the probability of exactly 7 successes d. the expected value of the random variable 5-30. A random variable follows a binomial distribution with a probability of success equal to 0.45. For n 11 find a. the probability of exactly 1 success b. the probability of 4 or fewer successes c. the probability of at least 8 successes 5-31. Use the binomial distribution table to determine the following probabilities: a. n 6 p 0.08 find P1x 22 b. n 9 p 0.80 determine P1x 6 42 c. n 11 p 0.65 calculate P12 6 x … 52 d. n 14 p 0.95 find P1x Ú 132 e. n 20 p 0.50 compute P1x 7 32 5-32. Use the binomial distribution in which n 6 and p 0.3 to calculate the following probabilities: a. x is at most 1. b. x is at least 2. c. x is more than 5. d. x is less than 6. 5- Exercises MyStatLab www.downloadslide.com

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Discrete Probability Distributions b. If Intel and Dell agree that Intel will not provide more than 5 defective chips calculate the probability that the entire shipment will be returned even though only 5 are defective. c. Calculate the probability that the entire shipment will be kept by Dell even though the shipment has 10 defective microprocessors. 5-37. In his article titled “Acceptance Sampling Solves Drilling Issues: A Case Study” published in Woodworking Magazine author Ken Wong discusses a problem faced by furniture manufacturing companies dealing with the quality of the drilling of dowel holes. Wong states “Incorrect sizing and distances with respect to dowel holes can cause many problems for the rest of the process especially when drilling is conducted early in the production process.” Consider the case of Dragon Wood Furniture in Bismarck North Dakota which believes that when the drilling process is operating at an acceptable rate the upper limit on the percentage of incorrectly drilled dowel holes is 4. To monitor its drilling process Dragon Wood Furniture randomly samples 20 products each hour and determines if the dowel hole in each product is correctly drilled or not. If in the sample of 20 holes 1 or more incorrectly drilled holes is discovered the production process is stopped and the drilling process is recalibrated. a. If the process is really operating correctly 1p 0.042 what is the probability that the sampling effort will produce x 0 defective holes and thus the process will properly be left to continue running b. Suppose the true defect rate has risen to 0.10. What is the probability the sample will produce results that properly tell the managers to halt production to recalibrate the drilling machine c. Prepare a short letter to the manufacturing manager at Dragon Wood Furniture discussing the effectiveness of the sampling process that her company is using. Base your response on the results to parts a and b. 5-38. Mooney Hileman Jones a marketing agency located in Cleveland has created an advertising campaign for a major retail chain which the agency’s executives believe is a winner. For an ad campaign to be successful at least 80 of those seeing a television commercial must be able to recall the name of the company featured in the commercial one hour after viewing the commercial. Before distributing the ad campaign nationally the company plans to show the commercial to a random sample of 20 people. It will also show the same people two additional commercials for different products or businesses. a. Assuming that the advertisement will be successful 80 will be able to recall the name of the company 5-33. Given a binomial distribution with n 8 and p 0.40 obtain the following: a. the mean b. the standard deviation c. the probability that the number of successes is larger than the mean d. the probability that the number of successes is within 2 standard deviations of the mean Business Applications 5-34. Magic Valley Memorial Hospital administrators have recently received an internal audit report that indicates that 15 of all patient bills contain an error of one form or another. After spending considerable effort to improve the hospital’s billing process the administrators are convinced that things have improved. They believe that the new error rate is somewhere closer to 0.05. a. Suppose that recently the hospital randomly sampled 10 patient bills and conducted a thorough study to determine whether an error exists. It found 3 bills with errors. Assuming that managers are correct that they have improved the error rate to 0.05 what is the probability that they would find 3 or more errors in a sample of 10 bills b. Referring to part a what conclusion would you reach based on the probability of finding 3 or more errors in the sample of 10 bills 5-35. The Center for the Study of the American Electorate indicated that 64 of the voting-age voters cast ballots in the 2008 presidential election. It also indicated in the west 34.6 of voting-age voters voted for a Democrat as a Representative an increase from 30.8 in 2004. A start-up company in San Jose California has 10 employees. a. How many of the employees would you expect to have voted for a Democrat as a Representative b. All of the employees indicated that they voted in the 2008 presidential election. Determine the probability of this assuming they followed the national trend. c. Eight of the employees voted for a Democratic Representative. Determine the probability that at least 8 of the employees would vote for the Democrat if they followed the national trend. d. Based on your calculations in parts b and c do the employees reflect the national trend Support your answer with statistical calculations and reasoning. 5-36. Dell Computers receives large shipments of micro- processors from Intel Corp. It must try to ensure the proportion of microprocessors that are defective is small. Suppose Dell decides to test five microprocessors out of a shipment of thousands of these microprocessors. Suppose that if at least one of the microprocessors is defective the shipment is returned. a. If Intel Corp.’s shipment contains 10 defective microprocessors calculate the probability the entire shipment will be returned. www.downloadslide.com

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Discrete Probability Distributions a. Suppose that the producers of NCIS commissioned a study that called for the consultants to randomly call 25 people immediately after the NCIS time slot and interview those who said that they had just watched NCIS. Suppose the consultant submits a report saying that it found no one in the sample of 25 homes who claimed to have watched the program and therefore did not do any surveys. What is the probability of this happening assuming that the Nielsen ratings for the show are accurate b. Assume the producers for The Big Bang Theory planned to survey 1000 people on the day following the broadcast of the program. The purpose of the survey was to determine what the reaction would be if one of the leading characters was retired from the show. Based on the Nielsen ratings what would be the expected number of people who would end up being included in the analysis assuming that all 1000 people could be reached 5-41. A small hotel in a popular resort area has 20 rooms. The hotel manager estimates that 15 of all confirmed reservations are “no-shows.” Consequently the hotel accepts confirmed reservations for as many as 25 rooms. If more confirmed reservations arrive than there are rooms the overbooked guests are sent to another hotel and given a complementary dinner. If the hotel currently has 25 confirmed reservations find a. the probability that no customers will be sent to another hotel b. the probability that exactly 2 guests will be sent to another hotel c. the probability that 3 or more guests will be sent to another hotel 5-42. A manufacturing firm produces a product that has a ceramic coating. The coating is baked on to the product and the baking process is known to produce 15 defective items for example cracked or chipped finishes. Every hour 20 products from the thousands that are baked hourly are sampled from the ceramic- coating process and inspected. a. What is the probability that 5 defective items will be found in the next sample of 20 b. On average how many defective items would be expected to occur in each sample of 20 c. How likely is it that 15 or more nondefective good items would occur in a sample due to chance alone 5-43. The Employee Benefit Research Institute reports that 69 of workers reported that they and/or their spouse had saved some money for retirement. a. If a random sample of 30 workers is taken what is the probability that fewer than 17 workers and/ or their spouses have saved some money for retirement b. If a random sample of 50 workers is taken what is the probability that more than 40 workers and/ or their spouses have saved some money for retirement in the ad what is the expected number of people in the sample who will recall the company featured in the Mooney Hileman Jones commercial one hour after viewing the three commercials b. Suppose that in the sample of 20 people 11 were able to recall the name of the company in the Mooney Hileman Jones commercial one hour after viewing. Based on the premise that the advertising campaign will be successful what is the probability of 11 or fewer people being able to recall the company name c. Based on your responses to parts a and b what conclusion might Mooney Hileman Jones executives make about this particular advertising campaign 5-39. A survey by KRC Research for U.S. News reported that 37 of people plan to spend more on eating out after they retire. If eight people are randomly selected then determine the a. expected number of people who plan to spend more on eating out after they retire b. standard deviation of the individuals who plan to spend more on eating out after they retire c. probability that two or fewer in the sample indicate that they actually plan to spend more on eating out after retirement 5-40. Nielsen is the major media measurement company and conducts surveys to determine household viewing choices. The following table shows the top 10 broadcast television programs for the week of January 23 2012. Rank Program Network HH Rating Viewers 1 American Idol— W ednesday FOX 11.1 19.671 2 American Idol— Thursday FOX 10.0 17.141 3 Big Bang Theory CBS 9.7 16.130 4 CSI CBS 9.0 14.257 5 Criminal Minds CBS 8.7 13.815 6 NCIS CBS 8.1 12.548 7 Undercover Boss CBS 7.9 13.151 8 AFC-NFC Pro-Bowl NBC 7.3 12.498 9 The Good Wife CBS 7.2 11.083 10 60 Minutes CBS 7.1 11.188 Rank is based on U.S. Household Rating from Nielsen’s National People Meter Sample. A household rating is the estimate of the size of a television audience relative to the total universe expressed as a percentage. As of August 27 2012 there are an estimated 114200000 television households in the United States. A single national household ratings point represents 1 or 1142000 households. Measured in millions includes all persons over the age of two. Source: www.nielsenmedia.com www.downloadslide.com

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Discrete Probability Distributions the percent reported by Ms. Borrus is correct a sample of 20 shareholders was asked if they held their stock in “street names.” Seventeen responded that they did. a. Supposing the true proportion of shareholders that hold stock under street names is 0.80 calculate the probability that 17 or more of the sampled individuals hold their stock under street names. b. Repeat the calculation in part a using proportions of 0.70 and 0.90. c. Based on your calculations in parts a and b which proportion do you think is most likely true Support your answer. Computer Database Exercises 5-47. USA Today has reported on the gender gap that exists between married spouses. One of the measures is the number of women who outearn their husbands. According to a study conducted by the Bureau of Labor Statistics 32.5 of female spouses outearn their male counterparts. The file titled Gendergap contains the incomes of 150 married couples in Utah. a. Determine the number of families in which the female outearns her husband. b. Calculate the expected number of female spouses who outearn their male counterparts in the sample of 150 married couples based on the Bureau of Labor Statistics study. c. If the percentage of married women in Utah who outearn their male spouses is the same as that indicated by the Bureau of Labor Statistics determine the probability that at least the number found in part a would occur. d. Based on your calculation in part c does the Bureau of Labor Statistics’ percentage seem plausible if Utah is not different from the rest of the United States 5-48. Tony Hsieh is CEO of e-tailer Zappos.com. His company sells shoes online. It differentiates itself by its selection of shoes and a devotion to customer service. It offers free shipping and free return shipping. An area in which costs could be cut back is the shipping charges for return shipping specifically those that result from the wrong size of shoes being sent. Zappos may try to keep the percentage of returns due to incorrect size to no more than 5. The file titled Shoesize contains a sample of 125 shoe sizes that were sent to customers and the sizes that were actually ordered. a. Determine the number of pairs of wrong-size shoes that were delivered to customers. b. Calculate the probability of obtaining at least that many pairs of wrong-sized shoes delivered to customers if the proportion of incorrect sizes is actually 0.05. c. On the basis of your calculation determine whether Zappos has kept the percentage of returns due to incorrect size to no more than 5. Support your answer with statistical reasoning. 5-44. Radio frequency identification RFID is an electronic scanning technology that can be used to identify items in a number of ways. One advantage of RFID is that it can eliminate the need to manually count inventory which can help improve inventory management. The technology is not infallible however and sometimes errors occur when items are scanned. If the probability that a scanning error occurs is 0.0065 use either Excel or Minitab to find a. the probability that exactly 20 items will be scanned incorrectly from the next 5000 items scanned b. the probability that more than 20 items will be scanned incorrectly from the next 5000 items scanned c. the probability that the number of items scanned incorrectly is between 10 and 25 from the next 5000 items scanned d. the expected number of items scanned incorrectly from the next 5000 items scanned 5-45. Peter S. Kastner director of the consulting firm Vericours Inc. reported that 40 of all rebates are not redeemed because consumers either fail to apply for them or their applications are rejected. TCA Fulfillment Services published its redemption rates: 50 for a 30 rebate on a 100 product 10 for a 10 rebate on a 100 product and 35 for a 50 rebate on a 200 product. a. Calculate the weighted average proportion of redemption rates for TCA Fulfillment using the size of the rebate to establish the weights. Does it appear that TCA Fulfillment has a lower rebate rate than that indicated by Vericours Explain. b. To more accurately answer the question posed in part a a random sample of 20 individuals who purchased an item accompanied by a rebate could be asked if they submitted their rebate. Suppose four of the questioned individuals said they did redeem their rebate. If Vericours’ estimate of the redemption rate is correct determine the expected number of rebates that would be redeemed. Does it appear that Vericours’ estimate may be too high c. Determine the likelihood that such an extreme sample result as indicated in part b or something more extreme would occur if the weighted average proportion provides the actual rebate rate. d. Repeat the calculations of part c assuming that Vericours’ estimate of the redemption rate is correct. e. Are you convinced that the redemption rate is smaller than that indicated by Vericours Explain. 5-46. Business Week reported that business executives want to break down the obstacles that keep them from communicating directly with stock owners. Ms. Borrus reports that 80 of shareholders hold stock in “street names” which are registered with their bank or brokerage. If the brokerage doesn’t furnish these names to the corporations executives cannot communicate with their shareholders. To determine if www.downloadslide.com

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Discrete Probability Distributions d. If Zappos sells 5 million pairs of shoes in one year and it costs an average of 4.75 a pair to return them calculate the expected cost associated with wrong-sized shoes being returned using the probability calculated from the sample data. 5-49. International Data Corp. IDC has shown that the average return on business analytics projects was almost four-and-a-half times the initial investment. Analytics consists of tools and applications that present better metrics to the user and to the probable future outcome of an event. IDC looked at how long it takes a typical company to recoup its investment in analytics. It determined that 29 of the U.S. corporations that adopted analytics took six months or less to recoup their investment. The file titled Analytics contains a sample of the time it might have taken 35 corporations to recoup their investment in analytics. a. Determine the number of corporations that recovered their investment in analytics in six months or less. b. Calculate the probability of obtaining at most the number of corporations that you determined in part a if the percent of those recovering their investment is as indicated by IDC. c. Determine the 70th percentile of the number of the 35 corporations that recovered their investment in analytics in six months or less. Hint: Recall and use the definition of percentiles from Section 1. END EXERCISES 5-2 Other Discrete Probability Distributions The binomial distribution is very useful in many business situations as indicated by the examples and applications presented in the previous section. However as we pointed out there are several requirements that must hold before we can use the binomial distribution to determine probabilities. If those conditions are not satisfied there may be other theoretical probability distributions that could be employed. In this section we introduce two other very useful discrete probability distributions: the Poisson distribution and the hypergeometric distribution. The Poisson Distribution To use the binomial distribution we must be able to count the number of successes and the number of failures. Although in many situations you may be able to count the number of successes you often cannot count the number of failures. For example suppose a company builds freeways in Vermont. The company could count the number of potholes that develop per mile here a pothole is referred to as a success because it is what we are looking for but how could it count the number of nonpotholes Or what about a hospital supplying emer- gency medical services in Los Angeles It could easily count the number of emergencies its units respond to in one hour but how could it determine how many calls it did not receive Obviously in these cases the number of possible outcomes of 1successes + failures2 is difficult if not impossible to determine. If the total number of possible outcomes cannot be determined the binomial distribution cannot be applied. In these cases you may be able to use the Poisson distribution. Characteristics of the Poisson Distribution The Poisson distribution 1 describes a process that extends over time space or any well-defined unit of inspection. The outcomes of interest such as emergency calls or potholes occur at random and we count the number of outcomes that occur in a given segment of time or space. We might count the number of emer- gency calls in a one-hour period or the number of potholes in a two-mile stretch of freeway. As we did with the binomial distribution we will call these outcomes successes even though like potholes they might be undesirable. The possible counts are the integers 0 1 2 c and we would like to know the probabil- ity of each of these values. For example what is the chance of getting exactly four emergency calls in a particular hour What is the chance that a chosen two-mile stretch of freeway will contain zero potholes Poisson Distribution The Poisson distribution describes a process that extends over space time or any well- defined interval or unit of inspection in which the outcomes of interest occur at random and the number of outcomes that occur in any given interval are counted. The Poisson distribution rather than the binomial distribution is used when the total number of possible outcomes cannot be determined. 1 The Poisson distribution can be derived as the limiting distribution of the binomial distribution as the number of trials n tends to infinity and the probability of success decreases to zero. It serves as a good approximation to the binomial when n is large. Chapter Outcome 3. www.downloadslide.com

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Discrete Probability Distributions We can use the Poisson probability distribution to answer these questions if we make the following assumptions: 1. We know l the average number of successes in one segment. For example we know that there is an average of 8 emergency calls per hour 1l 82 or an average of 15 pot- holes per mile of freeway 1l 152. 2. The probability of x successes in a segment is the same for all segments of the same size. For example the probability distribution of emergency calls is the same for any one-hour period of time at the hospital. 3. What happens in one segment has no influence on any nonoverlapping segment. For example the number of calls arriving between 9:30 p.m. and 10:30 p.m. has no influence on the number of calls between 11:00 p.m. and 12:00 midnight. 4. We imagine dividing time or space into tiny subsegments. Then the chance of more than one success in a subsegment is negligible and the chance of exactly one success in a tiny subsegment of length t is lt. For example the chance of two emergency calls in the same second is essentially 0 and if l 8 calls per hour the chance of a call in any given second is 18211/36002 ≈ 0.0022. Once l has been determined we can calculate the average occurrence rate for any number of segments t. This is lt. Note that l and t must be in compatible units. If we have l 20 arrivals per hour the segments must be in hours or fractional parts of an hour. That is if we have l 20 per hour and we wish to work with half-hour time periods the segment would be t 5 1 2 hour not t 30 minutes. Although the Poisson distribution is often used to describe situations such as the number of customers who arrive at a hospital emergency room per hour or the number of calls the Hewlett-Packard LaserJet printer service center receives in a 30-minute period the segments need not be time intervals. Poisson distributions are also used to describe such random vari- ables as the number of knots in a sheet of plywood or the number of contaminants in a gallon of lake water. The segments would be the sheet of plywood and the gallon of water. Another important point is that lt the average number in t segments is not necessarily the number we will see if we observe the process for t segments. We might expect an aver- age of 20 people to arrive at a checkout stand in any given hour but we do not expect to find exactly that number arriving every hour. The actual arrivals will form a distribution with an expected value or mean equal to lt. So for the Poisson distribution E3x4 m x lt Once l and t have been specified the probability for any discrete value in the Poisson distribution can be found using Equation 7. Poisson Probability Distribution Px te x xt − 7 where: t Number of segments of interest x Number of successes in t segments l Expected number of successes in one segment e Base of the natural logarithm system 12.71828 c2 BUSINESS APPLICATION POISSON DISTRIBUTION WHOLE FOODS GROCERY A study conducted at Whole Foods Grocery shows that the average number of arrivals to the checkout section of the store per hour is 16. Further the distribution for the number of arrivals is considered to be Poisson distributed. Figure 6 shows www.downloadslide.com

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Discrete Probability Distributions the shape of the Poisson distribution for l 16. The probability of each possible number of customers arriving can be computed using Equation 7. For example we can find the probability of x 12 customers in one hour 1t 12 as follows: Px te x e xt . 12 16 12 0 0661 12 16 Poisson Probability Distribution Table As was the case with the binomial distribu- tion a table of probabilities exists for the Poisson distribution. The Poisson table appears in Cumulative Poisson Probability Distribution Table. The Poisson table shows the cumulative probabilities for x or fewer occurrences for different lt values. We can use the following busi- ness application to illustrate how to use the Poisson table. BUSINESS APPLICATION USING THE POISSON DISTRIBUTION TABLE WHOLE FOODS GROCERY CONTINUED At Whole Foods Grocery customers are thought to arrive at the checkout section according to a Poisson distribution with l 16 customers per hour. See Figure 6. Based on previous studies the store manager believes that the service time for each customer is quite constant at six minutes. Suppose during each six- minute time period the store has three checkers available. This means that three customers can be served during each six-minute segment. The manager is interested in the probability that one or more customers will have to wait for service during a six-minute period. To determine this probability you will need to convert the mean arrivals from l 16 customers per hour to a new average for a six-minute segment. Six minutes corresponds to 0.10 hours so you will change the segment size t 0.10. Then the mean number of arrivals in six minutes is lt 1610.102 1.6 customers. Now because there are three checkers any time four or more customers arrive in a six- minute period at least one customer will have to wait for service. Thus P11 or more customers wait2 P142 + P152 + P162 + c or you can use the Complement Rule as follows: P11 or more customers wait2 1 - P1x … 32 The Poisson table in Cumulative Poisson Probability Distribution Table can be used to find the necessary probabilities. To use the table first go across the top of the table until you find the desired value of lt. In this case look for lt 1.6. Next go down the left-hand side to find the value of x corresponding to the number of occurrences of interest. For example consider x 3 customer arrivals. Because Cumulative Poisson Probability Distribution Table Poisson Probability 0 Number of Customers x Px Mean 16 0.1200 0.1000 0.0800 0.0600 0.0400 0.0200 0.0000 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 FIGURE 6 |  Poisson Distribution for Whole Foods Checkout Arrivals with l 16 www.downloadslide.com

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Discrete Probability Distributions is a cumulative Poisson table the probabilities are for x or fewer occurrences. This the prob- ability of x … 3 is given in the table as 0.9212. Thus P1x … 32 0.9212 Then the probability of four or more customers arriving is P14 or more customers2 1 - P1x … 32 P14 or more customers2 1 - 0.9212 0.0788 Given the store’s capacity to serve three customers in a six-minute period the probability of one or more customers having to wait is 0.0778. Suppose that when the store manager sees this probability she is somewhat concerned. She states that she wants enough checkout stands open so that the chance of a customer wait- ing does not exceed 0.05. To determine the appropriate number of checkers you can use the Poisson table to find the following: P14 or more customers2 1 - P1x … 2 … 0.05 In other words a customer will have to wait if more customers arrive than there are checkers. As long as the number of arrivals x is less than or equal to the number of checkers no one will wait. Then what value of x will provide the following 1 - P1x … 2 … 0.05 Therefore you want P1x … 2 Ú 0.95 You can go to the table for lt 1.6 and scan down the column starting with P1x 02 0.2019 until the cumulative probability listed is 0.95 or higher. When you reach x 4 the cumulative probability P1x … 42 Ú 0.9763. Then P14 or more customers2 1 - P1x … 42 1 - 0.9763 0.0237 Because 0.0237 is less than or equal to the 0.05 limit imposed by the manager she would have to schedule four checkers. EXAMPLE 7 USING THE POISSON DISTRIBUTION Fashion Leather Products Fashion Leather Products headquartered in Argentina makes leather clothing for export to many other countries around the world. Before shipping quality managers perform tests on the leather products. The industry standards call for the average number of defects per square meter of leather to not exceed five. During a recent test the inspector selected 3 square meters finding 18 defects. To determine the probability of this event occurring if the leather meets the industry standards assuming that the Poisson distribu- tion applies the company can perform the following steps: Step 1 Define the segment unit. Because the mean was stated as five defects per square meter the segment unit in this case is one meter. Step 2 Determine the mean of the random variable. In this case if the company meets the industry standards the mean will be l 5 Step 3 Determine the segment size t. The company quality inspectors analyzed 3 square meters which is equal to 3 units. So t 3.0 Then lt 15213.02 15.0 When looking at 3 square meters the company would expect to find at most 15.0 defects if the industry standards are being met. Step 4 Define the event of interest and use the Poisson formula or the Poisson tables to find the probability. Example 7 Using the Poisson Distribution The following steps are used to find probabilities using the Poisson distribution: Define the segment units. The segment units are usually blocks of time areas of space or volume. Determine the mean of the random variable. The mean is the parameter that defines the Poisson distribution and is referred to as l. It is the average number of successes in a segment of unit size. Determine t the number of the segments to be considered and then calculate lt. Define the event of interest and use the Poisson formula or the Poisson table to find the probability. www.downloadslide.com

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Discrete Probability Distributions In this case 18 defects were observed. Because 18 exceeds the expected number 1lt 15.02 the company would want to find P1x Ú 182 P1x 182 + P1x 192 + c The Poisson table in Cumulative Poisson Probability Distribution Table is used to determine these probabilities. Locate the desired probability under the column headed lt 15.0 Then find the values of x down the left-hand column. P1x Ú 182 1 - P1x … 172 1 - 0.7489 0.2511 There is about a 0.25 chance of finding 18 or more defects in 3 square meters of leather prod- ucts made by Fashion Leather if they are meeting the quality standard. END EXAMPLE TRY PROBLEM 50 The Mean and Standard Deviation of the Poisson Distribution The mean of the Poisson distribution is lt. This is the value we use to specify which Poisson distribu- tion we are using. We must know the mean before we can find probabilities for a Poisson distribution. Figure 6 illustrated that the outcome of a Poisson distributed variable is subject to vari- ation. Like any other discrete probability distribution the standard deviation for the Poisson can be computed using Equation 2: x xEx Px ∑− 2 However for a Poisson distribution the standard deviation also can be found using Equation 8. Standard Deviation of the Poisson Distribution t 8 The standard deviation of the Poisson distribution is simply the square root of the mean. Therefore if you are working with a Poisson process reducing the mean will reduce the vari- ability also. BUSINESS APPLICATION THE POISSON PROBABILITY DISTRIBUTION HERITAGE TILE To illustrate the importance of the relationship between the mean and standard deviation of the Poisson distribution consider Heritage Tile in New York City. The company makes ceramic tile for kitchens and bathrooms. The quality standards call for the number of imperfections in a tile to average 3 or fewer. The distribution of imperfections is thought to be Poisson. Many software packages including Excel will generate Poisson probabilities in much the same way as for the binomial distribution which was discussed in Section 2. If we assume that the company is meeting the standard Figure 7 shows the Poisson probability distribution generated using Excel when lt 3.0. Even though the average number of defects is 3 the manager is concerned about the high probabilities associated with the number of imperfections equal to 4 5 6 or more on a tile. The variability is too great. Using Equation 5 the standard deviation for this distribution is 30 1732 .. This large standard deviation means that although some tiles will have few if any imperfec- tions others will have several causing problems for installers and unhappy customers. Excel Tutorial Excel tutorials www.downloadslide.com

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Discrete Probability Distributions Minitab Instructions for similar results: 1. Create column Number with integers from 0 to 10. 2. Choose Calc Probability Distributions Poisson. 3. Select Probability. 4. In Mean enter 3. 5. Select Input column. 6. In Input column enter the column of integers. 7. In Optional storage enter column Probability. 8. Click OK. 9. Choose Graph Bar Chart. 10. In Bars represent select Values from a Table select Simple. 11. Click OK. 12. In Graph variables Insert Probability. 13. In Categorical variable insert Number. 14. Click OK. Excel 2010 Instructions: 1. Enter values for x ranging from 0 to 10. 2. Place the cursor in the frst blank cell in the next column. 3. Click on f x function wizard and then select the Statistical category. 4. Select the POISSON. DIST function. 5. Reference the cell with the desired x value and enter the mean. Enter False to choose noncumulative probabilities. 6. Copy function down for all values of x. 7. Graph using Insert Column. Remove gaps and add lines to the bars. Label axes and add titles. FIGURE 7 |  Excel 2010 Output for Heritage Tile Example A quality-improvement effort directed at reducing the average number of imperfections to 2.0 would also reduce the standard deviation to 20 1414 .. Further reductions in the average would also reduce variation in the number of imperfec- tions between tiles. This would mean more consistency for installers and higher customer satisfaction. The Hypergeometric Distribution Although the binomial and Poisson distributions are very useful in many business decision-making situations they both require that the trials be independent. For instance in binomial applications the probability of a success in one trial must be the same as the probability of a success in any other trial. Although there are certainly times when this assumption can be satisfied or at least approximated in instances in which the population is fairly small and we are sampling without replacement the condition of independence will not hold. In these cases a discrete probability distribution referred to as the hypergeo- metric distribution can be useful. Chapter Outcome 3. www.downloadslide.com

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Discrete Probability Distributions BUSINESS APPLICATION THE HYPERGEOMETRIC PROBABILITY DISTRIBUTION DOLBY INDUSTRIES Dolby Industries contracts with a Chinese manufacturer to make women’s handbags. Because of the intense competition in the marketplace for handbags Dolby has made every attempt to provide high-quality products. However a recent production run of 20 handbags of a particular model contained 2 units that tested out as defective. The problem was traced to a shipment of defective latches that Dolby’s Chinese partner received shortly before the production run started. The production manager ordered that the entire batch of 20 handbags be isolated from other production output until further testing could be completed. Unfortunately a new ship- ping clerk packaged 10 of these isolated handbags and shipped them to a California retailer to fill an order that was already overdue. By the time the production manager noticed what had happened the handbags were already in transit. The immediate concern was whether one or more of the defectives had been included in the shipment. The new shipping clerk thought there was a good chance that no defectives were included. Short of reinspecting the remaining handbags how might Dolby Industries determine the probability that no defectives were actually shipped At first glance it might seem that the question could be answered by employing the bino- mial distribution with n 10 p 220 0.10 and x 0. Using the binomial distribution table in Cumulative Binomial Distribution Table we get P1x 02 0.3487 There is a 0.3487 chance that no defectives were shipped assuming the selection process satisfied the requirements of a binomial distribution. However for the binomial distribution to be applica- ble the trials must be independent and the probability of a success p must remain constant from trial to trial. In order for this to occur when the sampling is from a “small” finite population as is the case here the sampling must be performed with replacement. This means that after each item is selected it is returned to the population and therefore may be selected again later in the sampling. In the Dolby example the sampling was performed without replacement because each handbag could only be shipped one time. Also the population of handbags is finite with size N 20 which is a “small” population. Thus p the probability of a defective handbag does not remain equal to 0.10 on each trial. The value of p on any particular trial depends on what has already been selected on previous trials. The event of interest is G G G G G G G G G G The probability that the first item selected for shipment would be good would be 18/20 because there were 18 good handbags in the batch of 20. Now assuming the first unit selected was good the probability the second unit was good is 17/19 because we then had only 19 handbags to select from and 17 of those would be good. The probability that all 10 items selected were good is 18 20 17 19 16 18 15 17 14 16 13 15 12 14 11 13 10 1 ×××× ×× × × 2 2 9 11 0 2368 × 5 . This value is not the same as the 0.3847 probability we got when the binomial distri- bution was used. This demonstrates that when sampling is performed without replacement from finite populations the binomial distribution produces inaccurate probabilities. To pro- tect against large inaccuracies the binomial distribution should only be used when the sample is small relative to the size of the population. Under that circumstance the value of p will not change very much as the sample is selected and the binomial distribution will be a reasonable approximation to the actual probability distribution. In cases in which the sample is large relative to the size of the population a discrete prob- ability distribution called the hypergeometric distribution is the correct distribution for computing probabilities for the random variable of interest. We use combinations see Section 2 to form the equation for computing probabilities for the hypergeometric distribution. When each trial has two possible outcomes success and failure hypergeometric probabilities are computed using Equation 9. Hypergeometric Distribution The hypergeometric distribution is formed by the ratio of the number of ways an event of interest can occur over the total number of ways any event can occur. Discovod/Fotolia www.downloadslide.com

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Discrete Probability Distributions Notice that the numerator of Equation 9 is the product of the number of ways you can select x successes in a random sample out of the X successes in the population and the number of ways you can select n - x failures in a sample from the N - X failures in the population. The denominator in the equation is the number of ways the sample can be selected from the population. In the earlier Dolby Industries example the probability of zero defectives being shipped 1x 02 is Px CC C Px CC C 55 55 0 0 10 0 20 2 0 2 10 20 10 18 0 2 1 − − . . 0 0 20 Carrying out the arithmetic we get Px . 0 43 758 1 184 756 0 2368 55 As we found before the probability that zero defectives were included in the shipment is 0.2368 or approximately 24. The probabilities of x 1 and x 2 defectives can also be found by using Equation 9 as follows: Px CC C . − − 1 0 5264 10 1 20 2 1 2 10 20 55 . and Px CC C . − − 2 0 2368 10 2 20 2 2 2 10 20 55 . Thus the hypergeometric probability distribution for the number of defective handbags in a random selection of 10 is xPx 0 0.2368 1 0.5264 2 0.2368 ΣP1x2 1.0000 Recall that when we introduced the hypergeometric distribution we said that it is used in situations when we are sampling without replacement from a finite population. However when the population size is large relative to the sample size decision makers typically use the binomial distribution as an approximation of the hypergeometric. This eases the computational Hypergeometric Distribution Two Possible Outcomes per Trial Px CC C nx NX x X n N . 9 where: N Population size X Number of successes in the population n Sample size x Number of successes in the sample n - x Number of failures in the sample www.downloadslide.com

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Discrete Probability Distributions burden and provides useful approximations in those cases. Although there is no exact rule for when the binomial approximation can be used we suggest that the sample should be less than 5 of the population size. Otherwise use the hypergeometric distribution when sampling is done without replacement from the finite population. EXAMPLE 8 THE HYPERGEOMETRIC DISTRIBUTION ONE OF TWO POSSIBLE OUTCOMES PER TRIAL Gender Equity One of the biggest changes in U.S. business practice in the past few decades has been the inclusion of women in the management ranks of companies. Tom Peters manage- ment consultant and author of such books as In Search of Excellence has stated that one of the reasons the Middle Eastern countries have suffered economically compared with countries such as the United States is that they have not included women in their economic system. How- ever there are still issues in U.S. business. Consider a situation in which a Maryland company needed to downsize one department having 30 people—12 women and 18 men. Ten people were laid off and upper management said the layoffs were done randomly. By chance alone 40 12/30 of the layoffs would be women. However of the 10 laid off 8 were women. This is 80 not the 40 due to chance. A labor attorney is interested in the probability of eight or more women being laid off by chance alone. This can be determined using the following steps: Step 1 Determine the population size and the combined sample size. The population size and sample size are N 30 and n 10 Step 2 Define the event of interest. The attorney is interested in the event: P1x Ú 82 What are the chances that eight or more women would be selected Step 3 Determine the number of successes in the population and the number of successes in the sample. In this situation a success is the event that a woman is selected. There are X 12 women in the population and x Ú 8 in the sample. We will break this down as x 8 x 9 x 10. Step 4 Compute the desired probabilities using Equation 9. Px CC C nx NX x X n N 5 − − . We want: 2 Px Px Px Px Px C 88 9 10 8 10 8 30 − −1 12 8 12 10 30 2 18 8 12 10 30 0 0025 9 .. C C CC C Px C . 1 1 18 9 12 10 30 0 18 10 12 10 3 0 0001 10 . . C C Px CC C . 0 0 0 0000 . Therefore P1x Ú 82 0.0025 + 0.0001 + 0.0000 0.0026 The chances that 8 or more women would have been selected among the 10 people chosen for layoff strictly due to chance is 0.0026. The attorney will likely wish to challenge the layoffs based on this extremely low probability. END EXAMPLE TRY PROBLEM 53 2 Note you can use Excel’s HYPGEOM.DIST function to compute these probabilities. www.downloadslide.com

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Discrete Probability Distributions The Hypergeometric Distribution with More Than Two Possible Outcomes per Trial Equation 9 assumes that on any given sample selection or trial only one of two pos- sible outcomes will occur. However the hypergeometric distribution can easily be extended to consider any number of possible categories of outcomes on a given trial by employing Equation 10. Hypergeometric Distribution k Possible Outcomes per Trial Px x x CC C C k x X x X x X x X k K ... ... 12 1 1 2 2 3 3 5 .. . . C C n N 10 where: XN xn i i k i i k 5 5 5 5 1 1 ∑ ∑ N Population size n Total sample size X i Number of items in the population with outcome i x i Number of items in the sample with outcome i EXAMPLE 9 THE HYPERGEOMETRIC DISTRIBUTION FOR MULTIPLE OUTCOMES Breakfast Cereal Preferences Consider a marketing study that involves placing break- fast cereal made by four different companies in a basket at the exit to a food store. A sign on the basket invites customers to take one box of cereal free of charge. At the beginning of the study the basket contains the following: 5 brand A 4 brand B 6 brand C 4 brand D The researchers were interested in the brand selection patterns for customers who could select without regard to price. Suppose six customers were observed and three selected brand B two selected brand D and one selected brand C. No one selected brand A. The probabil- ity of this selection mix assuming the customers were selecting entirely at random without replacement from a finite population can be found using the following steps: Step 1 Determine the population size and the combined sample size. The population size and sample size are N 19 and n 6 Step 2 Define the event of interest. The event of interest is P1x 1 0 x 2 3 x 3 1 x 4 22 Step 3 Determine the number in each category in the population and the number in each category in the sample. X 1 5 brand A x 1 0 X 2 4 brand B x 2 3 X 3 6 brand C x 3 1 X 4 4 brand D x 4 2 N 19 n 6 www.downloadslide.com

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Discrete Probability Distributions Skill Development 5-50. The mean number of errors per page made by a member of the word-processing pool for a large company is thought to be 1.5 with the number of errors distributed according to a Poisson distribution. If three pages are examined what is the probability that more than three errors will be observed 5-51. Arrivals to a bank automated teller machine ATM are distributed according to a Poisson distribution with a mean equal to three per 15 minutes. a. Determine the probability that in a given 15-minute segment no customers will arrive at the ATM. b. What is the probability that fewer than four customers will arrive in a 30-minute segment 5-52. Consider a situation in which a used-car lot contains five Fords four General Motors GM cars and five Toyotas. If five cars are selected at random to be placed on a special sale what is the probability that three are Fords and two are GMs 5-53. A population of 10 items contains 3 that are red and 7 that are green. What is the probability that in a random sample of 3 items selected without replacement 2 red and 1 green items are selected 5-54. If a random variable follows a Poisson distribution with l 20 and t 1 2 find the a. expected value variance and standard deviation of this Poisson distribution b. probability of exactly 8 successes 5-55. A corporation has 11 manufacturing plants. Of these seven are domestic and four are located outside the United States. Each year a performance evaluation is conducted for 4 randomly selected plants. a. What is the probability that a performance evaluation will include exactly 1 plant outside the United States b. What is the probability that a performance evaluation will contain 3 plants from the United States c. What is the probability that a performance evaluation will include 2 or more plants from outside the United States 5-56. Determine the following values associated with a Poisson distribution with l t equal to 3: a. P1x … 32 b. P1x 7 32 c. P12 6 x … 52 d. Find the smallest x′ so that P1x … x′2 7 0.50. 5-57. A random variable x has a hypergeometric distribution with N 10 X 7 and n 4. Calculate the following quantities: a. P1x 32 b. P1x 52 c. P1x Ú 42 d. Find the largest x′ so that P1x 7 x′2 7 0.25. Business Applications 5-58. A new phone-answering system installed by the Ohio Power Company is capable of handling five calls every 10 minutes. Prior to installing the new system company analysts determined that the incoming calls to the system are Poisson distributed with a mean 5- Exercises MyStatLab Step 4 Compute the desired probability using Equation 10 Px x x x CC C C k x X x X x X x ... ... 12 3 1 1 2 2 3 3 5 .. . . k k k X n N C P CCC C C 0312 14 0 5 3 4 1 6 2 4 6 19 5 5 .. . . 66 27 132 144 27 132 0 0053 5 5 There are slightly more than 5 chances in 1000 of this exact selection occurring by random chance. END EXAMPLE TRY PROBLEM 52 www.downloadslide.com

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Discrete Probability Distributions hardware software or computer peripheral crises. Masters-at-Work became highly successful with branches throughout the South and was purchased by Best Buy but continued to operate under the Masters- at-Work name. A shipment of 20 Intel ® Pentium ® 4 processors was sent to Masters-at-Work. Four of them were defective. One of the Masters-at-Work technicians selected 5 of the processors to put in his parts inventory and went on three service calls. a. Determine the probability that only 1 of the 5 processors is defective. b. Determine the probability that 3 of the 5 processors are not defective. c. Determine the probability that the technician will have enough processors to replace 3 defective processors at the repair sites. 5-64. John Thurgood founded a company that translates Chinese books into English. His company is currently testing a computer-based translation service. Since Chinese symbols are difficult to translate John assumes the computer program will make some errors but then so do human translators. The computer error rate is supposed to be an average of 3 per 400 words of translation. Suppose John randomly selects a 1200- word passage. Assuming that the Poisson distribution applies if the computer error rate is actually 3 errors per 400 words a. determine the probability that no errors will be found. b. calculate the probability that more than 14 errors will be found. c. find the probability that fewer than 9 errors will be found. d. If 15 errors are found in the 1200-word passage what would you conclude about the computer company’s claim Why 5-65. Beacon Hill Trees Shrubs currently has an inventory of 10 fruit trees 8 pine trees and 14 maple trees. It plans to give 4 trees away at next Saturday’s lawn and garden show in the city park. The 4 winners can select which type of tree they want. Assume they select randomly. a. What is the probability that all 4 winners will select the same type of tree b. What is the probability that 3 winners will select pine trees and the other tree will be a maple c. What is the probability that no fruit trees and 2 of each of the others will be selected 5-66. Fasteners used in a manufacturing process are shipped by the supplier to the manufacturer in boxes that contain 20 fasteners. Because the fasteners are critical to the production process their failure will cause the product to fail. The manufacturing firm and the supplier have agreed that a random sample of 4 fasteners will be selected from every box and tested to see if the fasteners meet the manufacturer’s specifications. The nature of the testing process is equal to two every 10 minutes. If this incoming call distribution is what the analysts think it is what is the probability that in a 10-minute period more calls will arrive than the system can handle Based on this probability comment on the adequacy of the new answering system. 5-59. The Weyerhauser Lumber Company headquartered in Tacoma Washington is one of the largest timber- and wood-product companies in the world. Weyerhauser manufactures plywood at one of its Oregon plants. Plywood contains minor imperfections that can be repaired with small “plugs.” One customer will accept plywood with a maximum of 3.5 plugs per sheet on average. Suppose a shipment was sent to this customer and when the customer inspected two sheets at random 10 plugged defects were counted. What is the probability of observing 10 or more plugged defects if in fact the 3.5 average per sheet is being satisfied Comment on what this probability implies about whether you think the company is meeting the 3.5 per sheet defect rate. 5-60. When things are operating properly E-Bank United an Internet bank can process a maximum of 25 electronic transfers every minute during the busiest periods of the day. If it receives more transfer requests than this then the bank’s computer system will become so overburdened that it will slow to the point that no electronic transfers can be handled. If during the busiest periods of the day requests for electronic transfers arrive at the rate of 170 per 10-minute period on average what is the probability that the system will be overwhelmed by requests Assume that the process can be described using a Poisson distribution. 5-61. A stock portfolio contains 20 stocks. Of these stocks 10 are considered “large-cap” stocks 5 are “mid cap” and 5 are “small cap.” The portfolio manager has been asked by his client to develop a report that highlights 7 randomly selected stocks. When she presents her report to the client all 7 of the stocks are large-cap stocks. The client is very suspicious that the manager has not randomly selected the stocks. She believes that the chances of all 7 of the stocks being large cap must be very low. Compute the probability of all 7 being large cap and comment on the concerns of the client. 5-62. College-Pro Painting does home interior and exterior painting. The company uses inexperienced painters that do not always do a high-quality job. It believes that its painting process can be described by a Poisson distribution with an average of 4.8 defects per 400 square feet of painting. a. What is the probability that a 400-square-foot painted section will have fewer than 6 blemishes b. What is the probability that six randomly sampled sections of size 400 square feet will each have 7 or fewer blemishes 5-63. Masters-at-Work was founded by two brothers in Atlanta to provide in-home computer and electronic installation services as well as tech support to solve www.downloadslide.com

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Discrete Probability Distributions such that tested fasteners become unusable and must be discarded. The supplier and the manufacturer have agreed that if 2 or more fasteners fail the test the entire box will be rejected as being defective. Assume that a new box has just been received for inspection. If the box has 5 defective fasteners what is the probability that a random sample of 4 will have 2 or more defective fasteners What is the probability the box will be accepted 5-67. Lucky Dogs sells spicy hot dogs from a pushcart. The owner of Lucky Dogs is open every day between 11:00 a.m. and 1:00 p.m. Assume the demand for spicy hot dogs follows a Poisson distribution with a mean of 50 per hour. a. What is the probability the owner will run out of spicy dogs over the two-hour period if he stocks his cart with 115 spicy dogs every day b. How many spicy hot dogs should the owner stock if he wants to limit the probability of being out of stock to less than 2.5 Hint: to solve this problem use Excel’s Formulas Statistics 7 POISSON.DIST function or Minitab’s Calc 7 Probability Distributions 7 Poisson option. 5-68. USA Today recently reported that about one-third of eligible workers haven’t enrolled in their employers’ 401k plans. Costco has been contemplating new incentives to encourage more participation from its employees. Of the 12 employees in one of Costco’s automotive departments 5 have enrolled in Costco’s 401k plan. The store manager has randomly selected 7 of the automotive department employees to receive investment training. a. Calculate the probability that all of the employees currently enrolled in the 401k program are selected for the investment training. b. Calculate the probability that none of the employees currently enrolled in the 401k program is selected for the investment training. c. Compute the probability that more than half of the employees currently enrolled in the 401k program are selected for the investment training. 5-69. The Small Business Administration’s Center for Women’s Business Research indicated 30 of private firms had female owners 52 had male owners and 18 had male and female co-owners. In one community there are 50 privately owned firms. Ten privately owned firms are selected to receive assistance in marketing their products. Assume the percentages indicated by the Small Business Administration apply to this community. a. Calculate the probability that one-half of the firms selected will be solely owned by a woman 3 owned by men and the rest co-owned by women and men. b. Calculate the probability that all of the firms selected will be solely owned by women. c. Calculate the probability that 6 will be owned by a woman and the rest co-owned. Computer Database Exercises 5-70. The National Federation of Independent Business NFIB survey contacted 130 small firms. One of the many inquiries was to determine the number of employees the firms had. The file titled Employees contains the responses by the firms. The number of employees was grouped into the following categories: 1 fewer than 20 2 20–99 3 100–499 and 4 500 or more. a. Determine the number of firms in each of these categories. b. If the NFIB contacts 25 of these firms to gather more information determine the probability that it will choose the following number of firms in each category: 1 22 2 2 3 1 and 4 0. c. Calculate the probability that it will choose all of the firms from those businesses with fewer than 20 workers. 5-71. Cliff Summey is the quality-assurance engineer for Sticks and Stones Billiard Supply a manufacturer of billiard supplies. One of the items that Sticks and Stones produces is sets of pocket billiard balls. Cliff has been monitoring the finish of the pocket billiard balls. He is concerned that sets of billiard balls have been shipped with an increasing number of scratches. The company’s goal is to have no more than an average of one scratch per set of pocket billiard balls. A set contains 16 balls. Over the last week Cliff selected a sample of 48 billiard balls and inspected them to determine the number of scratches. The data collected by Cliff are displayed in the file called Poolball. a. Determine the number of scratches in the sample. b. Calculate the average number of scratches for 48 pocket billiard balls if Sticks and Stones has met its goal. c. Determine the probability that there would be at least as many scratches observed per set of pocket billiard balls if Sticks and Stones has met its goal. d. Based on the sample evidence does it appear that Sticks and Stones has met its goal Provide statistical reasons for your conclusion. END EXERCISES 5-3 www.downloadslide.com

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Discrete Probability Distributions 1 Introduction to Discrete Probability Distributions 2 The Binomial Probability Distribution Outcome 2. Be able to apply the binomial distribution to business decision-making situations 3 Other Discrete Probability Distributions Outcome 3. Be able to compute probabilities for the Poisson and hypergeometric distributions and apply these distributions to decision-making situations Summary Although the binomial distribution may be the most often applied discrete distribution for business decision makers the Poisson distribution and the hypergeometric distribution are also frequently employed. The Poisson distribution is used in situations where the value of the random variable is found by counting the number of occurrences within a defned segment of time or space. If you know the mean number of occurrences per segment you can use the Poisson formula the Poisson tables in software such as Excel or Minitab to fnd the probability of any specifc number of occurrences within the segment. The Poisson distribution is often used to describe the number of customers who arrive at a service facility in a specifc amount of time. The hypergeometric distribution is used in situations where the sample size is large relative to the size of the population and the sampling is done without replacement. Conclusion Business applications involving discrete random variables are very common in business situations. The probabilities for each possible outcome of the discrete random variable form the discrete probability distribution. The expected value of a discrete probability distribution is the mean and represents the long-run average value of the random variable. This chapter has introduced three specifc discrete random variables that are frequently used in business situations: binomial distribution Poisson distribution and the hypergeometric distribution. A random variable can take on values that are either discrete or continuous. This chapter has focused on discrete random variables where the potential values are usually integer values. Examples of discrete random variables include the number of defects in a sample of twenty parts the number of customers who purchase Coca-Cola rather than Pepsi when 100 customers are observed the number of days late a shipment will be when the product is shipped from India to the United States or the number of female managers who are promoted from a pool of 30 females and 60 males at a Fortune 500 company. The probabilities associated with the individual values of a random variable form the probability distribution. The most frequently used discrete probability distributions are the binomial distribution and the Poisson distribution. Summary A discrete random variable can assume only a fnite number of values or an infnite sequence of values such as 0 1 2…. The mean of a discrete random variable is called the expected value and represents the long-run average value for the random variable. The graph of a discrete random variable looks like a histogram with the values of the random variable presented on the horizontal axis and the bars above the values having heights corresponding to the probability of the outcome occurring. The sum of the individual probabilities sum to one. Outcome 1. Be able to calculate and interpret the expected value of a discrete random variable Summary The binomial distribution applies when an experimental trial has only two possible outcomes called success and failure the probability of success remains constant from trial to trial the trials are independent and there are a fxed number of identical trials being considered. The probabilities for a binomial distribution can be calculated using Equation 4 or found using Excel or Minitab. The expected value of the binomial distribution is found by multiplying n the number of trials by p the probability of a success on any one trial. The shape of a binomial distribution depends on the sample size number of trials and p the probability of a success. When p is close to .50 the binomial distribution will be fairly symmetric and bell shaped. Even when p is near 0 or 1 if n the sample size is large the binomial distribution will still be fairly symmetric and bell shaped. www.downloadslide.com

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Discrete Probability Distributions 7 Poisson Probability Distribution Px te x xt − 8 Standard Deviation of the Poisson Distribution t 9 Hypergeometric Distribution Two Possible Outcomes per Trial Px CC C nx NX x X n N . 10 Hypergeometric Distribution k Possible Outcomes per Trial Px x x CC C C k x X x X x X x X k K ... ... 12 1 1 2 2 3 3 5 .. . . C C n N 1 Expected Value of a Discrete Probability Distribution E1x2 ΣxP1x2 2 Standard Deviation of a Discrete Probability Distribution x xEx Px ∑− 2 3 Counting Rule for Combinations C n xn x x n 4 Binomial Formula Px n xn x xn x pq 5 Expected Value of a Binomial Distribution m x E1x2 np 6 Standard Deviation of the Binomial Distribution npq Equations Key Terms Binomial Probability Distribution Characteristics Continuous random variable Counting rule for combinations Discrete random variable Expected value Hypergeometric distribution Random variable Poisson distribution Chapter Exercises MyStatLab Conceptual Questions 5-72. Three discrete distributions were discussed in this chapter. Each was defined by a random variable that measured the number of successes. To apply these distributions you must know which one to use. Describe the distinguishing characteristics for each distribution. 5-73. How is the shape of the binomial distribution changed for a given value of p as the sample size is increased Discuss. 5-74. Discuss the basic differences and similarities between the binomial distribution and the Poisson distribution. 5-75. Beginning statistics students are often puzzled by two characteristics of distributions in this chapter: 1 The trials are independent and 2 the probability of a success remains constant from trial to trial. Students often think these two characteristics are the same. The questions in this exercise point out the difference. Consider a hypergeometric distribution where N 3 X 2 and n 2. a. Mathematically demonstrate that the trials for this experiment are dependent by calculating the probability of obtaining a success on the second trial if the first trial resulted in a success. Repeat this calculation if the first trial was a failure. Use these two probabilities to prove that the trials are dependent. b. Now calculate the probability that a success is obtained on each of the three respective trials and therefore demonstrate that the trials are dependent but that the probability of a success is constant from trial to trial. 5-76. Consider an experiment in which a sample of size n 5 is taken from a binomial distribution. a. Calculate the probability of each value of the random variable for the probability of a success equal to 1 0.1 2 0.25 3 0.50 4 0.75 and 5 0.9. b. Which probabilities produced a right-skewed distribution Why c. Which probability of a success yielded a symmetric distribution Why d. Which probabilities produced a left-skewed distribution Discuss why. Business Applications 5-77. The McMillan Newspaper Company sometimes makes printing errors in its advertising and is forced to provide www.downloadslide.com

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Discrete Probability Distributions company concerning the return rate Make sure you support your statement with something other than opinion. 5-81. The Defense Department has recently advertised for bids for producing a new night-vision binocular. Vista Optical has decided to submit a bid for the contract. The first step was to supply a sample of binoculars for the army to test at its Kentucky development grounds. Vista makes a superior night-vision binocular. However the 4 sent to the army for testing were taken from a development-lab project of 20 units that contained 4 defectives. The army has indicated it will reject any manufacturer that submits 1 or more defective binoculars. What is the probability that this mistake has cost Vista any chance for the contract 5-82. VERCOR provides merger and acquisition consultants to assist corporations when owners decide to offer their business for sale. One of its news releases “Tax Audit Frequency Is Rising” written by David L. Perkins Jr. a VERCOR partner and which originally appeared in The Business Owner indicated that the proportion of the largest businesses those corporations with assets of 10 million and over that were audited was 0.17. a. One member of VERCOR’s board of directors is on the board of directors of four other large corporations. Calculate the expected number of these five corporations that should get audited assuming selection is random. b. Three of the five corporations were actually audited. Determine the probability that at least three of the five corporations would be audited if 17 of large corporations are audited. Assume random selection. c. The board member is concerned that the corporations have been singled out to be audited by the Internal Revenue Service IRS. Respond to these thoughts using probability and statistical logic. 5-83. Stafford Production Inc. is concerned with the quality of the parts it purchases that will be used in the end items it assembles. Part number 34-78D is used in the company’s new laser printer. The parts are sensitive to dust and can easily be damaged in shipment even if they are acceptable when they leave the vendor’s plant. In a shipment of four parts the purchasing agent has assessed the following probability distribution for the number of defective products: xP x 0 0.20 1 0.20 2 0.20 3 0.20 4 0.20 a. What is the expected number of defectives in a shipment of four parts Discuss what this value really means to Stafford Production Inc. corrected advertising in the next issue of the paper. The managing editor has done a study of this problem and found the following data: No. of Errors x Relative Frequency 0 0.56 1 0.21 2 0.13 3 0.07 4 0.03 a. Using the relative frequencies as probabilities what is the expected number of errors Interpret what this value means to the managing editor. b. Compute the variance and standard deviation for the number of errors and explain what these values measure. 5-78. The Ziteck Corporation buys parts from international suppliers. One part is currently being purchased from a Malaysian supplier under a contract that calls for at most 5 of the 10000 parts to be defective. When a shipment arrives Ziteck randomly samples 10 parts. If it finds 2 or fewer defectives in the sample it keeps the shipment otherwise it returns the entire shipment to the supplier. a. Assuming that the conditions for the binomial distribution are satisfied what is the probability that the sample will lead Ziteck to keep the shipment if the defect rate is actually 0.05 b. Suppose the supplier is actually sending Ziteck 10 defects. What is the probability that the sample will lead Ziteck to accept the shipment anyway c. Comment on this sampling plan sample size and accept/reject point. Do you think it favors either Ziteck or the supplier Discuss. 5-79. California-based Wagner Foods Inc. has a process that inserts fruit juice into 24-ounce containers. When the process is in control half the cans actually contain more than 24 ounces and half contain less. Suppose a quality inspector has just randomly sampled nine cans and found that all nine had more than 24 ounces. Calculate the probability that this result would occur if the filling process was actually still in control. Based on this probability what conclusion might be reached Discuss. 5-80. Your company president has told you that the company experiences product returns at the rate of two per month with the number of product returns distributed as a Poisson random variable. Determine the probability that next month there will be a. no returns b. one return c. two returns d. more than two returns e. In the last three months your company has had only one month in which the number of returns was at most two. Calculate the probability of this event occurring. What will you tell the president of your www.downloadslide.com

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Discrete Probability Distributions c. Recalling that the coefficient of variation is determined by the ratio of the standard deviation over the mean compute the coefficient of variation for each random variable. d. Referring to part c suppose the seminar director said that the first stock was riskier since its standard deviation was greater than the standard deviation of the second stock. How would you respond Hint: What do the coefficients of variation imply 5-86. Simmons Market Research conducted a national consumer study of 13787 respondents. The respondents were asked to indicate the primary source of the vitamins or mineral supplements they consume. Thirty- five percent indicated a multiple formula was their choice. A subset of 20 respondents who used multiple vitamins was selected for further questioning. Half of them used a One A Day vitamin the rest used generic brands. Of this subset 4 were asked to fill out a more complete health survey. a. Calculate the probability that the final selection of 4 subset members were all One A Day multiple vitamin users. b. Compute the number of One A Day users expected to be selected. c. Calculate the probability that fewer than half of the final selection were One A Day users. 5-87. The 700-room Westin Charlotte offers a premiere uptown location in the heart of the city’s financial district. On a busy weekend the hotel has 20 rooms that are not occupied. Suppose that smoking is allowed in 8 of the rooms. A small tour group arrives which has four smokers and six nonsmokers. The desk clerk randomly selects 10 rooms and gives the keys to the tour guide to distribute to the travelers. a. Compute the probability that the tour guide will have the correct mix of rooms so that all members of the tour group will receive a room that accommodates their smoking preferences. b. Determine the probability that the tour guide will have to assign at least one nonsmoker to a smoking room. c. Determine the probability that the tour guide will have to assign at least one smoker to a nonsmoking room. Computer Database Exercises 5-88. A 23-mile stretch of a two-lane highway east of Paso Robles California was once considered a “death trap” by residents of San Luis Obispo County. Formerly known as “Blood Alley” Highway 46 gained notoriety for the number of fatalities 29 and crashes over a 240-  week period. More than two-thirds involved head-on collisions. The file titled Crashes contains the simulated number of fatal crashes during this time period. a. Determine the average number of crashes in the 240 weeks. b. Compute and interpret the standard deviation of the number of defective parts in a shipment of four. c. Examine the probabilities as assessed and indicate why this probability distribution might be called a uniform distribution. Provide some reasons why the probabilities might all be equal as they are in this case. 5-84. Bach Photographs takes school pictures and charges only 0.99 for a sitting which consists of six poses. The company then makes up three packages that are offered to the parents who have a choice of buying 0 1 2 or all 3 of the packages. Based on his experience in the business Bill Bach has assessed the following probabilities of the number of packages that might be purchased by a parent: No. of Packages xP x 0 0.30 1 0.40 2 0.20 3 0.10 a. What is the expected number of packages to be purchased by each parent b. What is the standard deviation for the random variable x c. Suppose all of the picture packages are to be priced at the same level. How much should they be priced if Bach Photographs wants to break even Assume that the production costs are 3.00 per package. Remember that the sitting charge is 0.99. 5-85. The managing partner for Westwood One Investment Managers Inc. gave a public seminar in which she discussed a number of issues including investment risk analysis. In that seminar she reminded people that the coefficient of variation often can be used as a measure of risk of an investment. To demonstrate her point she used two hypothetical stocks as examples. She let x equal the change in assets for a 1000.00 investment in stock 1 and y reflect the change in assets for a 1000.00 investment in stock 2. She showed the seminar participants the following probability distributions: x P x YP y -+1000.00 0.10 -+1000.00 0.20 0.00 0.10 0.00 0.40 500.00 0.30 500.00 0.30 1000.00 0.30 1000.00 0.05 2000.00 0.20 2000.00 0.05 a. Compute the expected values for random variables x and y. b. Compute the standard deviations for random variables x and y. www.downloadslide.com

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Discrete Probability Distributions c. To make sure that the proportion of defectives does not change the quality control manager wants to establish control limits that are 3 standard deviations above the mean and 3 standard deviations below the mean. Calculate these limits. d. Determine the probability that a randomly chosen set of 20 smoke alarms would have a number of defectives that was beyond the control limits established in part c. 5-90. Covercraft manufactures covers to protect automobile interiors and finishes. Its Block-It 200 Series fabric has a limited two-year warranty. Periodic testing is done to determine if the warranty policy should be changed. One such study examined those covers that became unserviceable while still under warranty. Data that could be produced by such a study are contained in the file titled Covers. The data represent the number of months a cover was used until it became unserviceable. Covercraft might want to examine more carefully the covers that became unserviceable while still under warranty. Specifically it wants to examine those that became unserviceable before they had been in use one year. a. Determine the number of covers that became unserviceable before they had been in use less than a year and a half. b. If Covercraft quality-control staff selects 20 of the covers at random determine the probability that none of them will have failed before they had been in service a year and a half. c. If Covercraft quality control staff needs to examine at least 5 of the failed covers determine the probability that they will obtain this many. b. Calculate the probability that at least 19 crashes would occur over the 240-week period if the average number of crashes per week was as calculated in part a. c. Calculate the probability that at least 19 crashes would occur over a five-year period if the average number of crashes per week was as calculated in part a. d. A coalition of state local and private organizations devised a coordinated and innovative approach to dramatically reduce deaths and injuries on this road. During the 16 months before and after completion of the project fatal crashes were reduced to zero. Calculate the probability that there would be no fatal crashes if the mean number of fatal crashes was not changed by the coalition. Does it appear that the aver- age number of fatal accidents has indeed decreased 5-89. American Household SM Inc. produces a wide array of home safety and security products. One of its products is the First Alert SA302 Dual Sensor Remote Control Smoke Alarm. As part of its quality control program it constantly tests to assure that the alarms work. A change in the manufacturing process requires the company to determine the proportion of alarms that fail the quality control tests. Each day 20 smoke alarms are taken from the production line and tested and the number of defectives is recorded. A file titled Smokeless contains the possible results from the last 90 days of testing. a. Compute the proportion of defective smoke alarms. b. Calculate the expected number and the standard deviation of defectives for each day’s testing. Assume the proportion of defectives is what was computed in part a. Hint: Recall the formulas for the mean and the standard deviation for a binomial distribution. Case 1 SaveMor Pharmacies A common practice now is for large retail pharmacies to buy the customer base from smaller independent pharmacies. The way this works is that the buyer requests to see the customer list along with the buying history. The buyer then makes an offer based on its projection of how many of the seller’s customers will move their business to the buyer’s pharmacy and on how many dollars of new business will come to the buyer as a result of the purchase. Once the deal is made the buyer and seller usually send out a joint letter to the seller’s customers explaining the transaction and informing them that their prescription files have been transferred to the pur- chasing company. The problem is that there is no guarantee regarding what proportion of the existing customers will make the switch to the buying company. That is the issue facing Heidi Fendenand acqui- sitions manager for SaveMor Pharmacies. SaveMor has the oppor- tunity to purchase the 6780-person customer base from Hubbard Pharmacy in San Jose California. Based on previous acquisitions Heidi believes that if 70 or more of the customers will make the switch then the deal is favorable to SaveMor. However if 60 or less make the move to SaveMor then the deal will be a bad one and she would recommend against it. Quincy Kregthorpe a research analyst who works for Heidi has suggested that SaveMor take a new approach to this acquisi- tion decision. He has suggested that SaveMor contact a random sample of 20 Hubbard customers telling them of the proposed sale and asking them if they will be willing to switch their business to SaveMor. Quincy has suggested that if 15 or more of the 20 cus- tomers indicate that they would make the switch then SaveMor should go ahead with the purchase. Otherwise it should decline the deal or negotiate a lower purchase price. Heidi liked this idea and contacted Cal Hubbard Hubbard’s owner to discuss the idea of surveying 20 randomly selected cus- tomers. Cal was agreeable as long as only these 20 customers would be told about the potential sale. Before taking the next step Heidi met with Quincy to discuss the plan one more time. She was concerned that the www.downloadslide.com

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Discrete Probability Distributions even if the true proportion of all customers who would switch is actually 0.70. 2. Compute the probability that the sampling plan will provide a result that suggests that SaveMor should accept the deal even if the true proportion of all customers who would switch is actually only 0.60. 3. Write a short report to Heidi outlining the sampling plan the assumptions on which the evaluation of the sampling plan has been based and the conclusions regarding the potential effectiveness of the sampling plan. The report should make a recommendation about whether Heidi should go through with the idea of using the sampling plan. proposed sampling plan might have too high a probability of rejecting the purchase deal even if it was a positive one from SaveMor’s viewpoint. On the other hand she was concerned that the plan might also have a high probability of accept- ing the purchase deal when in fact it would be unfavorable to SaveMor. After discussing these concerns for over an hour Quincy finally offered to perform an evaluation of the sam- pling plan. Required Tasks: 1. Compute the probability that the sampling plan will provide a result that suggests that SaveMor should reject the deal Arrowmark Vending Arrowmark Vending has the contract to supply pizza at all home football games for a university in the Big 12 athletic conference. It is a constant challenge at each game to determine how many pizzas to have available at the games. Tom Kealey operations manager for Arrowmark has determined that his fixed cost of providing piz- zas whether he sells 1 pizza or 4000 pizzas is 1000. This cost includes hiring employees to work at the concession booths hiring extra employees to cook the pizzas the day of the game delivering them to the game and advertising during the game. He believes that this cost should be equally allocated between two types of pizzas. Tom has determined that he will supply only two types of piz- zas: plain cheese and pepperoni-and-cheese combo. His cost to make a plain cheese pizza is 4.50 each and his cost to make pep- peroni-and-cheese combo is 5.00 each. Both pizzas will sell for 9.00 at the game. Unsold pizzas have no value and are donated to a local shelter for the homeless. Experience has shown the following demand distributions for the two types of pizza at home games: Plain Cheese Demand Probability Pepperoni-and-Cheese Demand Probability 200 0.10 300 0.10 300 0.15 400 0.20 400 0.15 500 0.25 500 0.20 600 0.25 600 0.20 700 0.15 700 0.10 800 0.05 800 0.05 900 0.05 Required Tasks: 1. For each type of pizza determine the profit or loss associated with producing at each possible demand level. For instance determine the profit if 200 plain cheese pizzas are produced and 200 are demanded. What is the profit if 200 plain cheese pizzas are produced but 300 were demanded and so on 2. Compute the expected profit associated with each possible production level assuming Tom will only produce at one of the possible demand levels for each type of pizza. 3. Prepare a short report that provides Tom with the information regarding how many of each type of pizza he should produce if he wants to achieve the highest expected profit from pizza sales at the game. scale stations to have open during various times of the day. If he has too many stations open the scalers will have excessive idle time and the cost of scaling will be unnecessarily high. On the other hand if too few scale stations are open some log trucks will have to wait. Boise Cascade Corporation At the Boise Cascade Corporation lumber mill logs arrive by truck and are scaled measured to determine the number of board feet before they are dumped into a log pond. Figure C-3 illus- trates the basic flow. The mill manager must determine how many www.downloadslide.com

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Discrete Probability Distributions 0 to 6 trucks: open 1 scale station 7 to 12 trucks: open 2 scale stations etc. However the number of trucks is a random variable and is uncertain. Your task is to provide guidance for the decision. The manager has studied the truck arrival patterns and has determined that during the first open hour 7:00 a.m.–8:00 a.m. the trucks randomly arrive at 12 per hour on average. Each scale station can scale 6 trucks per hour 10 minutes each. If the man- ager knew how many trucks would arrive during the hour he would know how many scale stations to have open. FIGURE C-3 |  Truck Flow for Boise Cascade Mill Example Pond Scale Stations Trucks Enter Trucks Exit www.downloadslide.com

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Discrete Probability Distributions Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. 31. a. 0.0688 b. 0.0031 c. 0.1467 d. 0.8470 e. 0.9987 33. a. 3.2 b. 1.386 c. 0.4060 d. 0.9334 35. a. 3.46 b. 0.012 c. 0.004 d. It is quite unlikely. 37. a. 0.5580 b. 0.8784 c. An increase in sample size would be required. 39. a. 2.96 b. Variance 1.8648 Standard deviation 1.3656 c. 0.3811 41. a. 0.3179 b. 0.2174 c. 0.25374 43. a. 0.051987 b. 0.028989 45. a. 0.372 b. 12 estimate may be too high. c. 0.0832 d. 0.0003 e. Redemption rate is lower than either Vericours or TCA Fulfillment estimate. 49. a. 9 corporations b. 0.414324 c. 70th percentile is 12. 51. a. 0.0498 b. 0.1512 53. 0.175 55. a. 0.4242 b. 0.4242 c. 0.4696 57. a. Px 3 0.5 b. Px 5 0 c. 0.6667 d. Since 0.6667 7 0.25 then x 2. 59. Px Ú 10 1 - 0.8305 0.1695 61. 0.0015 63. a. Px 4 0.4696 b. Px 3 0.2167 c. 0.9680 65. a. 0.0355 b. 0.0218 c. 0.0709 67. a. 0.0632 b. 120 Spicy Dogs 69. a. 0.0274 b. 0.0000 c. 0.0001 1. a. discrete random variable b. The possible values for x are x 0 1 2 3 4 5 6. 3. a. number of children under 22 living in a household b. discrete 5. 3.7 days 7. a. 130 b. 412.50 c. 1412.50 20.31 9. a. 15.75 b. 20.75 c. 78.75 d. increases the expected value by an amount equal to the constant added e. the expected value being multiplied by that same constant 11. a. 3.51 b. s 2 1.6499 s 1.2845 13. a. 2.87 days b. s 11.4931 1.22 c. 1.65 days to 4.09 days 15. a. 58300 b. 57480 17. a. Small firm profits +135000 Midsized profits +155000 Large firm profits +160000 b. Small firm: s +30000 Mid-sized firm: s +90000 Large firm: s +156604.60 c. The large firm has the largest expected profit. 21. a. b. 19.168 s 2s 2 23.1634 1.7787 c. Median is 19 and the quality control department is correct. 23. 0.2668 25. a. Px 5 0.0746 b. Px Ú 7 0.2143 c. 4 d. s 1npq 120.20.80 1.7889 27. 0.1029 29. a. 0.1442 b. 0.8002 c. 4.55 x Px 14 0.008 15 0.024 16 0.064 17 0.048 18 0.184 19 0.216 20 0.240 21 0.128 22 0.072 23 0.016 www.downloadslide.com

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Discrete Probability Distributions 79. 0.0020 81. 0.6244 83. a. 2.0 b. 1.4142 c. because outcomes equally likely 85. a. Ex 750 Ey 100 b. StDevx 844.0972 StDevy 717.635 c. CVx 844.0972750 1.1255 CVy 717.635100 7.1764 87. a. 0.3501 b. 0.3250 89. a. 0.02 b. EX 0.6261 c. 0 2.2783 d. 1 - 0.9929 0.0071 71. a. 8 b. lt 13 3 c. 0.0119 d. It is very unlikely. Therefore we believe that the goal has not been met. 75. a. This means the trials are dependent. b. does not imply that the trials are independent 77. a. b. Standard deviation 1.0954 Variance 1.20 X Px xPx 0 0.56 0.00 1 0.21 0.21 2 0.13 0.26 3 0.07 0.21 4 0.03 0.12 0.80 Continuous Random Variables Random variables that can assume an uncountably infinite number of values. Discrete Random Variable A random variable that can only assume a finite number of values or an infinite sequence of values such as 0 1 2c. Expected Value The mean of a probability distribution. The average value when the experiment that generates values for the random variable is repeated over the long run. Hypergeometric Distribution The hypergeometric distribu- tion is formed by the ratio of the number of ways an event of interest can occur over the total number of ways any event can occur. Random Variable A variable that takes on different numerical values based on chance. Binomial Probability Distribution Characteristics A distri- bution that gives the probability of x successes in n trials in a process that meets the following conditions: 1. A trial has only two possible outcomes: a success or a f ailure. 2. There is a fixed number n of identical trials. 3. The trials of the experiment are independent of each other. This means that if one outcome is a success this does not influence the chance of another outcome being a success. 4. The process must be consistent in generating successes and failures. That is the probability p associated with a success remains constant from trial to trial. 5. If p represents the probability of a success then 1- p q is the probability of a failure. Glossary DeVeaux Richard D. Paul F. Velleman and David E. Bock Stats Data and Models 3rd ed. New York: Addison-Wesley 2012. Hogg R. V . and Elliot A. Tanis Probability and Statistical Infer- ence 8th ed. Upper Saddle River NJ: Prentice Hall 2010. Larsen Richard J. and Morris L. Marx An Introduction to Mathematical Statistics and Its Applications 5th ed. Upper Saddle River NJ: Prentice Hall 2012. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. References www.downloadslide.com

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Introduction to Continuous Probability Distributions From Chapter 6 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Why you need to know You will encounter many business situations in which the random variable of interest is discrete and in which prob- ability distributions such as the binomial the Poisson or the hypergeometric will be useful for analyzing decision situations. But there will also be many situations in which the random variable of interest is continuous rather than discrete. For instance the managers at Harley Davidson might be interested in a measure called throughput time which is the time it takes from when a motorcycle is started on the manufacturing line until it is completed. Lots of factors can affect the throughput time including breakdowns need for rework the type of accessories added to the motorcycle and worker productivity. The managers might be interested in determining the probability that the throughput time will be between 3.5 and 5.0 hours. In this case time is the random variable of interest and is continuous. A pharmaceutical company may be interested in the probability that a new drug will reduce blood pressure by more than 20 points for patients. Blood pressure is the continuous random variable of interest. The Kellogg’s Cereal company could be interested in the probability that cereal boxes labeled as containing 16 ounces will actually con- tain at least that much cereal. Here the variable of interest is weight which can be measured on a continuous scale. In each of these examples the value of the variable of interest is determined by measuring measuring the time required to make a motorcycle measuring the blood pressure reading measuring the weight of cereal in a box. In every instance the number of possible values for the variable is limited only by the capacity of the meas- uring device. The constraints imposed by the measuring devices produce a finite number of outcomes. In these and similar situations a continuous probability distribution can be used to approximate the distribution of possible outcomes for the random variables. The approximation is appropriate when the number of possible outcomes is large. This text introduces three specific continuous probability distributions of particular importance for decision making and the study of business statistics. The first of these the normal distribution is by far the most important because a great many applications involve random variables that possess the characteristics of the normal distri- bution. In addition many topics dealing with statistical estimation and hypothesis testing are based on the normal distribution. Review the concept of z-scores. Review the discussion of the mean and standard deviation. Quick Prep Links Review the methods for determining the probability for a discrete random variable. Outcome 4. Calculate probabilities associated with a uniformly distributed random variable. Outcome 5. Determine probabilities using an exponential probability distribution. Outcome 1. Convert a normal distribution to a standard normal distribution. Outcome 2. Determine probabilities using the standard normal distribution. Outcome 3. Calculate values of the random variable associated with specified probabilities from a normal distribution. 1 The Normal Probability Distribution Introduction to Continuous Probability Distributions 2 Other Continuous Probability Distributions Rafa Irusta/Shutterstock www.downloadslide.com

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Introduction to Continuous Probability Distributions In addition to the normal distribution you will be introduced to the uniform distribution and the exponential distribution. Both are important continuous probability distributions and have many applications in business decision making. You need to have a firm understanding and working knowledge of all three continuous probability distributions introduced in this chapter. 1 The Normal Probability Distribution A Pepsi-Cola can is supposed to contain 12 ounces but it might actually contain any amount between 11.90 and 12.10 ounces such as 11.9853 ounces. When the variable of interest such as the volume of soda in a can is approximately continuous the prob- ability distribution associated with the random variable is called a continuous probability distribution. One important difference between discrete and continuous probability distributions involves the calculation of probabilities associated with specific values of the random varia- ble. For instance in a market research example in which 100 people are surveyed and asked whether they have a positive view of a product we could use the binomial distribution to find the probability of any specific number of positive reviews such as P1x 752 or P1x 762. Although these individual probabilities may be small values they can be computed because the random variable is discrete. However if the random variable is continuous as in the Pepsi-Cola example there is an uncountable infinite number of possible outcomes for the random variable. Theoretically the probability of any one of these individual outcomes is zero. That is P1x 11.922 0 or P1x 12.052 0. Thus when you are working with continuous distributions you will need to find the probability for a range of possible values such as P1x … 11.922 or P1 11.92 … x … 12.02. You can also conclude that P1x … 11.922 P1x 6 11.922 because we assume that P1x 11.922 0. There are many different continuous probability distributions but the most important of these is the normal distribution. The Normal Distribution 1 Figure 1 illustrates a typical normal distribution and highlights the normal distribution’s characteristics. All normal distributions have the same general shape as the one shown in Figure 1. However they can differ in their mean value and their variation depending on the situation being considered. The process being represented determines the scale of the horizontal axis. It may be pounds inches dollars or any other attribute with a Normal Distribution The normal distribution is a bell-shaped distribution with the following properties: 1. It is unimodal that is the normal distribu- tion peaks at a single value. 2. It is symmetrical this means that the two areas under the curve between the mean and any two points equidistant on either side of the mean are identical. One side of the distribution is the mirror image of the other side. 3. The mean median and mode are equal. 4. The normal approaches the horizontal axis on either side of the mean toward plus and minus infinity ∞. In more formal terms the normal distribution is asymp- totic to the x axis. 5. The amount of variation in the random vari- able determines the height and spread of the normal distribution. 1 It is common to refer to the very large family of normal distributions as “the normal distribution.” x Mean Median Mode Probability 0.50 Probability 0.50 FIGURE 1 |  Characteristics of the Normal Distribution www.downloadslide.com

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Introduction to Continuous Probability Distributions FIGURE 2 |  Difference between Normal Distributions x x x Area 0.50 Area 0.50 Area 0.50 Area 0.50 Area 0.50 Area 0.50 a b c Normal Probability Density Function fx e x 1 2 2 2 2 1 where: x Any value of the continuous random variable s Population standard deviation p 3.14159 e Base of the natural log 2.71828 m Population mean To graph the normal distribution we need to know the mean m and the standard devia- tion s. Placing m s and a value of the variable x into the probability density function we can calculate a height fx of the density function. If we could try enough x values we could construct curves like those shown in Figures 1 and 2. The area under the normal curve corresponds to probability. Because x is a continuous random variable the probability Px is equal to 0 for any particular x. However we can find the probability for a range of values between x 1 and x 2 by finding the area under the curve between these two values. A special normal distribution called the standard normal distribu- tion is used to find areas probabilities for all normal distributions. The Standard Normal Distribution The trick to finding probabilities for a normal distribution is to convert the normal distribution to a standard normal distribution. To convert a normal distribution to a standard normal distribution the values x of the random variable are standardized. The conversion formula is shown as Equation 2. Standard Normal Distribution A normal distribution that has a mean 0.0 and a standard deviation 1.0 The horizontal axis is scaled in z-values that measure the number of standard deviations a point is from the mean. Values above the mean have positive z-values. Values below the mean have negative z-values. Chapter Outcome 1. continuous measurement. Figure 2 shows several normal distributions with different centers and different spreads. Note that the total area probability under each normal curve equals 1. The normal distribution is described by the rather complicated-looking probability den- sity function shown in Equation 1. www.downloadslide.com

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Introduction to Continuous Probability Distributions Standardized Normal z-Value z x 2 where: z Scaled value 1 the number of standard deviations a point x is from the mean2 x Any point on the horizontal axis m Mean of the specific normal distribution s Standard deviation of the specific normal distribution x Ounces 16 4 fx FIGURE 3 |  Distribution of Grapefruit Weights Equation 2 scales any normal distribution axis from its true units time weight dollars volume and so forth to the standard measure referred to as a z-value. Thus any value of the normally distributed continuous random variable can be represented by a unique z-value. Positive z-values represent corresponding values of the random variable x that are higher than the population mean. Values of x that are less than the population mean will have cor- responding z-values that are negative. BUSINESS APPLICATION STANDARD NORMAL DISTRIBUTION FRUIT PRODUCTION Fruit growers in California and Florida strive for consistency in their products in terms of quality and size. For example a grower in Florida that specializes in grapefruit has determined that in one orchard the mean weight of his “King” brand grapefruit is 16 ounces. Suppose after careful analysis he has determined the grapefruit weight distribution is approximated by a normal distribution with a standard deviation of 4 ounces. Figure 3 shows this normal distribution with m 16 and s 4. Three grapefruit were selected from a case in the grower’s cold storage. The weights of these three grapefruit were Grapefruit 1: x 16 ounces Grapefruit 2: x 18.5 ounces Grapefruit 3: x 9 ounces Equation 2 is used to convert these values from a normally distributed population with m 16 and s 4 to corresponding z-values in a standard normal distribution. For Grapefruit 1 we get z x 16 16 4 0 Note Grapefruit 1 weighed 16 ounces which happens to be equal to the population mean for all grapefruit. The standardized z-value corresponding to the population mean is zero. This indicates that the population mean is 0 standard deviations from itself. atoss/Fotolia www.downloadslide.com

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Introduction to Continuous Probability Distributions FIGURE 4 |  Standard Normal Distribution z –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 1.0 0.5 1.5 2.0 2.5 3.0 0 fz Chapter Outcome 2. For Grapefruit 2 we get z x−− 18 5 16 4 063 . . Thus a grapefruit that weighs 18 ounces is 0.63 standard deviations heavier than the mean for all grapefruit. The standardized z-value for Grapefruit 3 is z x−− − 916 4 175 . This means a grapefruit that weighs only 9 ounces is 1.75 standard deviations below the popu- lation mean. Note a negative z-value always indicates the x-value is less than the mean m. The z-value represents the number of standard deviations a point is above or below the population mean. Equation 2 can be used to convert any specified value x from the population distribution to a corresponding z-value. If the population distribution is normally distributed as shown in Figure 3 then the distribution of z-values will also be normally distributed and is called the standard normal distribution. Figure 4 shows a standard normal distribution where the horizontal axis represents z-values. Y ou can convert the normal distribution to a standard normal distribution and use the standard normal table to find the desired probability. Example 1 shows the steps required to do this. Using the Standard Normal Table The standard normal table provides probabilities or areas under the normal curve associated with many different z-values. The standard normal table is constructed so that the probabilities provided represent the chance of a value being between a positive z-value and its population mean 0. The standard normal table is also reproduced in Table 1. This table provides probabilities for z-values between z 0.00 and z 3.09. Note because the normal distribution is sym- metric the probability of a value being between a positive z-value and its population mean 0 is the same as that of a value being between a negative z-value and its population mean 0. So we can use one standard normal table for both positive and negative z-values. EXAMPLE 1 USING THE STANDARD NORMAL TABLE Airline Passenger Loading Times After completing a study the Chicago O’Hare Airport managers have concluded that the time needed to get passengers loaded onto an airplane is normally distributed with a mean equal to 15 minutes and a standard deviation equal to 3.5 minutes. Recently one airplane required 22 minutes to get passengers on board and ready for takeoff. To find the probability that a flight will take 22 or more minutes to get passengers loaded you can use the following steps: Step 1 Determine the mean and standard deviation for the random variable. The parameters of the probability distribution are m 15 and s 3.5 How to do it Example 1 Using the Normal Distribution If a continuous random variable is distributed as a normal distribution the distribution is symmetrically distributed around the mean and is described by the mean and standard deviation. T o find probabilities associ- ated with a normally distributed ran- dom variable use the following steps: 1. Determine the mean m and the standard deviation s. 2. Define the event of interest such as P1x Ú x 1 2. 3. Convert the normal distribution to the standard normal distribu- tion using Equation 2: 4. Use the standard normal distribution table to find the probability associated with the calculated z-value. The table gives the probability between the z-value and the mean. 5. Determine the desired probabil- ity using the knowledge that the probability of a value being on either side of the mean is 0.50 and the total probability under the normal distribution is 1.0. claudiozacc / Fotolia www.downloadslide.com

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Introduction to Continuous Probability Distributions Step 2 Define the event of interest. The flight load time is 22 minutes. We wish to find P1x Ú 222 Step 3 Convert the random variable to a standardized value using Equation 2. z x 22 15 35 200 . . Step 4 Find the probability associated with the z-value in the standard normal distribution table. To find the probability associated with z 2.00 3i.e. P10 … z … 2.0024 do the following: 1. Go down the left-hand column of the table to z 2.0. 2. Go across the top row of the table to the column 0.00 for the second decimal place in z 2.00. 3. Find the value where the row and column intersect. The value 0.4772 is the probability that a value in a normal distribution will lie between the mean and 2.00 standard deviations above the mean. Step 5 Determine the probability for the event of interest. P1x Ú 222 We know that the area on each side of the mean under the normal distribution is equal to 0.50. In Step 4 we computed the probability associated with z 2.00 to be 0.4772 which is the probability of a value falling between the mean and 2.00 standard deviations above the mean. Then the probability we are looking for is P1x Ú 222 P1z Ú 2.002 0.5000 - 0.4772 0.0228 END EXAMPLE TRY PROBLEM 2 BUSINESS APPLICATION THE NORMAL DISTRIBUTION FRUIT PRODUCTION CONTINUED Earlier we dis- cussed the situation involving the fruit grower in Florida. The grapefruit for this grower’s orchard have a mean weight of 16 ounces. We assumed the distribution for grapefruit weight was normally distributed with m 16 and s 4. A local television station that runs a consumer advocacy program reported that a grapefruit from this grower was selected and weighed only 14 ounces. The reporter said she thought it should have been heavier if the mean weight is supposed to be 16 ounces. The grower when interviewed said that he thought the probability was quite high that a grapefruit would weigh 14 or more ounces. To check his statement out we want to find P1x Ú 142 This probability corresponds to the area under a normal distribution to the right of x 14 ounces. This will be the sum of the area between x 14 and m 16 plus the area to the right of m 16. Refer to Figure 5. atoss/Fotolia www.downloadslide.com

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Introduction to Continuous Probability Distributions TABLE 1 | Standard Normal Distribution Table z 0 0.52 0.1985 Example: z 0.52 or – 0.52 P0 z 0.52 0.1985 or 19.85 z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990 To illustrate: 19.85 of the area under a normal curve lies between the mean m and a point 0.52 standard deviation units away. www.downloadslide.com

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Introduction to Continuous Probability Distributions To find this probability you first convert x 14 ounces to its corresponding z-value. This is equivalent to determining the number of standard deviations x 14 is from the popu- lation mean of m 16. Equation 2 is used to do this as follows: z x 14 16 4 050 . Because the normal distribution is symmetrical even though the z-value is –0.50 we find the desired probability by going to the standard normal distribution table for a positive z 0.50. The probability in the table for z 0.50 corresponds to the probability of a z-value occurring between z 0.50 and z 0.00. This is the same as the probability of a z -v alue falling between z- 0.50 and z 0.00. Thus from the standard normal table Table 1 we get P1- 0.50 … z … 0.002 0.1915 This is the area between x 14 and m 16 in Figure 5. We now add 0.1915 to 0.5000 3P1x 7 16 0.500024. Therefore the probability that a grapefruit will weigh 14 or more ounces is P1x Ú 142 0.1915 + 0.5000 0.6915 This is illustrated in Figure 5. Thus there is nearly a 70 chance that a grapefruit will weigh at least 14 ounces. BUSINESS APPLICATION USING THE NORMAL DISTRIBUTION GENERAL ELECTRIC COMPANY Several states includ ing California have passed legislation requiring automakers to sell a certain percentage of zero-emissions cars within their borders. One current alternative is battery-powered cars. The major problem with battery-operated cars is the limited time they can be driven before the batteries must be recharged. Suppose that General Electric GE has developed a Longlife battery pack it claims will power a car at a sustained speed of 45 miles per hour for an average of 8 hours. But of course there will be variations: Some battery packs will last longer and some less than 8 hours. Current data indicate that the standard deviation of battery operation time before a charge is needed is 0.4 hours. Data show a normal distribution of uptime on these battery FIGURE 5 |  Probabilities from the Normal Curve for Fruit Production x x 14 z z –.50 x 14 0.50 0.1915 Pixel Embargo / Fotolia www.downloadslide.com

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Introduction to Continuous Probability Distributions packs. Automakers are concerned that batteries may run short. For example drivers might find an “8-hour” battery that lasts 7.5 hours or less unacceptable. What are the chances of this happening with the Longlife battery pack To calculate the probability the batteries will last 7.5 hours or less find the appropriate area under the normal curve shown in Figure 6. There is approximately 1 chance in 10 that a battery will last 7.5 hours or less when the vehicle is driven at 45 miles per hour. Suppose this level of reliability is unacceptable to the automakers. Instead of a 10 chance of an “8-hour” battery lasting 7.5 hours or less the automakers will accept no more than a 2 chance. GE managers ask what the mean uptime would have to be to meet the 2 requirement. Assuming that uptime is normally distributed we can answer this question by using the standard normal distribution. However instead of using the standard normal table to find a probability we use it in reverse to find the z-value that corresponds to a known probability. Figure 7 shows the uptime distribution for the battery packs. Note the 2 probability is shown in the left tail of the distribution. This is the allowable chance of a battery lasting 7.5 hours or less. We must solve for m the mean uptime that will meet this requirement. 1. Go to the body of the standard normal table where the probabilities are located and find the probability as close to 0.48 as possible. This is 0.4798. 2. Determine the z-value associated with 0.4798. This is z 2.05. Because we are below the mean the z is negative. Thus z- 2.05. FIGURE 6 |  Longlife Battery z z –1.25 x 7.5 0.3944 0.1056 z From the normal table P–1.25 z 0 0.3944 Ten we fnd Px 7.5 hours 0.5000 – 0.3944 0.1056 –1.25 7.5 – 8 0.4 FIGURE 7 |  Longlife Battery Solving for the Mean x Battery uptime hours 7.5 z –2.05 0.48 0.02 fx z –2.05 www.downloadslide.com

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Introduction to Continuous Probability Distributions 3. The formula for z is z x 4. Substituting the known values we get 205 75 . . 0.4 5. Solve for m: m 7.5 -1-2.05210.42 8.32 hours General Electric will need to increase the mean life of the battery pack to 8.32 hours to meet the automakers’ requirement that no more than 2 of the batteries fail in 7.5 hours or less. BUSINESS APPLICATION USING THE NORMAL DISTRIBUTION STATE BANK AND TRUST The director of operations for the State Bank and Trust recently performed a study of the time bank customers spent from when they walk into the bank until they complete their banking. The data file State Bank contains the data for a sample of 1045 customers randomly observed over a four-week period. The customers in the survey were limited to those who were there for basic bank business such as making a deposit or a withdrawal or cashing a check. The histogram in Figure 8 shows that the banking times are distributed as an approximate normal distribution. 2 Minitab Instructions for similar results: 1. Open fle: State Bank. MTW. 2. Choose Graph Histogram. 3. Click Simple. 4. Click OK. 5. In Graph Variables enter data column Service Time. 6. Click OK. Excel 2010 Instructions: 1. Open fle: State Bank.xlsx. 2. Create bins upper limit of each class. 3. Select Data Data Analysis. 4. Select Histogram. 5. Defne data and bin ranges. 6. Check Chart Output. 7. Defne Output Location and click OK. 8. Select the chart and right click. 9. Click on Format Data Series and set gap width to zero. Add lines to the bars and label axes and title appropriately. FIGURE 8 |  Excel 2010 Output for State Bank and Trust Service Times Chapter Outcome 3. Excel tutorials Excel Tutorial 2 A statistical technique known as the chi-square goodness-of-fit test can be used to determine statistically whether the data follow a normal distribution. www.downloadslide.com

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Introduction to Continuous Probability Distributions The mean service time for the 1045 customers was 22.14 minutes with a standard deviation equal to 6.09 minutes. On the basis of these data the manager assumes that the service times are normally distributed with m 22.14 and s 6.09. Given these assumptions the manager is considering providing a gift certificate to a local restaurant to any customer who is required to spend more than 30 minutes to complete basic bank business. Before doing this she is interested in the probability of having to pay off on this offer. Figure 9 shows the theoretical distribution with the area of interest identified. The manager is interested in finding P1x 7 30 minutes2 This can be done manually or with Excel. Figure 10 shows the computer output for Excel. The cumulative probability is P1x … 302 0.9016 Then to find the probability of interest we subtract this value from 1.0 giving P1x … 30 minutes2 1.0 - 0.9016 0.0984 FIGURE 9 |  Normal Distribution for the State Bank and Trust Example x Time x 30 Area of interest 0.0984 FIGURE 10 |  Excel 2010 Output for State Bank and Trust Minitab Instructions for similar results: 1. Choose Calc Probability Distribution Normal. 2. Choose Cumulative probability. 3. In Mean enter m. 4. In Standard deviation enter s. 5. In Input constant enter x. 6. Click OK. Excel 2010 Instructions: 1. Open a blank worksheet. 2. Select Formulas. 3. Click on f x function wizard. 4. Select the Statistical category. 5. Select the NORM.DIST function. 6. Fill in the requested information in the template. 7. True indicates cumulative probabilities. 8. Click OK. www.downloadslide.com

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Introduction to Continuous Probability Distributions Thus there are just under 10 chances in 100 that the bank would have to give out a gift certificate. Suppose the manager believes this policy is too liberal. She wants to set the time limit so that the chance of giving out the gift is at most only 5. You can use the standard normal table or the NORM.INV function in Excel to find the new limit. 3 To use the table we first consider that the manager wants a 5 area in the upper tail of the normal distribution. This will leave 0.50 - 0.05 0.45 between the new time limit and the mean. Now go to the body of the standard normal table where the probabilities are and locate the value as close to 0.45 as possible 0.4495 or 0.4505. Next determine the z-value that corresponds to this probability. Because 0.45 lies midway between 0.4495 and 0.4505 we interpolate halfway between z 1.64 and z 1.65 to get z 1.645 Now we know z x We then substitute the known values and solve for x: 1 645 22 14 22 14 1 645 6 09 32 15 . . .. . . x x x 6.09 8 8 minutes Therefore any customer required to spend more than 32.158 minutes will receive the gift. This should result in no more than 5 of the customers getting the restaurant certificate. Obviously the bank will work to reduce the average service time or standard deviation so even fewer customers will have to be in the bank for more than 32 minutes. EXAMPLE 2 USING THE NORMAL DISTRIBUTION Lockheed Martin Lockheed Martin the defense contractor has a project underway involv- ing the design and construction of communication satellite systems to be used by the U.S. military. Because of the very high cost more than 1 billion each the company performs numerous tests on every component. These tests tend to extend the component assembly time. The time required to construct and test called build time a particular component part is thought to be normally distributed with a mean equal to 30 hours and a standard deviation equal to 4.7 hours. To keep the assembly flow moving on schedule this component needs to have a build time of between 26 to 35 hours. To determine the probability of this happening use the following steps: Step 1 Determine the mean m and the standard deviation s. The mean build time for this step in the process is thought to be 30 hours and the standard deviation is thought to be 4.7 hours. Step 2 Define the event of interest. We are interested in determining the following: P126 … x … 352 Step 3 Convert values of the specified normal distribution to corresponding values of the standard normal distribution using Equation 2: z x 3 The function is NORM.INV1probability mean standard deviation2. For this example NORM.INV .9522.146.09 32.157. www.downloadslide.com

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Introduction to Continuous Probability Distributions We need to find the z-value corresponding to x 26 and to x 35. z x z 26 30 47 085 35 30 47 . . . and106 . Step 4 Use the standard normal table to find the probabilities associated with each z value. For z- 0.85 the probability is 0.3023. For z 1.06 the probability is 0.3554. Step 5 Determine the desired probability for the event of interest. P126 … x … 352 0.3023 + 0.3554 0.6577 Thus there is a 0.6577 chance that the build time will be such that assembly will stay on schedule. Using Excel’s NORM.DIST function we find NORM. DIST35304.7true - NORM.DIST26304.7true 0.6589. The difference in the two probabilities is due to rounding of the calculated z-values when done using the table. END EXAMPLE TRY PROBLEM 13 Approximate Areas under the Normal Curve For the normal distribution we can make this rule more precise. Knowing the area under the normal curve between 1s 2s and 3s provides a useful benchmark for estimating probabilities and checking reasonableness of results. Figure 11 shows these benchmark areas for any normal distribution. FIGURE 11 |  Approximate Areas under the Normal Curve 68.26 95.44 99.74 MyStatLab 6-1: Exercises Skill Development 6-1. For a normally distributed population with m 200 and s 20 determine the standardized z-value for each of the following: a. x 225 b. x 190 c. x 240 6-2. For a standardized normal distribution calculate the following probabilities: a. P1z 6 1.52 b. P1z Ú 0.852 c. P1-1.28 6 z 6 1.752 6-3. For a standardized normal distribution calculate the following probabilities: a. P10.00 6 z … 2.332 b. P1-1.00 6 z … 1.002 c. P11.78 6 z 6 2.342 www.downloadslide.com

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Introduction to Continuous Probability Distributions a. What is the probability that a randomly selected value will be greater than 1550 b. What is the probability that a randomly selected value will be less than 1485 c. What is the probability that a randomly selected value will be either less than 1475 or greater than 1535 6-12. A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution a. What value will be exceeded 10 of the time b. What value will be exceeded 85 of the time c. Determine two values of which the smallest has 25 of the values below it and the largest has 25 of the values above it. d. What value will 15 of the observations be below 6-13. A random variable is normally distributed with a mean of 60 and a standard deviation of 9. a. What is the probability that a randomly selected value from the distribution will be less than 46.5 b. What is the probability that a randomly selected value from the distribution will be greater than 78 c. What is the probability that a randomly selected value will be between 51 and 73.5 Business Applications 6-14. A global financial institution transfers a large data file every evening from offices around the world to its London headquarters. Once the file is received it must be cleaned and partitioned before being stored in the company’s data warehouse. Each file is the same size and the time required to transfer clean and partition a file is normally distributed with a mean of 1.5 hours and a standard deviation of 15 minutes. a. If one file is selected at random what is the probability that it will take longer than 1 hour and 55 minutes to transfer clean and partition the file b. If a manager must be present until 85 of the files are transferred cleaned and partitioned how long will the manager need to be there c. What percentage of the data files will take between 63 minutes and 110 minutes to be transferred cleaned and partitioned 6-15. Doggie Nuggets Inc. DNI sells large bags of dog food to warehouse clubs. DNI uses an automatic filling process to fill the bags. Weights of the filled bags are approximately normally distributed with a mean of 50 kilograms and a standard deviation of 1.25 kilograms. a. What is the probability that a filled bag will weigh less than 49.5 kilograms b. What is the probability that a randomly sampled filled bag will weigh between 48.5 and 51 kilograms c. What is the minimum weight a bag of dog food could be and remain in the top 15 of all bags filled d. DNI is unable to adjust the mean of the filling process. However it is able to adjust the standard 6-4. For a standardized normal distribution determine a value say z 0 so that a. P10 6 z 6 z 0 2 0.4772 b. P1- z 0 … z 6 02 0.45 c. P1-z 0 … z … z 0 2 0.95 d. P1z 7 z 0 2 0.025 e. P1z … z 0 2 0.01 6-5. Consider a random variable z that has a standardized normal distribution. Determine the following probabilities: a. P10 6 z 6 1.962 b. P1z 7 1.6452 c. P11.28 6 z … 2.332 d. P1-2 … z … 32 e. P1z7-12 6-6. A random variable x has a normal distribution with m 13.6 and s 2.90. Determine a value x 0 so that a. P1x 7 x 0 2 0.05. b. P1x … x 0 2 0.975. c. P1m - x 0 … x …m + x 0 2 0.95. 6-7. For the following normal distributions with parameters as specified calculate the required probabilities: a. m 5 s 2 calculate P10 6 x 6 82. b. m 5 s 4 calculate P10 6 x 6 82. c. m 3 s 2 calculate P10 6 x 6 82. d. m 4 s 3 calculate P1x 7 12. e. m 0 s 3 calculate P1x 7 12. 6-8. A population is normally distributed with m 100 and s 20. a. Find the probability that a value randomly selected from this population will have a value greater than 130. b. Find the probability that a value randomly selected from this population will have a value less than 90. c. Find the probability that a value randomly selected from this population will have a value between 90 and 130. 6-9. A random variable is known to be normally distributed with the following parameters: m 5.5 and s 0.50 a. Determine the value of x such that the probability of a value from this distribution exceeding x is at most 0.10. b. Referring to your answer in part a what must the population mean be changed to if the probability of exceeding the value of x found in part a is reduced from 0.10 to 0.05 6-10. A randomly selected value from a normal distribution is found to be 2.1 standard deviations above its mean. a. What is the probability that a randomly selected value from the distribution will be greater than 2.1 standard deviations above the mean b. What is the probability that a randomly selected value from the distribution will be less than 2.1 standard deviations from the mean 6-11. Assume that a random variable is normally distributed with a mean of 1500 and a variance of 324. www.downloadslide.com

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Introduction to Continuous Probability Distributions of operating hours before failure is approximately normally distributed. a. What is the probability that a wash-down motor will have to be replaced free of charge b. What percentage of FG wash-down motors can be expected to operate for more than 17500 hours c. If FG wants to design a wash-down motor so that no more than 1 are replaced free of charge what would the average hours of operation before failure have to be if the standard deviation remains at 1250 hours 6-20. A private equity firm is evaluating two alternative investments. Although the returns are random each investment’s return can be described using a normal distribution. The first investment has a mean return of 2000000 with a standard deviation of 125000. The second investment has a mean return of 2275000 with a standard deviation of 500000. a. How likely is it that the first investment will return 1900000 or less b. How likely is it that the second investment will return 1900000 or less c. If the firm would like to limit the probability of a return being less than 1750000 which investment should it make 6-21. L.J. Raney Associates is a financial planning group in Kansas City Missouri. The company specializes in financial planning for schoolteachers in the Kansas City area. As such it administers a 403b tax shelter annuity program in which public schoolteachers can participate. The teachers can contribute up to 20000 per year on a pretax basis to the 403b account. Very few teachers have incomes sufficient to allow them to make the maximum contribution. The lead analyst at L.J. Raney Associates has recently analyzed the company’s 403b clients and determined that the annual contribution is approximately normally distributed with a mean equal to 6400. Further he has determined that the probability a customer will contribute more than 13000 is 0.025. Based on this information what is the standard deviation of contributions to the 403b program 6-22. No Leak Plumbing and Repair provides customers with firm quotes for a plumbing repair job before actually starting the job. To be able to do this No Leak has been very careful to maintain time records over the years. For example it has determined that the time it takes to remove a broken sink disposal and to install a new unit is normally distributed with a mean equal to 47 minutes and a standard deviation equal to 12 minutes. The company bills at 75.00 for the first 30 minutes and 2.00 per minute for anything beyond 30 minutes. Suppose the going rate for this procedure by other plumbing shops in the area is 85.00 not including the cost of the new equipment. If No Leak bids the disposal job at 85 on what percentage of such jobs will the actual time required exceed the time for which it will be getting paid deviation of the filling process. What would the standard deviation need to be so that no more than 2 of all filled bags weigh more than 52 kilograms 6-16. LaCrosse Technology is one of many manufacturers of atomic clocks. It makes an atomic digital watch that is radio controlled and that maintains its accuracy by reading a radio signal from a WWVB radio signal from Colorado. It neither loses nor gains a second in 20 million years. It is powered by a 3-volt lithium battery expected to last three years. Suppose the life of the battery has a standard deviation of 0.3 years and is normally distributed. a. Determine the probability that the watch’s battery will last longer than 3.5 years. b. Calculate the probability that the watch’s battery will last more than 2.75 years. c. Compute the length-of-life value for which 10 of the watch’s batteries last longer. 6-17. The average number of acres burned by forest and range fires in a large Wyoming county is 4300 acres per year with a standard deviation of 750 acres. The distribution of the number of acres burned is normal. a. Compute the probability that more than 5000 acres will be burned in any year. b. Determine the probability that fewer then 4000 acres will be burned in any year. c. What is the probability that between 2500 and 4200 acres will be burned d. In those years when more than 5500 acres are burned help is needed from eastern-region fire teams. Determine the probability help will be needed in any year. 6-18. An Internet retailer stocks a popular electronic toy at a central warehouse that supplies the eastern United States. Every week the retailer makes a decision about how many units of the toy to stock. Suppose that weekly demand for the toy is approximately normally distributed with a mean of 2500 units and a standard deviation of 300 units. a. If the retailer wants to limit the probability of being out of stock of the electronic toy to no more than 2.5 in a week how many units should the central warehouse stock b. If the retailer has 2750 units on hand at the start of the week what is the probability that weekly demand will be greater than inventory c. If the standard deviation of weekly demand for the toy increases from 300 units to 500 units how many more toys would have to be stocked to ensure that the probability of weekly demand exceeding inventory is no more than 2.5 6-19. FG Industries manufactures a wash-down motor that is used in the food processing industry. The motor is marketed with a warranty that guarantees it will be replaced free of charge if it fails within the first 13000 hours of operation. On average FG wash- down motors operate for 15000 hours with a standard deviation of 1250 hours before failing. The number www.downloadslide.com

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Introduction to Continuous Probability Distributions usage would have enough capacity with the new 1-GB player 6-26. According to the Federal Reserve Board the average credit card debt per U.S. household was 8565 in 2008. Assume that the distribution of credit card debt per household has a normal distribution with a standard deviation of 3000. a. Determine the percentage of households that have a credit card debt of more than 13000. b. One household has a credit card debt that is at the 95th percentile. Determine its credit card debt. c. If four households were selected at random determine the probability that at least half of them would have credit card debt of more than 13000. 6-27. Georgia-Pacific is a major forest products company in the United States. In addition to timberlands the company owns and operates numerous manufacturing plants that make lumber and paper products. At one of their plywood plants the operations manager has been struggling to make sure that the plywood thickness meets quality standards. Specifically all sheets of their 3/4-inch plywood must fall within the range 0.747 to 0.753 inches in thickness. Studies have shown that the current process produces plywood that has thicknesses that are normally distributed with a mean of 0.751 inches and a standard deviation equal to 0.004 inches. a. Use Excel to determine the proportion of plywood sheets that will meet quality specifications 0.747 to 0.753 given how the current process is performing. b. Referring to part a suppose the manager is unhappy with the proportion of product meeting specifications. Assuming that he can get the mean adjusted to 0.75 inches what must the standard deviation be if he is going to have 98 of his product meet specifications 6-28. A senior loan officer for Whitney National Bank has recently studied the bank’s real estate loan portfolio and found that the distribution of loan balances is approximately normally distributed with a mean of 155600 and a standard deviation equal to 33050. As part of an internal audit bank auditors recently randomly selected 100 real estate loans from the portfolio of all loans and found that 80 of these loans had balances below 170000. The senior loan officer is concerned that the sample selected by the auditors is not representative of the overall portfolio. In particular he is interested in knowing the expected proportion of loans in the portfolio that would have balances below 170000. You are asked to conduct an appropriate analysis and write a short report to the senior loan officers with your conclusion about the sample. Computer Database Exercises 6-29. The PricewaterhouseCoopers Human Capital Index Report indicated that the average cost for an American company to fill a job vacancy during the study period 6-23. According to Business Week Maternity Chic a purveyor of designer maternity wear sells dresses and pants priced around 150 each for an average total sale of 1200. The total sale has a normal distribution with a standard deviation of 350. a. Calculate the probability that a randomly selected customer will have a total sale of more than 1500. b. Compute the probability that the total sale will be within 2 standard deviations of the mean total sales. c. Determine the median total sale. 6-24. The Aberdeen Coca-Cola Bottling plant located in Aberdeen North Carolina is the bottler and distributor for Coca-Cola products in the Aberdeen area. The company’s product line includes 12-ounce cans of Coke products. The cans are filled by an automated filling process that can be adjusted to any mean fill volume and that will fill cans according to a normal distribution. However not all cans will contain the same volume due to variation in the filling process. Historical records show that regardless of what the mean is set at the standard deviation in fill will be 0.035 ounces. Operations managers at the plant know that if they put too much Coke in a can the company loses money. If too little is put in the can customers are short changed and the North Carolina Department of Weights and Measures may fine the company. a. Suppose the industry standards for fill volume call for each 12-ounce can to contain between 11.98 and 12.02 ounces. Assuming that the Aberdeen manager sets the mean fill at 12 ounces what is the probability that a can will contain a volume of Coke product that falls in the desired range b. Assume that the Aberdeen manager is focused on an upcoming audit by the North Carolina Department of Weights and Measures. She knows the process is to select one Coke can at random and that if it contains less than 11.97 ounces the company will be reprimanded and potentially fined. Assuming that the manager wants at most a 5 chance of this happening at what level should she set the mean fill level Comment on the ramifications of this step assuming that the company fills tens of thousands of cans each week. 6-25. MP-3 players and most notably the Apple iPod have become an industry standard for people who want to have access to their favorite music and videos in a portable format. The iPod can store massive numbers of songs and videos with its 120-GB hard drive. Although owners of the iPod have the potential to store lots of data a recent study showed that the actual disk storage being used is normally distributed with a mean equal to 1.95 GB and a standard deviation of 0.48 GB. Suppose a competitor to Apple is thinking of entering the market with a low-cost iPod clone that has only 1.0 GB of storage. The marketing slogan will be “Why Pay for Storage Capacity that You Don’t Need” Based on the data from the study of iPod owners what percentage of owners based on their current www.downloadslide.com

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Introduction to Continuous Probability Distributions randomly chosen transaction would yield a price of 2.12 or smaller even though the population mean was 2.51. 6-31. USA Today’s annual survey of public flagship universities Arienne Thompson and Breanne Gilpatrick “Double-Digit Hikes Are Down” October 5 2005 indicates that the median increase in in-state tuition was 7 for the 2005–2006 academic year. A file titled Tuition contains the percentage change for the 67 flagship universities. a. Produce a relative frequency histogram for these data. Does it seem plausible that the data are from a population that has a normal distribution b. Suppose the decimal point of the three largest numbers had inadvertently been moved one place to the right in the data. Move the decimal point one place to the left and reconstruct the relative frequency histogram. Now does it seem plausible that the data have an approximate normal distribution c. Use the normal distribution of part b to approximate the proportion of universities that raised their in-state tuition more than 10. Use the appropriate parameters obtained from this population. d. Use the normal distribution of part b to approximate the fifth percentile for the percent of tuition increase. was 3270. Sample data similar to those used in the study are in a file titled Hired. a. Produce a relative frequency histogram for these data. Does it seem plausible the data were sampled from a normally distributed population b. Calculate the mean and standard deviation of the cost of filling a job vacancy. c. Determine the probability that the cost of filling a job vacancy would be between 2000 and 3000. d. Given that the cost of filling a job vacancy was between 2000 and 3000 determine the probability that the cost would be more than 2500. 6-30. A recent article in USA Today discussed prices for the 200 brand-name drugs most commonly used by Americans over age 50. Atrovent a treatment for lung conditions such as emphysema was one of the drugs. The file titled Drug contains daily cost data similar to those obtained in the research. a. Produce a relative frequency histogram for these data. Does it seem plausible the data were sampled from a population that was normally distributed b. Compute the mean and standard deviation for the sample data in the file Drug. c. Assuming the sample came from a normally distributed population and the sample standard deviation is a good approximation for the population standard deviation determine the probability that a END EXERCISES 6-1 2 Other Continuous Probability Distributions This section introduces two additional continuous probability distributions that are used in business decision making: the uniform distribution and the exponential distribution. Uniform Probability Distribution The uniform distribution is sometimes referred to as the distribution of little information because the probability over any interval of the continuous random variable is the same as for any other interval of the same width. Equation 3 defines the continuous uniform density function. Chapter Outcome 4. Continuous Uniform Density Function fx ba axb 1 0 if otherwise 3 where: f1x2 Value of the density function at any x-value a The smallest value assumed by the uniform random variable of interest b The largest value assumed by the uniform random variable of interest www.downloadslide.com

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Introduction to Continuous Probability Distributions Figure 12 illustrates two examples of uniform probability distributions with different a to b intervals. Note the height of the probability density function is the same for all values of x between a and b for a given distribution. The graph of the uniform distribution is a rec- tangle. EXAMPLE 3 USING THE UNIFORM DISTRIBUTION Georgia Pacific The Georgia Pacific Company owns and operates several tree farms in dif- ferent parts of the United States and South America. The lead botanist for the company has stated that pine trees on one parcel of land will increase in diameter between one and four inches per year according to a uniform distribution. Suppose the company is interested in the probability that a given tree will have an increased diameter of more than 2 inches. The prob- ability can be determined using the following steps: Step 1 Define the density function. The height of the probability rectangle fx for the tree growth interval of one to four inches is determined using Equation 3 as follows: fx ba fx . 1 1 41 1 3 033 Step 2 Define the event of interest. The botanist is specifically interested in a tree that has increased by more than two inches in diameter. This event of interest is x 7 2.0. Step 3 Calculate the required probability. We determine the probability as follows: P1x 7 2.02 1 - P1x … 2.02 1 - f1x212.0 - 1.02 1 - 0.331 1.02 1 - 0.33 0.67 Thus there is a 0.67 probability that a tree will increase by more than two inches in diameter. END EXAMPLE TRY PROBLEM 32 Like the normal distribution the uniform distribution can be further described by specifying the mean and the standard deviation. These values are computed using Equations 4 and 5. FIGURE 12 |  Uniform Distributions fx 0.50 0.25 2 a 5 b fx 0.50 0.25 3 a 8 b fx xx 1 5 – 2 0.33 for 2 x 5 1 3 fx 1 8 – 3 0.2 for 3 x 8 1 5 a b www.downloadslide.com

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Introduction to Continuous Probability Distributions EXAMPLE 4 THE MEAN AND STANDARD DEVIATION OF A UNIFORM DISTRIBUTION Surgery Recovery The chief administrator of a San Francisco–area surgical center has analyzed data from a large number of shoulder surgeries conducted by her center and others in a medical association in California. The analysis shows that the recovery time for shoulder surgery ranges between 15 and 45 weeks. Without any further information the administrator will apply a uniform distribution to surgery times. Based on this she can determine the mean and standard deviation for the recovery duration using the following steps: Step 1 Define the density function. Equation 3 can be used to define the distribution: fx ba . 11 45 15 1 30 0 0333 Step 2 Compute the mean of the probability distribution using Equation 4. ab 2 15 45 2 30 Thus the mean recovery time is 30 weeks. Step 3 Compute the standard deviation using Equation 5. . ba 22 12 45 15 12 75 8 66 The standard deviation is 8.66 weeks. END EXAMPLE TRY PROBLEM 34 The Exponential Probability Distribution Another continuous probability distribution frequently used in business situations is the exponential distribution. The exponential distribution is used to measure the time that elapses between two occurrences of an event such as the time between “hits” on an Internet home page. The exponential distribution might also be used to describe the time between arrivals of customers at a bank drive-in teller window or the time between failures of an electronic component. Equation 6 shows the probability density function for the exponen- tial distribution. Mean and Standard Deviation of a Uniform Distribution Mean Expected Value: Ex ab 2 4 Standard Deviation: ba 2 12 5 where: a The smallest value assumed by the uniform random variable of interest b The largest value assumed by the uniform random variable of interest Chapter Outcome 5. araraadt/Fotolia www.downloadslide.com

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Introduction to Continuous Probability Distributions Note the parameter that defines the exponential distribution is l lambda. You should recall that l is the mean value for the Poisson distribution. If the number of occurrences per time period is known to be Poisson distributed with a mean of l then the time between occur- rences will be exponentially distributed with a mean time of 1l. If we select a value for l we can graph the exponential distribution by substituting l and different values for x into Equation 6. For instance Figure 13 shows exponential density functions for l 0.5 l 1.0 l 2.0 and l 3.0. Note in Figure 13 that for any expo- nential density function f1x2 f102 l as x increases fx approaches zero. It can also be shown that the standard deviation of any exponential distribution is equal to the mean 1l. As with any continuous probability distribution the probability that a value will fall within an interval is the area under the graph between the two points defining the interval. Equation 7 is used to find the probability that a value will be equal to or less than a particular value for an exponential distribution. Exponential Density Function A continuous random variable that is exponentially distributed has the probability density function given by fx le -lx x Ú 0 6 where: e 2.71828c 1l The mean time between events1l 7 02 fx Probability Density Function 3.0 2.5 2.0 1.5 1.0 0.5 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Values of x l 3.0 Mean 0.3333 l 2.0 Mean 0.50 l 1.0 Mean 1.0 l 0.50 Mean 2.0 FIGURE 13 |  Exponential Distributions Exponential Probability P10 … x … a2 1 - e -la 7 where: a the value of interest 1l Mean e Base of natural log 2.71828 www.downloadslide.com

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Introduction to Continuous Probability Distributions Exponential Distribution Table contains a table of e -la values for different values of l a. You can use this table and Equation 7 to find the probabilities when the l a of interest is con- tained in the table. You can also use Excel to find exponential probabilities as the following application illustrates. BUSINESS APPLICATION USING EXCEL TO CALCULATE EXPONENTIAL PROBABILITIES HAINES INTERNET SERVICES The Haines Internet Services Company has determined that the number of customers who attempt to connect to the Internet per hour is Poisson distributed with l 30 per hour. The time between connect requests is exponentially distributed with a mean time between requests of 2.0 minutes computed as follows: l 30 attempts per 60 minutes 0.50 attempts per minute The mean time between attempted connects then is 1 1 050 20 / . . minutes Because of the system that Haines uses if customer requests are too close together—45 seconds 0.75 minutes or less—the connection will fail. The managers at Haines are analyzing whether they should purchase new equipment that will eliminate this problem. They need to know the probability that a customer will fail to connect. Thus they want P1x … 0.75 minutes2 Excel tutorials Excel Tutorial FIGURE 14 |  Excel 2010 Exponential Probability Output for Haines Internet Services Minitab Instructions for similar results: 1. Choose Calc Probability Distributions Exponential. 2. Choose Cumulative probability. 3. In Scale enter m . 4. In Input constant enter value for x. 5. Click OK. Excel 2010 Instructions: 1. On the Formula tab click on f x function wizard. 2. Select Statistical category. 3. Select EXPON.DIST function. 4. Supply x and l. 5. Set Cumulative TRUE for cumulative probability. Inputs: x 0.75 minutes 45 seconds l 0.50 per minute True output is the cumulative probability www.downloadslide.com

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Introduction to Continuous Probability Distributions To find this probability using a calculator we need to first determine la. In this example l 0.50 and a 0.75. Then l a 10.50210.752 0.3750 We find that the desired probability is 1 - e -la 1 - e - 0.3750 0.3127 The managers can also use the EXPON.DIST function in Excel to compute the precise value for the desired probability. 4 Using Excel Figure 14 shows that the chance of failing to connect is 0.3127. This means that nearly one third of the customers will experience a problem with the current system. 4 The Excel EXPON.DIST function requires that l be inputted rather than 1l. Skill Development 6-32. A continuous random variable is uniformly distributed between 100 and 150. a. What is the probability a randomly selected value will be greater than 135 b. What is the probability a randomly selected value will be less than 115 c. What is the probability a randomly selected value will be between 115 and 135 6-33. Determine the following: a. the probability that a uniform random variable whose range is between 10 and 30 assumes a value in the interval 10 to 20 or 15 to 25 b. the quartiles for a uniform random variable whose range is from 4 to 20 c. the mean time between events for an exponential random variable that has a median equal to 10 d. the 90th percentile for an exponential random variable that has the mean time between events equal to 0.4. 6-34. Suppose a random variable x has a uniform distribution with a 5 and b 9. a. Calculate P15.5 … x … 82. b. Determine P1x 7 72. c. Compute the mean m and standard deviation s of this random variable. d. Determine the probability that x is in the interval 1m 2s2. 6-35. Let x be an exponential random variable with l 0.5. Calculate the following probabilities: a. P1x 6 52 b. P1x 7 62 c. P15 … x … 62 d. P1x Ú 22 e. the probability that x is at most 6 6-36. The useful life of an electrical component is exponentially distributed with a mean of 2500 hours. MyStatLab 6-2: Exercises a. What is the probability the circuit will last more than 3000 hours b. What is the probability the circuit will last between 2500 and 2750 hours c. What is the probability the circuit will fail within the first 2000 hours 6-37. The time between telephone calls to a cable television payment processing center follows an exponential distribution with a mean of 1.5 minutes. a. What is the probability that the time between the next two calls will be 45 seconds or less b. What is the probability that the time between the next two calls will be greater than 112.5 seconds Business Applications 6-38. Suppose you are traveling on business to a foreign country for the first time. You do not have a bus schedule or a watch with you. However you have been told that buses stop in front of your hotel every 20 minutes throughout the day. If you show up at the bus stop at a random moment during the day determine the probability that a. you will have to wait for more than 10 minutes b. you will only have to wait for 6 minutes or less c. you will have to wait between 8 and 15 minutes 6-39. When only the value-added time is considered the time it takes to build a laser printer is thought to be uniformly distributed between 8 and 15 hours. a. What are the chances that it will take more than 10 value-added hours to build a printer b. How likely is it that a printer will require less than 9 value-added hours c. Suppose a single customer orders two printers. Determine the probability that the first and second printer each will require less than 9 value-added hours to complete. 6-40. The time required to prepare a dry cappuccino using whole milk at the Daily Grind Coffee House is www.downloadslide.com

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Introduction to Continuous Probability Distributions a. What is the probability that a customer will arrive within the next 3 minutes b. What is the probability that the time between the arrivals of customers is 12 minutes or more c. What is the probability that the next customer will arrive within 4 and 6 minutes 6-45. The average amount spent on electronics each year in U.S. households is 1250 according to an article in USA Today Michelle Kessler “Gadget Makers Make Mad Dash to Market” January 4 2006. Assume that the amount spent on electronics each year has an exponential distribution. a. Calculate the probability that a randomly chosen U.S. household would spend more than 5000 on electronics. b. Compute the probability that a randomly chosen U.S. household would spend more than the average amount spent by U.S. households. c. Determine the probability that a randomly chosen U.S. household would spend more than 1 standard deviation below the average amount spent by U.S. households. 6-46. Charter Southeast Airlines states that the flight between Fort Lauderdale Florida and Los Angeles takes 5 hours and 37 minutes. Assume that the actual flight times are uniformly distributed between 5 hours and 20 minutes and 5 hours and 50 minutes. a. Determine the probability that the flight will be more than 10 minutes late. b. Calculate the probability that the flight will be more than 5 minutes early. c. Compute the average flight time between these two cities. d. Determine the variance in the flight times between these two cities. 6-47. A corrugated container company is testing whether a computer decision model will improve the uptime of its box production line. Currently knives used in the production process are checked manually and replaced when the operator believes the knives are dull. Knives are expensive so operators are encouraged not to change the knives early. Unfortunately if knives are left running for too long the cuts are not made properly which can jam the machines and require that the entire process be shut down for unscheduled maintenance. Shutting down the entire line is costly in terms of lost production and repair work so the company would like to reduce the number of shutdowns that occur daily. Currently the company experiences an average of 0.75 knife-related shutdowns per shift exponentially distributed. In testing the computer decision model reduced the frequency of knife-related shutdowns to an average of 0.20 per shift exponentially distributed. The decision model is expensive but the company will install it if it can help achieve the target of four consecutive shifts without a knife-related shutdown. uniformly distributed between 25 and 35 seconds. Assuming a customer has just ordered a whole-milk dry cappuccino a. What is the probability that the preparation time will be more than 29 seconds b. What is the probability that the preparation time will be between 28 and 33 seconds c. What percentage of whole-milk dry cappuccinos will be prepared within 31 seconds d. What is the standard deviation of preparation times for a dry cappuccino using whole milk at the Daily Grind Coffee House 6-41. The time to failure for a power supply unit used in a particular brand of personal computer PC is thought to be exponentially distributed with a mean of 4000 hours as per the contract between the vendor and the PC maker. The PC manufacturer has just had a warranty return from a customer who had the power supply fail after 2100 hours of use. a. What is the probability that the power supply would fail at 2100 hours or less Based on this probability do you feel the PC maker has a right to require that the power supply maker refund the money on this unit b. Assuming that the PC maker has sold 100000 computers with this power supply approximately how many should be returned due to failure at 2100 hours or less 6-42. A delicatessen located in the heart of the business district of a large city serves a variety of customers. The delicatessen is open 24 hours a day every day of the week. In an effort to speed up take-out orders the deli accepts orders by fax. If on the average 20 orders are received by fax every two hours throughout the day find the a. probability that a faxed order will arrive within the next 9 minutes b. probability that the time between two faxed orders will be between 3 and 6 minutes c. probability that 12 or more minutes will elapse between faxed orders 6-43. Dennis Cauchon and Julie Appleby reported in USA Today that the average patient cost per stay in American hospitals was 8166. Assume that this cost is exponentially distributed. a. Determine the probability that a randomly selected patient’s stay in an American hospital will cost more than 10000. b. Calculate the probability that a randomly selected patient’s stay in an American hospital will cost less than 5000. c. Compute the probability that a randomly selected patient’s stay in an American hospital will cost between 8000 and 12000. 6-44. During the busiest time of the day customers arrive at the Daily Grind Coffee House at an average of 15 customers per 20-minute period. www.downloadslide.com

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Introduction to Continuous Probability Distributions e. Using only the information obtained in parts c and d describe the shape of this distribution. Does this agree with the findings in part a 6-49. Although some financial institutions do not charge fees for using ATMs many do. A recent study found the average fee charged by banks to process an ATM transaction was 2.91. The file titled ATM Fees contains a list of ATM fees that might be required by banks. a. Produce a relative frequency histogram for these data. Does it seem plausible the data came from a population that has an exponential distribution b. Calculate the mean and standard deviation of the ATM fees. c. Assume that the distribution of ATM fees is exponentially distributed with the same mean as that of the sample. Determine the probability that a randomly chosen bank’s ATM fee would be greater than 3.00. 6-50. The San Luis Obispo California Transit Program provides daily fixed-route transit service to the general public within the city limits and to Cal Poly State University’s staff and students. The most heavily traveled route schedules a city bus to arrive at Cal Poly at 8:54 a.m. The file titled Late lists plausible differences between the actual and scheduled time of arrival rounded to the nearest minute for this route. a. Produce a relative frequency histogram for these data. Does it seem plausible the data came from a population that has a uniform distribution b. Provide the density for this uniform distribution. c. Classes start 10 minutes after the hour and classes are a 5-minute walk from the drop-off point. Determine the probability that a randomly chosen bus on this route would cause the students on board to be late for class. Assume the differences form a continuous uniform distribution with a range the same as the sample. d. Determine the median difference between the actual and scheduled arrival times. a. Under the current system what is the probability that the plant would run four or more consecutive shifts without a knife-related shutdown b. Using the computer decision model what is the probability that the plant could run four or more consecutive shifts without a knife-related shutdown Has the decision model helped the company achieve its goal c. What would be the maximum average number of shutdowns allowed per day such that the probability of experiencing four or more consecutive shifts without a knife-related shutdown is greater than or equal to 0.70 Computer Database Exercises 6-48. Rolls-Royce PLC provides forecasts for the business jet market and covers the regional and major aircraft markets. In a recent release Rolls Royce indicated that in both North America and Europe the number of delayed departures has declined since a peak in 1999/2000. This is partly due to a reduction in the number of flights at major airports and the younger aircraft fleets but it also results from improvements in air traffic management capacity especially in Europe. Comparing January–April 2003 with the same period in 2001 for similar traffic levels the average en route delay per flight was reduced by 65 from 2.2 minutes to 0.7 minutes. The file titled Delays contains a possible sample of the en route delay times in minutes for 200 flights. a. Produce a relative frequency histogram for this data. Does it seem plausible the data come from a population that has an exponential distribution b. Calculate the mean and standard deviation of the en route delays. c. Determine the probability that this exponential random variable will be smaller than its mean. d. Determine the median time in minutes for the en route delays assuming they have an exponential distribution with a mean equal to that obtained in part b. END EXERCISES 6-2 www.downloadslide.com

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Introduction to Continuous Probability Distributions Visual Summary 1 The Normal Probability Distribution 2 Other Continuous Probability Distributions Outcome 4. Calculate probabilities associated with a uniformly distributed random variable. Outcome 5. Determine probabilities for an exponential probability distribution. Conclusion The normal distribution has wide application throughout the study of business statistics. You will be making use of the normal distribution. The normal distribution has very special properties. It is a symmetric bell-shaped distribution. To fnd probabilities for a normal distribution you will frst standardize the distribution by converting values of the random variable to standardized z-values. Other continuous distributions introduced in this chapter are the exponential distribution and the uniform distribution. Figure 15 summarizes the discrete probability distributions and the continuous probability distributions introduced in this chapter. A random variable can take on values that are either discrete or continuous. This chapter has focused on continuous random variables where the potential values of the variable can be any value on a continuum. Examples of continuous random variables include the time it takes a worker to assemble a part the weight of a potato the distance it takes to stop a car once the brakes have been applied and the volume of waste water emitted from a food processing facility. Values of a continuous random variable are generally determined by measuring. One of the most frequently used continuous probability distributions is called the normal distribution. Summary The normal distribution is a symmetric bell-shaped probability distribution. Half the probability lies to the right and half lies to the left of the mean. To fnd probabilities associated with a normal distribution you will want to convert to a standard normal distribution by frst converting values of the random variables to standardizedz-values. The probabilities associated with a range of values for the random variable are found using Excel. Outcome 1. Convert a normal distribution to a standard normal distribution. Outcome 2. Determine probabilities using the standard normal distribution. Outcome 3. Calculate values of the random variable associated with specifed probabilities from a normal distribution. Summary Although the normal distribution is by far the most frequently used continuous probability distribution two other continuous distributions are introduced in this section. These are the uniform distribution and the exponential distribution. With the uniform distribution the probability over any interval is the same as any other interval of the same width. The probabilities for the uniform distribution are computed using Equation 3. The exponential distribution is based on a single parameter lambda and is often used to describe random service times or the time between customer arrivals in waiting line applications. The probability over a range of values for an exponential distribution can be computed using Equation 7. Also Excel and Minitab have functions for calculating the exponential probabilities. www.downloadslide.com

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Introduction to Continuous Probability Distributions 4 Mean of the Uniform Distribution Ex ab 2 5 Standard Deviation of the Uniform Distribution ba 2 12 6 Exponential Density Function f1x2 l e -lx x Ú 0 7 Exponential Probability P10 … x … a2 1 - e -la 1 Normal Probability Density Function fx e x 1 2 2 2 2 2 Standardized Normalz-Value z x 3 Continuous Uniform Density Function fx ba axb 1 0 if otherwise FIGURE 15 | Probability Distribution Summary Exponential Distribution Hypergeometric Distribution Random Variable Values Are Determined by Counting Random Variable Values Are Determined by Measuring Random Variable Binomial Distribution Discrete Continuous Normal Distribution Uniform Distribution Poisson Distribution Equations Key Terms Normal distribution Standard normal distribution Chapter Exercises MyStatLab Conceptual Questions 6-51. Discuss the difference between discrete and continuous probability distributions. Discuss two situations in which a variable of interest may be considered either continuous or discrete. 6-52. Recall the Empirical Rule. It states that if the data distribution is bell shaped then the interval m s contains approximately 68 of the values m 2s contains approximately 95 and m 3s contains virtually all of the data values. The bell-shaped distribution referenced is the normal distribution. a. Verify that a standard normal distribution contains approximately 68 of the values in the interval m s. b. Verify that a standard normal distribution contains approximately 95 of the values in the interval m 2s. c. Verify that a standard normal distribution contains virtually all of the data in the interval m 3s. www.downloadslide.com

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Introduction to Continuous Probability Distributions Business Applications 6-59. The manager for Select-a-Seat a company that sells tickets to athletic games concerts and other events has determined that the number of people arriving at the Broadway location on a typical day is Poisson distributed with a mean of 12 per hour. It takes approximately four minutes to process a ticket request. Thus if customers arrive in intervals that are less than four minutes they will have to wait. Assuming that a customer has just arrived and the ticket agent is starting to serve that customer what is the probability that the next customer who arrives will have to wait in line 6-60. The Four Brothers Lumber Company is considering buying a machine that planes lumber to the correct thickness. The machine is advertised to produce “6-inch lumber” having a thickness that is normally distributed with a mean of 6 inches and a standard deviation of 0.1 inch. a. If building standards in the industry require a 99 chance of a board being between 5.85 and 6.15 inches should Four Brothers purchase this machine Why or why not b. To what level would the company that manufactures the machine have to reduce the standard deviation for the machine to conform to industry standards 6-61. Two automatic dispensing machines are being considered for use in a fast-food chain. The first dispenses an amount of liquid that has a normal distribution with a mean of 11.9 ounces and a standard deviation of 0.07 ounces. The second dispenses an amount of liquid that has a normal distribution with a mean of 12.0 ounces and a standard deviation of 0.05 ounces. Acceptable amounts of dispensed liquid are between 11.9 and 12.0 ounces. Calculate the relevant probabilities and determine which machine should be selected. 6-62. A small private ambulance service in Kentucky has determined that the time between emergency calls is exponentially distributed with a mean of 41 minutes. When a unit goes on call it is out of service for 60 minutes. If a unit is busy when an emergency call is received the call is immediately routed to another service. The company is considering buying a second ambulance. However before doing so the owners are interested in determining the probability that a call will come in before the ambulance is back in service. Without knowing the costs involved in this situation does this probability tend to support the need for a second ambulance Discuss. 6-63. Assume that after the first 12 hours the average remaining useful life of a particular battery before recharging is required is 9 hours and that the remaining time is exponentially distributed. What is the probability that a randomly sampled battery of this type will last between 15 and 17 hours 6-53. The probability that a value from a normally distributed random variable will exceed the mean is 0.50. The same is true for the uniform distribution. Why is this not necessarily true for the exponential distribution Discuss and show examples to illustrate your point. 6-54. Suppose you tell one of your fellow students that when working with a continuous distribution it does not make sense to try to compute the probability of any specific value since it will be zero. She says that when the experiment is performed some value must occur the probability can’t be zero. Y our task is to respond to her statement and in doing so explain why it is appropriate to find the probability for specific ranges of values for a continuous distribution. 6-55. The exponential distribution has a characteristic that is called the “memoryless” property. This means P1X 7 x2 P1X 7 x + x 0 X 7 x 0 2. To illustrate this consider the calls coming into 911. Suppose that the distribution of the time between occurrences has an exponential distribution with a mean of one half hour 1 0.52. a. Calculate the probability that no calls come in during the first hour. b. Now suppose that you are monitoring the call frequency and you note that a call does not come in during the first two hours. Determine the probability that no calls will come in during the next hour. 6-56. Revisit Problem 55 but examine whether it would matter when you started monitoring the 911 calls if the time between occurrences had a uniform distribution with a mean of 2 and a range of 4. a. Calculate the probability that no call comes in during the first hour. b. Now suppose that you are monitoring the call frequency and you note that no call comes in during the first two hours. Determine the probability that no calls will arrive during the next hour. 6-57. Suppose that on average 20 customers arrive every hour at a twenty-four-hour coffee shop. Assume that the time between customer arrivals is exponentially distributed. Determine a. The probability that a customer arrives within the next 2 minutes. b. The probability that the time between two arriving customers will be between 1 and 4 minutes. c. The probability that 5 or more minutes will pass between customer arrivals. 6-58. Assume that the time required to receive a confirmation that an electronic transfer has occurred is uniformly distributed between 30 and 90 seconds. a. What is the probability that a randomly selected transfer will take between 30 and 45 seconds b. What is the probability that a randomly selected transfer will take between 50 and 90 seconds c. What proportion of transfers will take between 40 and 75 seconds www.downloadslide.com

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Introduction to Continuous Probability Distributions b. How likely is it that no traffic violations will be recorded within the next 7 minutes 6-68. The St. Maries plywood plant is part of the Potlatch Corporation’s Northwest Division. The plywood superintendent organized a study of the tree diameters that are being shipped to the mill. After collecting a large amount of data on diameters he concluded that the distribution is approximately normally distributed with a mean of 14.25 inches and a standard deviation of 2.92 inches. Because of the way plywood is made there is a certain amount of waste on each log because the peeling process leaves a core that is approximately 3 inches thick. For this reason he feels that any log less than 10 inches in diameter is not profitable for making plywood. a. Based on the data the superintendent has collected what is the probability that a log will be unprofitable b. An alternative is to peel the log and then sell the core as “peeler logs.” These peeler logs are sold as fence posts and for various landscape projects. There is not as much profit in these peeler logs however. The superintendent has determined that he can make a profit if the peeler log’s diameter is not more than 32 of the diameter of the log. Using this additional information calculate the proportion of logs that will be unprofitable. 6-69. The personnel manager for a large company is interested in the distribution of sick-leave hours for employees of her company. A recent study revealed the distribution to be approximately normal with a mean of 58 hours per year and a standard deviation of 14 hours. An office manager in one division has reason to believe that during the past year two of his employees have taken excessive sick leave relative to everyone else. The first employee used 74 hours of sick leave and the second used 90 hours. What would you conclude about the office manager’s claim and why 6-70. If the number of hours between servicing required for a particular snowmobile engine is exponentially distributed with an average of 118 hours determine the probability that a randomly selected engine a. Will run at least 145 hours before servicing is needed. b. Will run at most 161 hours before servicing is needed. 6-71. Assume that the amount of time in minutes for eighth graders to compete an assessment examination is 78 minutes with a standard deviation of 12 minutes. a. What proportion of eighth graders completes the assessment examination in 72 minutes or less b. What proportion of eighth graders completes the assessment examination in 82 minutes or more c. For what number of minutes would 90 of all eighth graders complete the assessment examination 6-64. An online article http://beauty.about.com by Julyne Derrick “Shelf Lives: How Long Can Y ou Keep Makeup” suggests that eye shadow and eyeliner each have a shelf life of up to three years. Suppose the shelf lives of these two products are exponentially distributed with an average shelf life of one year. a. Calculate the probability that the shelf life of eye shadow will be longer than three years. b. Determine the probability that at least one of these products will have a shelf life of more than three years. c. Determine the probability that a purchased eyeliner that is useful after one year will be useful after three years. 6-65. The Shadow Mountain Golf Course is preparing for a major LPGA golf tournament. Since parking near the course is extremely limited room for only 500 cars the course officials have contracted with the local community to provide parking and a bus shuttle service. Sunday the final day of the tournament will have the largest crowd and the officials estimate there will be between 8000 and 12000 cars needing parking spaces but think no value is more likely than another. The tournament committee is discussing how many parking spots to contract from the city. If they want to limit the chance of not having enough provided parking to 10 how many spaces do they need from the city on Sunday 6-66. One of the products of Pittsburg Plate Glass Industries PPG is laminated safety glass. It is made up of two pieces of glass 0.125 inch thick with a thin layer of vinyl sandwiched between them. The average thickness of the laminated safety glass is 0.25 inch. The thickness of the glass does not vary from the mean by more than 0.10 inch. Assume the thickness of the glass has a uniform distribution. a. Provide the density for this uniform distribution. b. If the glass has a thickness that is more than 0.05 inch below the mean it must be discarded for safety considerations. Determine the probability that a randomly selected automobile glass is discarded due to safety considerations. c. If the glass is more than 0.075 above the mean it will create installation problems and must be discarded. Calculate the probability that a randomly selected automobile glass will be rejected due to installation concerns. d. Given that a randomly selected automobile glass is not rejected for safety considerations determine the probability that it will be rejected for installation concerns. 6-67. A traffic control camera at a busy intersection records on average 5 traffic violations per hour. Assume that the random variable number of recorded traffic violations follow a Poisson distribution. a. What is the probability that the next recorded violation will occur within 5 minutes www.downloadslide.com

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Introduction to Continuous Probability Distributions 1 users can expect to lose as much as 12 pounds in four weeks Discuss. 6-74. Midwest Fan Manufacturing Inc. was established in 1986 as a manufacturer and distributor of quality ventilation equipment. Midwest Fan’s products include the AXC range hood exhaust fans. The file titled Fan Life contains the length of life of 125 randomly chosen AXC fans that were used in an accelerated life-testing experiment. a. Produce a relative frequency histogram for the data. Does it seem plausible the data came from a population that has an exponential distribution b. Calculate the mean and standard deviation of the fans’ length of life. c. Calculate the median length of life of the fans. d. Determine the probability that a randomly chosen fan will have a life of more than 25000 hours. 6-75. Team Marketing Report TMR produces the Fan Cost Index™ FCI survey now in its 16th year which tracks the cost of attendance for a family of four at National Football League NFL games. The FCI includes four average-price tickets four small soft drinks two small beers four hot dogs two game programs parking and two adult-size caps. The league’s average FCI in 2008 was 396.36. The file titled NFL Price is a sample of 175 randomly chosen fans’ FCIs. a. Produce a relative frequency histogram for these data. Does it seem plausible the data were sampled from a population that was normally distributed b. Calculate the mean and standard deviation of the league’s FCI. c. Calculate the 90th percentile of the league’s fans’ FCI. d. The San Francisco 49ers had an FCI of 376.71. Determine the percentile of the FCI of a randomly chosen family whose FCI is the same as that of the 49ers’ average FCI. 6-76. The Future-Vision Cable TV Company recently surveyed its customers. A total of 548 responses were received. Among other things the respondents were asked to indicate their household income. The data from the survey are found in a file named Future-Vision. a. Develop a frequency histogram for the income variable. Does it appear from the graph that income is approximately normally distributed Discuss. b. Compute the mean and standard deviation for the income variable. c. Referring to parts a and b and assuming that income is normally distributed and the sample mean and standard deviation are good substitutes for the population values what is the probability that a Future-Vision customer will have an income exceeding 40000 d. Suppose that Future-Vision managers are thinking about offering a monthly discount to customers who have a household income below a certain level. Computer Database Exercises 6-72. The Cozine Corporation runs the landfill operation outside Little Rock Arkansas. Each day each of the company’s trucks makes several trips from the city to the landfill. On each entry the truck is weighed. The data file Cozine contains a sample of 200 truck weights. Determine the mean and standard deviation for the garbage truck weights. Assuming that these sample values are representative of the population of all Cozine garbage trucks and assuming that the distribution is normally distributed a. Determine the probability that a truck will arrive at the landfill weighing in excess of 46000 pounds. b. Compare the probability in part a to the proportion of trucks in the sample that weighed more than 46000 pounds. What does this imply to you c. Suppose the managers are concerned that trucks are returning to the landfill before they are fully loaded. If they have set a minimum weight of 38000 pounds before the truck returns to the landfill what is the probability that a truck will fail to meet the minimum standard 6-73. The Hydronics Company is in the business of developing health supplements. Recently the company’s research and development department came up with two weight-loss products that included products produced by Hydronics. To determine whether these products are effective the company has conducted a test. A total of 300 people who were 30 pounds or more overweight were recruited to participate in the study. Of these 100 people were given a placebo supplement 100 people were given product 1 and 100 people were given product 2. As might be expected some people dropped out of the study before the four-week study period was completed. The weight loss or gain for each individual is listed in the data file called Hydronics. Note positive values indicate that the individual actually gained weight during the study period. a. Develop a frequency histogram for the weight loss or gain for those people on product 1. Does it appear from this graph that weight loss is approximately normally distributed b. Referring to part a assuming that a normal distribution does apply compute the mean and standard deviation weight loss for the product 1 subjects. c. Referring to parts a and b assume that the weight- change distribution for product 1 users is normally distributed and that the sample mean and standard deviation are used to directly represent the population mean and standard deviation. What is the probability that a plan 1 user will lose over 12 pounds in a four-week period d. Referring to your answer in part c would it be appropriate for the company to claim that plan www.downloadslide.com

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Introduction to Continuous Probability Distributions a. Produce a relative frequency histogram for these data. Does it seem plausible the data came from a population that has a uniform distribution b. Provide the density f x for this uniform distribution. c. A billiard retailer Sticks Stones Billiard Supply is going to recover the pool tables in the local college pool hall which has eight tables. It takes approximately 3.8 yards per table. If Championship ships a randomly chosen half-bolt determine the probability that it will contain enough cloth to recover the eight tables. If the management wants to grant discounts to no more than 7 of the customers what income level should be used for the cutoff 6-77. Championship Billiards owned by D R Industries in Lincolnwood Illinois provides some of the finest billiard fabrics cushion rubber and component parts in the industry. It sells billiard cloth in bolts and half-bolts. A half-bolt of billiard cloth has an average length of 35 yards with widths of either 62 or 66 inches. The file titled Half Bolts contains the lengths of 120 randomly selected half-bolts. Case 1 State Entitlement Programs Franklin Joiner director of health education and welfare had just left a meeting with the state’s newly elected governor and several of the other recently appointed department heads. One of the gov- ernor’s campaign promises was to try to halt the rising cost of a certain state entitlement program. In several speeches the gover- nor indicated the state of Idaho should allocate funds only to those individuals ranked in the bottom 10 of the state’s income dis- tribution. Now the governor wants to know how much one could earn before being disqualified from the program and he also wants to know the range of incomes for the middle 95 of the state’s income distribution. Frank had mentioned in the meeting that he thought incomes in the state could be approximated by a normal distribution and that mean per capita income was about 33000 with a standard deviation of nearly 9000. The governor was expecting a memo in his office by 3:00 p.m. that afternoon with answers to his questions. Required Tasks: 1. Assuming that incomes can be approximated using a normal distribution with the specified mean and standard deviation calculate the income that cut off the bottom 10 of incomes. 2. Assuming that incomes can be approximated using a normal distribution with the specified mean and standard deviation calculate the middle 95 of incomes. Hint: This requires calculating two values. 3. Write a short memo describing your results and how they were obtained. Your memo should clearly state the income that would disqualify people from the program as well as the range of incomes in the middle 95 of the state’s income distribution. Case 2 Credit Data Inc. Credit Data Inc. has been monitoring the amount of time its bill collectors spend on calls that produce contacts with consumers. Management is interested in the distribution of time a collector spends on each call in which he or she initiates contact informs a consumer about an outstanding debt discusses a payment plan and receives payments by phone. Credit Data is mostly interested in how quickly a collector can initiate and end a conversation to move on to the next call. For employees of Credit Data time is money in the sense that one account may require one call and 2 minutes to collect whereas another account may take five calls and 10 minutes per call to collect. The company has discovered that the time collectors spend talking to consumers about accounts is approximated by a normal distribution with a mean of 8 min- utes and a standard deviation of 2.5 minutes. The managers believe that the mean is too high and should be reduced by more efficient phone call methods. Specifically they wish to have no more than 10 of all calls require more than 10.5 minutes. Required Tasks: 1. Assuming that training can affect the average time but not the standard deviation the managers are interested in knowing to what level the mean call time needs to be reduced in order to meet the 10 requirement. 2. Assuming that the standard deviation can be affected by training but the mean time will remain at 8 minutes to what level must the standard deviation be reduced in order to meet the 10 requirement 3. If nothing is done what percent of all calls can be expected to require more than 10.5 minutes www.downloadslide.com

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Introduction to Continuous Probability Distributions the equipment’s ability to detect oil. The enhancement requires 800 capacitors which must operate within ±0.50 microns from the specified standard of 12 microns. The problem is that the supplier can provide capacitors that operate according to a normal distribution with a mean of 12 microns and a standard deviation of 1 micron. Thus Chad knows that not all capacitors will meet the specifications required by the new piece of exploration equipment. This will mean that to have at least 800 usable capacitors American Oil will have to order more than 800 from the supplier. However these items are very expen- sive so he wants to order as few as possible to meet their needs. At the meeting the group agreed that they wanted a 98 chance that any order of capacitors would contain the sufficient number of usable items. If the project is to remain on schedule Chad must place the order by tomorrow. He wants the new equipment ready to go by the time he leaves for an exploration trip in Australia. As he reclined in his seat sipping a cool lemonade he wondered whether a basic statistical technique could be used to help determine how many capacitors to order. American Oil Company Chad Williams field geologist for the American Oil Company settled into his first-class seat on the Sun-Air flight between Los Angeles and Oakland California. Earlier that afternoon he had attended a meeting with the design engineering group at the Los Angeles New Product Division. He was now on his way to the home office in Oakland. He was looking forward to the one-hour flight because it would give him a chance to reflect on a prob- lem that surfaced during the meeting. It would also give him a chance to think about the exciting opportunities that lay ahead in Australia. Chad works with a small group of highly trained people at American Oil who literally walk the earth looking for new sources of oil. They make use of the latest in electronic equipment to take a wide range of measurements from many thousands of feet below the earth’s surface. It is one of these electronic machines that is the source of Chad’s current problem. Engineers in Los Angeles have designed a sophisticated enhancement that will greatly improve Case 3 www.downloadslide.com

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Introduction to Continuous Probability Distributions 33. a. 0.75 b. Q 1 4.250.0625 8 Q 2 4.500.0625 12 Q 3 4.750.0625 16 c. 14.43 d. 0.92 35. a. 0.9179 b. 0.0498 c. 0.0323 d. 0.9502 37. a. 0.3935 b. 0.2865 39. a. 0.7143 b. 0.1429 c. 0.0204 41. a. 0.4084 yes b. 40840 43. a. 0.2939 b. 0.4579 c. 0.1455 45. a. 0.0183 b. 0.3679 47. a. 0.0498 b. 0.4493 c. approximately l 0.08917 49. a. positively skewed b. Descriptive Statistics: ATM FEES Variable Mean StDev ATM FEES 2.907 2.555 c. 1 - 0.6433 0.3567 55. a. 0.1353 b. 0.1353 57. a. 0.486583 b. 0.452934 c. 0.18888 59. 0.5507 61. Machine 1: 0.4236 Machine 2: 0.4772 63. 0.142778 65. Need an additional 11500 - 400 11100 parking spaces. Px 6 60 0.7686 shou 67. a. 0.3406 b. 0.5580 71. a. 0.3085 b. 0.3707 c. 93.36 minutes. 73. a. approximately normally distributed b. Mean 2.453 Standard deviation 4.778 c. 0.0012 d. No 1. a. 225 - 200 20 25 20 1.25 b. 190 - 200 20 -10 20 -0.50 c. 240 - 200 20 40 20 2.00 3. a. 0.4901 b. 0.6826 c. 0.0279 5. a. 0.4750 b. 0.05 c. 0.0904 d. 0.97585 e. 0.8513 7. a. 0.9270 b. 0.6678 c. 0.9260 d. 0.8413 e. 0.3707 9. a. x 1.290.50 + 5.5 6.145 b. m 6.145 - 1.650.50 5.32 11. a. 0.0027 b. 0.2033 c. 0.1085 13. a. 0.0668 b. 0.228 c. 0.7745 15. a. 0.3446 b. 0.673 c. 51.30 d. 0.9732 17. a. 0.1762 b. 0.3446 c. 0.4401 d. 0.0548 19. The mean and standard deviation of the random variable are 15000 and 1250 respectively. a. 0.0548 b. 0.0228 c. m 15912 approximately 21. about 3367.35 23. a. 0.1949 b. 0.9544 c. Mean Median symmetric distribution 25. Px 6 1.0 0.5000 - 0.4761 0.0239 27. a. P0.747 … x … 0.753 0.6915 - 0.1587 0.5328 b. s 0.753 - 0.75 2.33 0.001 31. a. skewed right b. approximate normal distribution c. 0.1230 d. 2.034 Answers to Selected Odd-Numbered Problems This section contains summary answers to most of the odd-numbered problems in the text. The Student Solutions Manual contains fully developed solutions to all odd-numbered problems and shows clearly how each answer is determined. www.downloadslide.com

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Introduction to Continuous Probability Distributions 77. a. Uniform distribution. Sampling error could account for differences in this sample. b. fx 1 b - a 1 35 - 24.8 0.098 c. 0.451 75. a. The histogram seems to be “bell shaped.” b. Mean 396.36 Standard Deviation 112.41 c. The 90th percentile is 540.419. d. 376.71 is the 43rd percentile. Albright Christian S. Wayne L. Winston and Christopher Zappe Data Analysis for Managers with Microsoft Excel Pacific Grove CA: Duxbury 2003. DeVeaux Richard D. Paul F. Velleman and David E. Bock Stats Data and Models 3rd ed. New York: Addison-Wesley 2012. Hogg R. V. and Elliot A. Tanis Probability and Statistical Inference 8th ed. Upper Saddle River NJ: Prentice Hall 2010. Larsen Richard J. and Morris L. Marx An Introduction to Mathematical Statistics and Its Applications 5th ed. Upper Saddle River NJ: Prentice Hall 2012. Microsoft Excel 2010 Redmond WA: Microsoft Corp. 2010. Siegel Andrew F. Practical Business Statistics 5th ed. Burr Ridge IL: Irwin 2002. References Normal Distribution The normal distribution is a bell-shaped distribution with the following properties: 1. It is unimodal that is the normal distribution peaks at a single value. 2. It is symmetrical this means that the two areas under the curve between the mean and any two points equidistant on either side of the mean are identical. One side of the distribution is the mirror image of the other side. 3. The mean median and mode are equal. 4. The normal approaches the horizontal axis on either side of the mean toward plus and minus infinity 1∞2. In more formal terms the normal distribution is asymptotic to the x axis. 5. The amount of variation in the random variable deter- mines the height and spread of the normal distribution. Standard Normal Distribution A normal distribution that has a mean 0.0 and a standard deviation 1.0. The horizontal axis is scaled in z-values that measure the number of stand- ard deviations a point is from the mean. Values above the mean have positive z-values. Values below the mean have negative z-values. Glossary www.downloadslide.com

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1 Sampling Error: What It Is and Why It Happens 2 Sampling Distribution of the Mean 3 Sampling Distribution of a Proportion Outcome 2. Determine the mean and standard deviation for the sampling distribution of the sample mean x. Outcome 3. Understand the importance of the Central Limit Theorem. Why you need to know The Jamaica Director of Tourism has recently conducted a study that shows that the mean daily expenditure for adult visitors to the country is 318.69. The mean value is based on a statistical sample of 780 adult visitors to Jamaica. The 318.69 is a statistic not a parameter because it is based on a sample rather than an entire popu- lation. If you were this official you might have several questions: Is the actual population mean equal to 318.69 If the population mean is not 318.69 how close is 318.69 to the true population mean Is a sample of 780 taken from a population of almost 2 million annual visitors to the country sufficient to provide a “good” estimate of the population mean A furniture manufacturer that makes made-to-assemble furniture kits selects a random sample of kits boxed and ready for shipment to custom- ers. These kits are unboxed and inspected to see whether what is in the box matches exactly what is supposed to be in the box. This past week 150 kits were sampled and 15 had one or more discrepancies. This is a 10 defect rate. Should the quality engineer conclude that exactly 10 of the 6900 fur- niture kits made since the first of the year reached the customer with one or more order discrepancies Is the actual percentage higher or lower than 10 and if so by how much Should the quality engineer request that more furni- ture kits be sampled The questions facing the tourism director and the furniture quality engi- neer are common to those faced by people in business everywhere. You will almost assuredly find yourself in a similar situation many times in the future. To help answer these questions you need to have an understanding of sampling distributions. Whenever decisions are based on samples rather than an entire Outcome 4. Determine the mean and standard deviation for the sampling distribution of the sample proportion p. Quick Prep Links Review the discussion of random sampling. Review the steps for computing means and standard deviations. Make sure you are familiar with the normal distribution and how to compute standardized z-values. Review the concepts associated with finding probabilities with a standard normal distribution. Introduction to Sampling Distributions Outcome 1. Understand the concept of sampling error. gdvcom/Shutterstock From Chapter 7 of Business Statistics A Decision-Making Approach Ninth Edition. David F. Groebner Patrick W. Shannon and Phillip C. Fry. Copyright © 2014 by Pearson Education Inc. All rights reserved. www.downloadslide.com

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Introduction to Sampling Distributions population questions about the sample results exist. Anytime we sample from a population there are many many possible samples that could have been selected. Each sample will contain different items. Because of this the sample means for each possible sample can be different or the sample percentages can be different. The sampling distribu- tion describes the distribution of possible sample outcomes. Knowing what this distribution looks like will help you understand the specific result you obtained from the one sample you selected. This chapter introduces you to the important concepts of sampling error and sampling distributions and dis- cusses how you can use this knowledge to help answer the questions facing the tourism director and the quality engineer. The information presented here provides an essential building block to understanding statistical estimation and hypothesis testing. 1 Sampling Error: What It Is and Why It Happens You will encounter many situations in business in which a sample will be taken from a popu- lation and you will be required to analyze the sample data. You have learned about several different statistical sampling techniques including random sampling. The objective of ran- dom sampling is to gather data that accurately represent a population. Then when analysis is performed on the sample data the results will be as though we had worked with all the population data. Calculating Sampling Error Regardless of how careful we are in using random sampling methods the sample may not be a perfect representation of the population. For example a statistic such as x might be com- puted for sample data. Unless the sample is a perfect replication of the population the statistic will likely not equal the parameter m. In this case the difference between the sample mean and the population mean is called sampling error. In the case in which we are interested in the mean value the sampling error is computed using Equation 1. Sampling Error The difference between a measure computed from a sample a statistic and the corresponding measure computed from the population a parameter. Sampling Error of the Sample Mean Sampling error x 1 where: x 5 5 Sample mean Population mean BUSINESS APPLICATION SAMPLING ERROR HUMMEL DEVELOPMENT CORPORATION The Hummel Development Corporation has built 12 office complexes. Table 1 shows a list of the 12 projects and the total square footage of each project. Because these 12 projects are all the office complexes the company has worked on the square-feet area for all 12 projects shown in Table 1 is a population. Equation 2 is used to compute the mean square feet in the population of projects. Chapter Outcome 1. Population Mean 5 x N 2 where: 5 5 Population mean Values in the population x N N 5 Population size TABLE 1 | Square Feet for Office Complex Projects Complex Square Feet 1 114560 2 202300 3 78600 4 156700 5 134600 6 88200 7 177300 8 155300 9 214200 10 303800 11 125200 12 156900 www.downloadslide.com

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Introduction to Sampling Distributions The mean square feet for the 12 office complexes is 114 560 125 200 156 900 12 15 202300 ... 8 8 972 square feet The average square footage of the offices built by the firm is 158972 square feet. This value is a parameter. No matter how many times we compute the value assuming no arithmetic mistakes we will get the same value for the population mean. Hummel is a finalist to be the developer of a new office building in Madison Wisconsin. The client who will hire the firm plans to select a simple random sample of n 5 projects from those the finalists have completed. The client plans to travel to these office buildings to see the quality of the construction and to interview owners and occupants. You may want to review the material on simple random samples. Referring to the office complex data in Table 1 suppose the client randomly selects the following five Hummel projects from the population: Complex Square Feet 5 134600 4 156700 1 114560 8 155300 9 214200 Key in the selection process is the finalists’ past performance on large projects so the client might be interested in the mean size of the office buildings that the Hummel Development Company has built. Equation 3 is used to compute the sample mean. Parameter A measure computed from the entire population. As long as the population does not change the value of the parameter will not change. Simple Random Sample A sample selected in such a manner that each possible sample of a given size has an equal chance of being selected. Sample Mean x x n 5 3 where: x x 5 5 Sample mean Sample values selected from t the population Sample size n 5 The sample mean is x 134 600 156 700 114 560 155 300 214 200 5 7 775 360 5 155 072 The average number of square feet in the random sample of five Hummel office buildings selected by the client is 155072. This value is a statistic based on the sample. Recall the mean for the population: m 158972 square feet The sample mean is x 5155 072 square feet As you can see the sample mean does not equal the population mean. This difference is called the sampling error. Using Equation 1 we compute the sampling error as follows. Sampling error s x 155 072 158 972 3 900 q quare feet The sample mean for the random sample of n 5 office buildings is 3900 square feet less than the population mean. Regardless of how carefully you construct your sampling plan you www.downloadslide.com

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Introduction to Sampling Distributions can expect to see sampling error. A random sample will almost never be a perfect representa- tion of its population. The sample value and the population value will most likely be different. Suppose the client who selected the random sample throws these five projects back into the stack and selects a second random sample of five as follows: Complex Square Feet 9 214200 6 88200 5 134600 12 156900 10 303800 The mean for this sample is x 214 200 88 200 134 600 156 900 303 800 5 8 9 97 700 5 179 540 square feet This time the sample mean is larger than the population mean. This time the sampling error is x 179 540 158 972 20 568 square feet This illustrates some useful fundamental concepts: ● The size of the sampling error depends on which sample is selected. ● The sampling error may be positive or negative. ● There is potentially a different x for each possible sample. If the client wanted to use these sample means to estimate the population mean in one case they would be 3900 square feet too small and in the other they would be 20568 square feet too large. EXAMPLE 1 COMPUTING THE SAMPLING ERROR High-Definition Televisions The website for a major seller of consumer electronics has 10 different brands of HD televisions avail- able. The stated prices for the 46-inch size for the 10 brands are listed as follows: 479 569 599 649 649 699 699 749 799 799 Suppose a competitor who is monitoring this company has randomly sampled n 4 HD brands and recorded the prices from the population of N 10. The selected HD prices were 569 649 799 799 The sampling error can be computed using the following steps: Step 1 Determine the population mean using Equation 2. x N 479 569 599 799 799 10 6 690 10 66 ... 9 9 Step 2 Compute the sample mean using Equation 3. x x n 569 649 799 799 4 2 816 4 704 Step 3 Compute the sampling error using Equation 1. x 704 669 35 nexusseven/Fotolia www.downloadslide.com

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Introduction to Sampling Distributions This sample of four has a sampling error of 35. The sample of TV prices has a slightly larger mean price than the mean for the population. END EXAMPLE TRY PROBLEM 1 The Role of Sample Size in Sampling Error BUSINESS APPLICATION SAMPLING ERROR HUMMEL DEVELOPMENT CORPORATION CONTINUED We selected random samples of 5 office complexes from the 12 projects Hummel Development Corporation has built. We then computed the resulting sampling error. There are actually 792 possible samples of size 5 taken from 12 projects. This value is found using the counting rule for combinations. 1 In actual situations only one sample is selected and the decision maker uses the sample measure to estimate the population measure. A “small” sampling error may be acceptable. How- ever if the sampling error is too “large” conclusions about the population could be misleading. We can look at the extremes on either end to evaluate the potential for extreme sampling error. The population of square feet for the 12 projects is Complex Square Feet Complex Square Feet 1 114560 7 177300 2 202300 8 155300 3 78600 9 214200 4 156700 10 303800 5 134600 11 125200 6 88200 12 156900 Suppose by chance the developers ended up with the fve smallest offce complexes in their sample. These would be Complex Square Feet 3 78600 6 88200 1 114560 11 125200 5 134600 The mean of this sample is x 5108 232 square feet Of all the possible random samples of fve this one provides the smallest sample mean. The sampling error is x 108 232 158 972 50 740 squarefeet On the other extreme suppose the sample contained the five largest office complexes as follows: Complex Square Feet 10 303800 9 214200 2 202300 7 177300 12 156900 1 The number of combinations of items from a sample of is n xn x 2 www.downloadslide.com

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Introduction to Sampling Distributions The mean for this sample is x 210900. This is the largest possible sample mean from all the possible samples of fve complexes. The sampling error in this case would be x 210 900 158 972 51 928 squarefeet The potential for extreme sampling error ranges from -50740 to +51928 square feet The remaining possible random samples of fve will provide sampling errors between these limits. What happens if the size of the sample selected is larger or smaller Suppose the client scales back his sample size to n 3 office complexes. Table 2 shows the extremes. By reducing the sample size from five to three the range of potential sampling error has increased from 1-50740 to +51928 square feet2 to 1-65185.33 to +81128 square feet2 This illustrates that the potential for extreme sampling error is greater when smaller-sized samples are used. Likewise larger sample sizes will reduce the range of potential sampling error. Although larger sample sizes reduce the potential for extreme sampling error there is no guarantee that a larger sample size will always give a smaller sampling error. For example Table 3 shows two further applications of the office complex data. As illustrated this random sample of three has a sampling error of - 2672 square feet whereas the larger random sample of size five has a sampling error of 16540 square feet. In this case the smaller sample was TABLE 2 | Hummel Office Building Example for n 5 3 Extreme Samples Smallest Office Buildings Largest Office Buildings Complex Square Feet Complex Square Feet 3 78600 10 303800 6 88200 9 214200 1 114560 2 202300 x 93786.67 sq. feet x 240100 sq. feet Sampling Error: Sampling Error: 93786.67 - 158972-65185.33 square feet 240100 - 158972 81128 square feet TABLE 3 | Hummel Office Building Example with Different Sample Sizes n 5 n 3 Complex Square Feet Complex Square Feet 4 156700 12 156900 1 114560 8 155300 7 177300 4 156700 11 125200 10 303800 x 175512 sq. feet x 156300 sq. feet Sampling Error: Sampling Error: 175512 - 158972 16540 square feet 156300 - 158972- 2672 square feet www.downloadslide.com

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Introduction to Sampling Distributions “better” than the larger sample. However in Section 2 you will learn that on average the sam- pling error produced by large samples will be less than the sampling error from small samples. Skill Development 7-1. A population has a mean of 125. If a random sample of 8 items from the population results in the following sampled values what is the sampling error for the sample 103 123 99 107 121 100 100 99 7-2. The following data are the 16 values in a population: 10 5192010 8 10 2 14 18 7 814 2 310 a. Compute the population mean. b. Suppose a simple random sample of 5 values from the population is selected with the following results: 5 10202 3 Compute the mean of this sample. c. Based on the results for parts a and b compute the sampling error for the sample mean. 7-3. The following population is provided: 17 15 812 9 7 911 12 14 16 9 5101413 12 12 11 914 81412 Further a simple random sample from this population gives the following values: 12 9 5101411 Compute the sampling error for the sample mean in this situation. 7-4. Consider the following population: 18 26 32 17 34 17 17 22 29 24 24 35 13 29 38 The following sample was drawn from this population: 35 18 24 17 24 32 17 29 a. Determine the sampling error for the sample mean. b. Determine the largest possible sampling error for this sample of n 8. 7-5. Assume that the following represent a population of N 24 values: 10 14 32 9 34 19 31 24 33 11 14 30 6 27 33 32 28 30 10 31 19 13 6 35 a. If a random sample of n 10 items includes the following values compute the sampling error for the sample mean: 32 19 6 11 10 19 28 9 13 33 b. For a sample of size n 6 compute the range for the possible sampling error. Hint: Find the sampling error for the 6 smallest sample values and the 6 largest sample values. c. For a sample of size n 12 compute the range for the possible sampling error. How does sample size affect the potential for extreme sampling error 7-6. Assume that the following represent a population of N 16 values. 25 12 21 13 19 17 15 18 23 16 18 15 22 14 23 17 a. Compute the population mean. b. If a random sample of n 9 includes the following values 12 18 13 17 23 14 16 25 15 compute the sample mean and calculate the sampling error for this sample. c. Determine the range of extreme sampling error for a sample of size n 4. Hint: Calculate the lowest possible sample mean and highest possible sample mean. 7-7. Consider the following population: 369 a. Calculate the population mean. b. Select with replacement and list each possible sample of size 2. Also calculate the sample mean for each sample. 7-1: Exercises MyStatLab www.downloadslide.com

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Introduction to Sampling Distributions c. Calculate the sampling error associated with each sample mean. d. Assuming that each sample is equally likely produce the distribution of the sampling errors. Business Applications 7-8. Hillman Management Services manages apartment complexes in Tulsa Oklahoma. They currently have 30 units available for rent. The monthly rental prices in dollars for this population of 30 units are 455 690 450 495 550 780 800 395 500 405 675 550 490 495 700 995 650 550 400 750 600 780 650 905 415 600 600 780 575 750 a. What is the range of possible sampling error if a random sample of size n 6 is selected from the population b. What is the range of possible sampling error if a random sample of size n 10 is selected Compare your answers to parts a and b and explain why the difference exists. 7-9. A previous report from the Centers for Disease Control and Prevention CDC indicates that smokers on average miss 6.16 days of work per year due to sickness including smoking-related acute and chronic conditions. Nonsmokers miss an average of 3.86 days of work per year. If two years later the CDC believes that the average days of work missed by smokers has not changed it could confirm this by sampling. Consider the following sample: 13445 1289 11 1 5 69 14635 10 7 0 14 6 150253 10 8 6 7 00 15 146221 4 15 10 12 3 0 14 10 0 1 9 14 Determine the sampling error of this sample assuming that the CDC supposition is correct. 7-10. An Internet service provider states that the average number of hours its customers are online each day is 3.75. Suppose a random sample of 14 of the company’s customers is selected and the average number of hours that they are online each day is measured. The sample results are 3.11 1.97 3.52 4.56 7.19 3.89 7.71 2.12 4.68 6.78 5.02 4.28 3.23 1.29 Based on the sample of 14 customers how much sampling error exists Would you expect the sampling error to increase or decrease if the sample size was increased to 40 7-11. The Anasazi Real Estate Company has 20 listings for homes in Santa Fe New Mexico. The number of days each house has been on the market without selling is as follows: 26 45 16 77 33 50 19 23 55 107 88 15 7193060806631 17 a. Considering these 20 values to be the population of interest what is the mean of the population b. The company is making a sales brochure and wishes to feature 5 homes selected at random from the list. The number of days the 5 sampled homes have been on the market is 77 60 15 31 23 If these 5 houses were used to estimate the mean for all 20 what would the sampling error be c. What is the range of possible sampling error if 5 homes are selected at random from the population 7-12. The administrator at Saint Frances Hospital is concerned about the amount of overtime the nursing staff is incurring and wonders whether so much overtime is really necessary. The hospital employs 60 nurses. Following is the number of hours of overtime reported by each nurse last week. These data are the population of interest. Nurse Overtime Nurse Overtime Nurse Overtime 1 2 214413 2 1 222423 3 7 233432 4 0 245441 5 4 255453 6 2 266463 76 27 2 47 3 8 4 282483 9 2 297494 10 5304506 11 5314510 12 4323523 13 5333534 14 0344546 15 6355550 16 0365563 17 2370573 18 4380587 19 2394595 20 5403607 Using the Random Numbers Table with a starting point in column digit 14 and row 10 select a random sample of 6 nurses. Go down the table from the starting point. Determine the mean hours of overtime for these 6 nurses and calculate the sampling error associated with this particular sample mean. 7-13. Princess Cruises recently offered a 16-day voyage from Beijing to Bangkok during the time period from May to August. The announced price excluding airfare for a room with an ocean view or a balcony was listed as 3475. Cruise fares usually are quite variable due to discounting by the cruise line and travel agents. A sample of 20 passengers who www.downloadslide.com

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Introduction to Sampling Distributions c. What is the range of possible sampling error if a random sample size of 7 computers is taken to estimate the mean scan time for all 25 machines Computer Database Exercises 7-16. USA Today reports salaries for National Football League NFL teams. The file Jaguars contains the salaries for the 2011 Jacksonville Jaguars. a. Calculate the average total salary for the Jacksonville Jaguars for 2011. b. Calculate the smallest sample mean for total salary and the largest sample mean for total salary using a sample size of 10. Calculate the sampling error for each sample mean. c. Repeat the calculations in part b for samples of size 5 and 2. d. What effect does a change in the sample size appear to have on the dispersion of the sampling errors 7-17. The file titled Clothing contains the monthly retail sales millions of U.S. women’s clothing stores for 70 months. A sample taken from this population to estimate the average sales in this time period follows: 2942 2574 2760 2939 2642 2905 2568 2677 2572 3119 2697 2884 2632 2742 2671 2884 2946 2825 2987 2729 2676 2846 3112 2924 2676 a. Calculate the population mean. b. Calculate the sample mean. c. How much sampling error is present in this sample d. Determine the range of possible sampling error if 25 sales figures are sampled at random from this population. 7-18. The Dow-Jones Industrial Average DJIA Index is a well-known stock index. The index was originally developed in 1884 and has been in place ever since as a gauge of how the U.S. stock market is performing. The file Dow Jones contains date open high low close and volume for the DJIA for almost eight years of the trading days. a. Assuming that the data in the file Dow Jones constitute the population of interest what is the population mean closing value for the DJIA b. Using Excel select a random sample of 10 days’ closing values make certain not to include duplicate days and calculate the sample mean and the sampling error for the sample. c. Repeat part b with a sample size of 50 days’ closing values. d. Repeat part b with a sample size of 100 days’ closing values. e. Write a short statement describing your results. Were they as expected Explain. 7-19. Welco Lumber Company is based in Shelton Washington and is a privately held company that makes cedar siding cedar lumber and purchased this cruise paid the following amounts in dollars: 3559 3005 3389 3505 3605 3545 3529 3709 3229 3419 3439 3375 3349 3559 3419 3569 3559 3575 3449 3119 a. Calculate the sample mean cruise fare. b. Determine the sampling error for this sample. c. Would the results obtained in part b indicate that the average cruise fare during this period for this cruise is different from the listed price Explain your answer from a statistical point of view. 7-14. An investment advisor has worked with 24 clients for the past five years. Following are the percentage rates of average five-year returns that these 24 clients experienced over this time frame on their investments: 11.2 11.2 15.9 2.7 4.6 7.6 15.6 1.3 3.3 4.8 12.8 14.9 10.1 10.9 4.9 -2.1 12.5 3.7 7.6 4.9 10.2 0.4 9.6 -0.5 This investment advisor plans to introduce a new investment program to a sample of his customers this year. Because this is experimental he plans to randomly select 5 of the customers to be part of the program. However he would like those selected to have a mean return rate close to the population mean for the 24 clients. Suppose the following 5 values represent the average five-year annual return for the clients that were selected in the random sample: 11.2 -2.1 12.5 1.3 3.3 Calculate the sampling error associated with the mean of this random sample. What would you tell this advisor regarding the sample he has selected 7-15. A computer lab at a small college has 25 computers. Twice during the day a full scan for viruses is performed on each computer. Because of differences in the configuration of the computers the times required to complete the scan are different for each machine. Records for the scans are kept and indicate that the time in seconds required to perform the scan for each machine is as shown here. Time in Seconds to Complete Scan 1500 1347 1552 1453 1371 1362 1447 1362 1216 1378 1647 1093 1350 1834 1480 1522 1410 1446 1291 1601 1365 1575 1134 1532 1534 a. What is the mean time required to scan all 25 computers b. Suppose a random sample of 5 computers is taken and the scan times for each are as follows: 1534 1447 1371 1410 and 1834. If these 5 randomly sampled computers are used to estimate the mean scan time for all 25 computers what would the sampling error be www.downloadslide.com

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Introduction to Sampling Distributions cedar fencing products for sale and distribution throughout North America. The major cost of production is the cedar logs that are the raw material necessary to make the finished cedar products. Thus it is very important to the company to get the maximum yield from each log. Of course the dollar value to be achieved from a log depends initially on the diameter of the log. Each log is 8 feet long when it reaches the mill. The file called Welco contains a random sample of logs of various diameters and the potential value of the finished products that could be developed from the log if it is made into fence boards. a. Calculate the sample mean potential value for each diameter of logs in the sample. b. Discuss whether there is a way to determine how much sampling error exists for a given diameter log based on the sample. Can you determine whether the sampling error will be positive or negative Discuss. 7-20. Maher Barney and White LLC is a legal firm with 40 employees. All of the firm’s employees are eligible to participate in the company’s 401k plan and the firm is proud of its 100 participation rate. The file MBW 401 contains the most recent year- end 401k account balance for each of the firm’s 40 employees. a. Compute the population mean and population standard deviation for the most recent year-end 401k account balances at Maher Barney and White. b. Suppose that an audit of the firm’s 401k plan is being conducted and 12 randomly selected employee account balances are to be examined. If the following employees indicated by employee number are randomly selected to be included in the study what is the estimate for the most recent year-end mean 401k account balance How much sampling error is present in this estimate Employee 26 831 3383017 921391811 c. Calculate the range of possible sampling error if a random sample of 15 employees is used to estimate the most recent year-end mean 401k account balance. 7-21. The Badke Foundation was set up by the Fred Badke family following his death in 2001. Fred had been a very successful heart surgeon and real estate investor in San Diego and the family wanted to set up an organization that could be used to help less fortunate people. However one of the concepts behind the Badke Foundation is to use the Badke money as seed money for gathering contributions from middle-class families. To help in the solicitation of contributions the foundation was considering the idea of hiring a consulting company that specialized in this activity. Leaders of the consulting company maintained in their presentation that the mean contribution from families who actually contribute after receiving a specially prepared letter would be 20.00. Before actually hiring the company the Badke Foundation sent out the letter and request materials to many people in the San Diego area. They received contributions from 166 families. The contribution amounts are in the data file called Badke. a. Assuming that these data reflect a random sample of the population of contributions that would be received compute the sampling error based on the claim made by the consulting firm. b. Comment on any issues you have with the assumption that the data represent a random sample. Does the calculation of the sampling error matter if the sample is not a random sample Discuss. END EXERCISES 7-1 2 Sampling Distribution of the Mean Section 1 introduced the concept of sampling error. A random sample selected from a popula- tion will not perfectly match the population. Thus the sample statistic likely will not equal the population parameter. If this difference arises because the random sample is not a perfect representation of the population it is called sampling error. In business applications decision makers select a single random sample from a popula- tion. They compute a sample measure and use it to make decisions about the entire population. For example Nielsen Media Research takes a single random sample of television viewers to determine the percentage of the population who are watching a particular program during a particular week. Of course the sample selected is only one of many possible samples that could have been selected from the same population. The sampling error will differ depending on which sample is selected. If in theory you were to select all possible random samples of a given size and compute the sample means for each one these means would vary above and below the true population mean. If we graphed these values as a histogram the graph would be the sampling distribution. Sampling Distribution The distribution of all possible values of a statistic for a given sample size that has been randomly selected from a population. www.downloadslide.com

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Introduction to Sampling Distributions In this section we introduce the basic concepts of sampling distributions. We will use an Excel tool to select repeated samples from the same population for demonstration purposes only. Simulating the Sampling Distribution for x BUSINESS APPLICATION SAMPLING DISTRIBUTIONS AIMS INVESTMENT COMPANY Aims Investment Company handles employee retirement funds primarily for small companies. The file named AIMS contains data on the number of mutual funds in each client’s portfolio. The file contains data for all 200 Aims customers so it is considered a population. Figure 1 shows a histogram for the population. The mean number of mutual funds in a portfolio is 2.505 funds. The standard deviation is 1.507 funds. The graph in Figure 1 indicates that the popula- tion is spread between zero and six funds with more customers owning two funds than any other number. Suppose the controller at Aims plans to select a random sample of 10 accounts. In Excel we can use the Sampling tool to generate the random sample. 2 Figure 2 shows the num- ber of mutual funds owned for a random sample of 10 clients. The sample mean of 2.1 is also shown. To illustrate the concept of a sampling distribution we repeat this process 500 times generating 500 different random samples of 10. For each sample we compute the sample mean. Figure 3 shows the frequency histogram for these sample means. Note that the horizontal axis represents the x-values. The graph in Figure 3 is not a complete sampling distribution because it is based on only 500 samples out of the many possible samples that could be selected. However this simulation gives us an idea of what the sampling distribu- tion looks like. Look again at the population distribution in Figure 1 and compare it with the shape of the frequency histogram in Figure 3. Although the population distribution is some- what skewed the distribution of sample means is taking the shape of a normal distribu- tion. Note also the population mean number of mutual funds owned by the 200 Aims Invest- ment customers in the population is 2.505. If we average the 500 sample means in Figure 3 we get 2.41. This value is the mean of the 500 sample means. It is reasonably close to the population mean. 2 The same thing can be achieved in Minitab by using the Sample from Columns option under the Calc 7 Probability Data command. Number of Customers 06 5 4 3 2 1 Number of Mutual Funds 60 50 40 30 20 10 0 POPULATION OF FUNDS OWNED FIGURE 1 |  Distribution of Mutual Funds for the Aims Investment Company Excel Tutorial Excel tutorials Chapter Outcome 2. Gunnar Pippel/Shutterstock www.downloadslide.com

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Introduction to Sampling Distributions Frequency 1.4 x Sample Mean 100 80 60 40 20 0 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 DISTRIBUTION OF SAMPLE MEANS n 10 FIGURE 3 |  Aims Investment Company Histogram of 500 Sample Means from Sample Size n 10 Excel 2010 Instructions: 1. Open fle: Aims.xlsx. 2. Select Data Data Analysis. 3. Select Sampling. 4. Defne the population data range B2:B201. 5. Select Random Number of Samples: 10. 6. Select Output Range: D2. 7. Compute sample mean using Excel Average function using the range D2:D11. Minitab Instructions for similar results: 1. Open fle: AIMS.MTW. 2. Choose Calc Random Data Sample From Columns. 3. In Number of rows to Sample enter the sample size. 4. In box following From columns enter data column: Number of Mutual Fund Accounts. 5. In Store Samples in enter sample’s storage column. 6. Click OK. 7. Choose Calc Calculator. 8. In Store Result in Variable enter column to store mean. 9. Choose Mean from Functions. Expression: Mean Sample Column. Repeat steps 3–10 to form sample. Click OK. 10. 11. FIGURE 2 |  Excel 2010 Output for the Aims Investment Company First Sample Size n 10 www.downloadslide.com

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Introduction to Sampling Distributions Had we selected all possible random samples of size 10 from the population and com- puted all possible sample means the average of all the possible sample means would be equal to the population mean. This concept is expressed as Theorem 1. Theorem 1 For any population the average value of all possible sample means computed from all possible random samples of a given size from the population will equal the population mean. This is expressed as x 5 When the average of all possible values of the sample statistic equals the corresponding parameter no matter the value of the parameter we call that statistic an unbiased estimator of the parameter. Also the population standard deviation is 1.507 mutual funds. This measures the variation in the number of mutual funds between individual customers. When we compute the standard deviation of the 500 sample means we get 0.421 which is considerably smaller than the popu- lation standard deviation. If all possible random samples of size n are selected from the popula- tion the distribution of possible sample means will have a standard deviation that is equal to the population standard deviation divided by the square root of the sample size as Theorem 2 states. Unbiased Estimator A characteristic of certain statistics in which the average of all possible values of the sample statistic equals a parameter no matter the value of the parameter. Theorem 2 For any population the standard deviation of the possible sample means computed from all possible random samples of size n is equal to the population standard deviation divided by the square root of the sample size. This is shown as x n 5 Recall the population standard deviation is s 1.507. Then based on Theorem 2 had we selected all possible random samples of size n 10 rather than only 500 samples the stand- ard deviation for the possible sample means would be x n 55 5 1 507 10 0 477 . . Our simulated value of 0.421 is fairly close to 0.477. The standard deviation of the sampling distribution will be less than the popula- tion standard deviation. To further illustrate suppose we increased the sample size from n 10 to n 20 and selected 500 new samples of size 20. Figure 4 shows the distribution of the 500 different sample means. The distribution in Figure 4 is even closer to a normal distribution than what we observed in Figure 3. As sample size increases the distribution of sample means will become shaped more like a normal distribution. The average sample mean for these 500 samples is 2.53 and the standard deviation of the different sample means is 0.376. Based on Theorems 1 and 2 for a sample size of 20 we would expect the following: x x n 55 5 5 5 2 505 1 507 20 0 337 . . . and Thus our simulated values are quite close to the theoretical values we would expect had we selected all possible random samples of size 20. Sampling from Normal Populations The previous discussion began with the population of mutual funds shown in Figure 1. The population was not normally distributed but as we increased the sample size the sampling distribution of possible sample means began to approach a normal distribution. We will return to this situation shortly but what happens if the population itself is normally distributed To help answer this question Theorem 3 can be applied. www.downloadslide.com

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Introduction to Sampling Distributions Frequency 1.4 Sample Means 140 120 100 80 60 40 20 0 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 DISTRIBUTION OF SAMPLE MEANS n 20 x FIGURE 4 |  Aims Investment Company Histogram of Sample Means from Sample Size n5 20 Theorem 3 If a population is normally distributed with mean m and a standard deviation s the sampling distribution of the sample mean x is also normally distributed with a mean equal to the population mean 1m x m2 and a standard deviation equal to the popula- tion standard deviation divided by the square root of the sample size 1s x s1n2. In Theorem 3 the quantity 1s x s1n2 is the standard deviation of the sampling dis- tribution. Another term that is given to this is the standard error of x because it is the measure of the standard deviation of the potential sampling error. We can again use simulation to demonstrate Theorem 3. We begin by using Excel to generate a normally distributed population. 3 Figure 5 shows a simulated population that is approximately normally distributed with a mean equal to 1000 and a standard deviation equal to 200. The data range is from 250 to 1800. Next we simulate the selection of 2000 random samples of size n 10 from the normally distributed population and compute the sample mean for each sample. These sample means can then be graphed as a frequency histogram as shown in Figure 6. This histogram represents the sampling distribution. Note that it too is approximately normally distributed. 3 The same task can be performed in Minitab using the Calc. 7 Random Data command. However you will have to generate each sample individually which will take time. x 1600 1400 1200 1000 800 600 400 200 250 1800 0 Frequency 1000 200 FIGURE 5 |  Simulated Normal Population Distribution www.downloadslide.com

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Introduction to Sampling Distributions We next compute the average of the 2000 sample means and use it to approximate m x as follows: x x 2 000 2 000 178 2 000 1000 5 The mean of these sample means is approximately 1000. This is the same value as the popu- lation mean. We also approximate the standard deviation of the sample means as follows: x x x . 2 2 000 62 10 We see the standard deviation of the sample means is 62.10. This is much smaller than the population standard deviation which is 200. The largest sample mean was just more than 1212 and the smallest sample mean was just less than 775. Recall however that the popula- tion ranged from 250 to 1800. The variation in the sample means always will be less than the variation for the population as a whole. Using Theorem 3 we would expect the sample means to have a standard deviation equal to x n 55 5 200 10 63 25 . Our simulated standard deviation of 62.10 is fairly close to the theoretical value of 63.25. Suppose we again use the simulated population shown in Figure 5 with m 1 000 and s 200. We are interested in seeing what the sampling distribution will look like for different size samples. For a sample size of 5 Theorem 3 indicates that the sampling distri- bution will be normally distributed and have a mean equal to 1000 and a standard deviation equal to x 55 200 5 89 44 . If we were to take a random sample of 10 as simulated earlier Theorem 3 indicates the sampling distribution would be normal with a mean equal to 1000 and a standard deviation equal to x 55 200 10 63 25 . 500 750 1250 1500 x 62.10 140 120 100 80 60 40 20 0 Frequency x 1000 x Sample Means DISTRIBUTION OF SAMPLE MEANS FIGURE 6 |  Approximated Sampling Distribution n5 10 www.downloadslide.com

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Introduction to Sampling Distributions For a sample size of 20 the sampling distribution will be centered at m x 1000 with a standard deviation equal to x 55 200 20 44 72 . Notice as the sample size is increased the standard deviation of the sampling distribution is reduced. This means the potential for extreme sampling error is reduced when larger sam- ple sizes are used. Figure 7 shows sampling distributions for sample sizes of 5 10 and 20. However when the population is normally distributed the sampling distribution of x will always be normal and centered at the population mean. Only the spread in the distribution will change as the sample size changes. This illustrates a very important statistical concept referred to as consistency. Earlier we defined a statistic as unbiased if the average value of the statistic equals the parameter to be estimated. Theorem 1 asserted that the sample mean is an unbiased estimator of the population mean no matter the value of the parameter. However just because a statistic is unbiased does not tell us whether the statistic will be close in value to the parameter. But if as the sample size is increased we can expect the value of the statistic to become closer to the parameter then we say that the statistic is a consistent estimator of the parameter. Figure 7 illustrates that the sample mean is a consistent estimator of the popu- lation mean. The sampling distribution is composed of all possible sample means of the same size. Half the sample means will lie above the center of the sampling distribution and half will lie below. The relative distance that a given sample mean is from the center can be determined by standardizing the sampling distribution. A standardized value is deter- mined by converting the value from its original units into a z-value. A z-value measures the number of standard deviations a value is from the mean. This same concept can be used when working with a sampling distribution. Equation 4 shows how the z-values are computed. Consistent Estimator An unbiased estimator is said to be a consistent estimator if the difference between the estimator and the parameter tends to become smaller as the sample size becomes larger. FIGURE 7 |  Theorem 3 Examples Population s 200 Sample Size n 5 200 5 89.44 s x s x Sample Size n 10 200 10 63.25 s x Sample Size n 20 200 20 44.72 x x x x www.downloadslide.com

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Introduction to Sampling Distributions Note if the sample being selected is large relative to the size of the population greater than 5 of the population size of if the sampling is being done without replacement we need to modify how we compute the standard deviation of the sampling distribution and z-value using what is known as the finite population correction factor as shown in Equation 5. z-Value for Sampling Distribution of x z x n 4 where: x 5 5 5 Sample mean Population mean Population s standard deviation Sample size n 5 z-Value Adjusted for the Finite Population Correction Factor z x n Nn N 1 5 where: N n Nn N Population size Sample size Finite 1 population correction factor The finite population correction factor is used to calculate the standard deviation of the sam- pling distribution when the sampling is performed without replacement or when the sample size is greater than 5 of the population size. EXAMPLE 2 FINDING THE PROBABILITY THAT X IS IN A GIVEN RANGE Scribner Products Scribner Products manufactures floor- ing materials for the residential and commercial construction industries. One item they make is a mosaic tile for bathrooms and showers. When the production process is operating accord- ing to specifications the diagonal dimension of a tile used for decorative purposes is normally distributed with a mean equal to 1.5 inches and a standard deviation of 0.05 inches. Before shipping a large batch of these tiles Scribner quality analysts have selected a random sample of eight tiles with the following diameters: 1.57 1.59 1.48 1.60 1.59 1.62 1.55 1.52 The analysts want to use these measurements to determine if the process is no longer operat- ing within the specifications. The following steps can be used: Step 1 Determine the mean for this sample. x n 55 5 12 52 8 1 565 . . inches x Wollwerth Imagery/Fotolia www.downloadslide.com

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Introduction to Sampling Distributions Step 2 Define the sampling distribution for x using Theorem 3. Theorem 3 indicates that if the population is normally distributed the sampling distribution for x will also be normally distributed with xx n 55 and Thus in this case the mean of the sampling distribution should be 1.50 inches and the standard deviation should be 0.0518 0.0177 inches. Step 3 Define the probability statement of interest. Because the sample mean is x 1.565 which is greater than the mean of the sampling distribution we want to find Px. 1 565 inches Step 4 Convert the sample mean to a standardized z-value using Equation 4. z x n 1 565 1 50 005 8 0 065 0 0177 367 .. . . . . Step 5 Use the standard normal distribution table to determine the desired probability. P1z Ú 3.672 The Standard Normal Distribution Table does not show z-values as high as 3.67. This implies that P1z Ú 3.672 ≈ 0.00. So if the production process is working properly there is virtually no chance that a random sample of eight tiles will have a mean diameter of 1.565 inches or greater. Because the analysts at Scribner Products did find this sample result there is a very good chance that something is wrong with the process. END EXAMPLE TRY PROBLEM 26 The Central Limit Theorem Theorem 3 applies when the population is normally distributed. Although there are many situ- ations in business in which this will be the case there are also many situations in which the population is not normal. For example incomes in a region tend to be right skewed. Some dis- tributions such as people’s weight are bimodal a peak weight group for males and another peak weight group for females. What does the sampling distribution of x look like when a population is not normally dis- tributed The answer is . . . it depends. It depends on what the shape of the population is and what size sample is selected. To illustrate suppose we have a U-shaped population such as the one in Figure 8 with mean 14.00 and standard deviation 3.00. Now we select 3000 10 x-Values 11 12 13 14 15 16 17 18 FIGURE 8 |  Simulated Nonnormal Population Chapter Outcome 3. www.downloadslide.com

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Introduction to Sampling Distributions simple random samples of size 3 and compute the mean for each sample. These x - v alues are graphed in the histogram shown in Figure 9. The average of these 3000 sample means is x x 3 000 14 02 . Notice this value is approximately equal to the population mean of 14.00 as Theorem 1 would suggest. 4 Next we compute the standard deviation as x x x . 2 3 000 182 The standard deviation of the sampling distribution is less than the standard deviation for the population which was 3.00. This will always be the case. The frequency histogram of x-values for the 3000 samples of 3 looks different from the population distribution which is U-shaped. Suppose we increase the sample size to 10 and take 3000 samples from the same U-shaped population. The resulting frequency histogram of x -values is shown in Figure 10. Now the frequency distribution looks much like a normal distri- bution. The average of the sample means is still equal to 14.02 which is virtually equal to the population mean. The standard deviation for this sampling distribution is now reduced to 0.97. 10 11 12 13 14 15 16 17 18 Sample Means x FIGURE 9 |  Frequency Histogram of x1n 32 4 Note if we had selected all possible samples of three the average of the samples means would have been equal to the population mean. FIGURE 10 |  Frequency Histogram of x1n 102 10 11 12 13 14 15 16 17 18 x 14.02 Average of sample means 0.97 Standard deviation of sample means www.downloadslide.com

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Introduction to Sampling Distributions The Central Limit Theorem is very important because with it we know the shape of the sampling distribution even though we may not know the shape of the population distribution. The one catch is that the sample size must be “sufficiently large.” What is a sufficiently large sample size The answer depends on the shape of the population. If the population is quite symmetric then sample sizes as small as 2 or 3 can provide a normally distributed sampling distribution. If the population is highly skewed or otherwise irregularly shaped the required sample size will be larger. Recall the example of the U-shaped population. The frequency distribution obtained from samples of 3 was shaped differently than the population but not like a normal distribution. However for samples of 10 the frequency distribution was a very close approxi- mation to a normal distribution. Figures 11 12 and 13 show some examples of the Central Limit Theorem concept. Simulation studies indicate that even for very strange-looking popu- lations samples of 25 to 30 produce sampling distributions that are approximately normal. Thus a conservative definition of a sufficiently large sample size is n Ú 30. The Central Limit Theorem is illustrated in the following example. Theorem 4: The Central Limit Theorem For simple random samples of n observations taken from a population with mean m and standard deviation s regardless of the population’s distribution provided the sample size is sufficiently large the distribution of the sample means x will be approximately normal with a mean equal to the population mean 1m x m2 and a standard deviation equal to the population standard deviation divided by the square root of the sample size 1s x s1n2. The larger the sample size the better the approximation to the normal distribution. FIGURE 11 |  Central Limit Theorem With Uniform Population Distribution b Sampling Distribution n 2 a x x fx Population Sampling Distribution n 5 c x fx fx This example is not a special case. Instead it illustrates a very important statistical concept called the Central Limit Theorem. www.downloadslide.com

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Introduction to Sampling Distributions FIGURE 12 |  Central Limit Theorem With Triangular Population Sampling Distribution n 5 Sampling Distribution n 30 a x fx Population b x c x fx fx FIGURE 13 |  Central Limit Theorem With a Skewed Population Sampling Distribution n 4 Sampling Distribution n 25 a x fx Population c x b x fx fx www.downloadslide.com

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Introduction to Sampling Distributions EXAMPLE 3 FINDING THE PROBABILITY THAT X IS IN A GIVEN RANGE Westside Drive-In. Past sales records indicate that the dollar value of lunch orders at the State Street restaurant are right skewed with a population mean of 12.50 per customer and a standard deviation of 5.50. The Westside Drive-In manager has selected a random sample of 100 lunch receipts. She is interested in determining the probability that the mean lunch order for this sample from this population will fall between 12.25 and 13.00. To find this probability she can use the following steps. Step 1 Determine the sample mean. In this case two sample means are being considered: xx 55 . . 12 25 13 00 and Step 2 Define the sampling distribution. The Central Limit Theorem can be used because the sample size is large enough1n 1002to determine that the sampling distribution will be approximately normal even though the population is right skewed with xx 555 . . . 12 50 550 100 055 and Step 3 Define the probability statement of interest. The manager is interested in Px . . 12 25 13 00 Step 4 Use the standard normal distribution to find the probability of interest. Assuming the population of lunch receipts is quite large we use Equation 4 to convert the sample means to corresponding z-values. x n x 12 25 12 50 550 100 046 .. . .and n 13 00 12 50 550 100 091 .. . . From the standard normal table the probability associated with z- 0.46 is 0.1772 and the probability for z 0.91 is 0.3186. Therefore Px P . . . . . 12 25 13 00 0 46 0 91 0 1772 z 0 0 3186 0 4958 .. There is nearly a 0.50 chance that the sample mean will fall in the range 12.25 to 13.00. END EXAMPLE TRY PROBLEM 30 How to do it Example 3 Sampling Distribution of x To find probabilities associated with a sampling distribution of x for samples of size n from a popu- lation with mean m and standard deviation s use the following steps. 1. Compute the sample mean using x x n ∑ 2. Define the sampling distribution. If the population is normally distributed the sampling dis- tribution also will be normally distributed for any size sample. If the population is not normally distributed but the sample size is sufficiently large the sampling distribution will be approxi- mately normal. In either case the sampling distribution will have n xx and 55 3. Define the probability statement of interest. We are interested in finding the probability of some range of sample means such as Px 25 4. Use the standard normal distri- bution to find the probability of interest using Equation 4 or 5 to convert the sample mean to a corresponding z-value. z x n z x n Nn N m s m s or 1 Then use the standard normal table to find the probability asso- ciated with the calculated z-value. Skill Development 7-22. A population with a mean of 1250 and a standard deviation of 400 is known to be highly skewed to the right. If a random sample of 64 items is selected from the population what is the probability that the sample mean will be less than 1325 7-23. Suppose that a population is known to be normally distributed with m 2000 and s 230. If a random sample of size n 8 is selected calculate the probability that the sample mean will exceed 2100. MyStatLab 7-2: Exercises 7-24. A normally distributed population has a mean of 500 and a standard deviation of 60. a. Determine the probability that a random sample of size 16 selected from this population will have a sample mean less than 475. b. Determine the probability that a random sample of size 25 selected from the population will have a sample mean greater than or equal to 515. 7-25. If a population is known to be normally distributed with m 250 and s 40 what will be the characteristics www.downloadslide.com

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Introduction to Sampling Distributions a. What is the probability of having a sample mean equal to or smaller than the sample mean for this sample if the population mean is 12 processed returns daily with a standard deviation of 3 returns per day b. What is the probability of having a sample mean larger than the one obtained from this sample if the population mean is 12 processed returns daily with a standard deviation of 3 returns per day c. Explain how it is possible to answer parts a and b when the population distribution of daily tax returns at Many Happy Returns is not known. 7-32. SeaFair Fashions relies on its sales force of 220 to do an initial screening of all new fashions. The company is currently bringing out a new line of swimwear and has invited 40 salespeople to its Orlando home office. An issue of constant concern to the SeaFair sales office is the volume of orders generated by each salesperson. Last year the overall company average was 417330 with a standard deviation of 45285. Hint: The finite population correction factor Equation 5 is required. a. Determine the probability the sample of 40 will have a sales average less than 400000. b. What shape do you think the distribution of all possible sample means of 40 will have Discuss. c. Determine the value of the standard deviation of the distribution of the sample mean of all possible samples of size 40. d. How would the answers to parts a b and c change if the home office brought 60 salespeople to Orlando Provide the respective answers for this sample size. e. Each year SeaFair invites the sales personnel with sales above the 85th percentile to enjoy a complementary vacation in Hawaii. Determine the smallest average salary for the sales personnel that were in Hawaii last year. Assume the distribution of sales was normally distributed last year. 7-33. Suppose the life of a particular brand of calculator battery is approximately normally distributed with a mean of 75 hours and a standard deviation of 10 hours. a. What is the probability that a single battery randomly selected from the population will have a life between 70 and 80 hours b. What is the probability that 16 randomly sampled batteries from the population will have a sample mean life of between 70 and 80 hours c. If the manufacturer of the battery is able to reduce the standard deviation of battery life from 10 to 9 hours what would be the probability that 16 batteries randomly sampled from the population will have a sample mean life of between 70 and 80 hours 7-34. Sands Inc. makes particleboard for the building industry. Particleboard is built by mixing wood chips and resins together and pressing the sheets under extreme heat and pressure to form a 4-feet 8-feet sheet that is used as a substitute for plywood. The strength of the particleboards is tied to the board’s weight. Boards that are too light are brittle and do not of the sampling distribution for x based on a random sample of size 25 selected from the population 7-26. Suppose nine items are randomly sampled from a normally distributed population with a mean of 100 and a standard deviation of 20. The nine randomly sampled values are 125 95 66 116 99 91 102 51 110 Calculate the probability of getting a sample mean that is smaller than the sample mean for these nine sampled values. 7-27. A random sample of 100 items is selected from a population of size 350. What is the probability that the sample mean will exceed 200 if the population mean is 195 and the population standard deviation equals 20 Hint: Use the finite population correction factor since the sample size is more than 5 of the population size. 7-28. Given a distribution that has a mean of 40 and a standard deviation of 13 calculate the probability that a sample of 49 has a sample mean that is a. greater than 37 b. at most 43 c. between 37 and 43 d. between 43 and 45 e. no more than 35 7-29. Consider a normal distribution with mean 12 and standard deviation 90. Calculate P1x 7 362 for each of the following sample sizes: a. n 1 b. n 9 c. n 16 d. n 25 Business Applications 7-30. SeeClear Windows makes windows for use in homes and commercial buildings. The standards for glass thickness call for the glass to average 0.375 inches with a standard deviation equal to 0.050 inches. Suppose a random sample of n 50 windows yields a sample mean of 0.392 inches. a. What is the probability of x Ú 0.392 if the windows meet the standards b. Based on your answer to part a what would you conclude about the population of windows Is it meeting the standards 7-31. Many Happy Returns is a tax preparation service with offices located throughout the western United States. Suppose the average number of returns processed by employees of Many Happy Returns during tax season is 12 per day with a standard deviation of 3 per day. A random sample of 36 employees taken during tax season revealed the following number of returns processed daily: 11 17 13 91313131215 15 13 15 10 15 13 13 15 13 9 9 9 15 13 14 9 14 11 11 17 16 9 8 10 15 12 11 www.downloadslide.com

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Introduction to Sampling Distributions airline was 34.3 pounds with a standard deviation of 5.7 pounds. Further it stated that the distribution of weights was approximately normally distributed. This memo was leaked to a consumers’ group in Atlanta. This group had selected and weighed a random sample of 14 bags to be checked on a flight departing from Atlanta. The following data pounds were recorded: 29 27 40 34 30 30 35 44 33 28 36 33 30 40 What is the probability that a sample mean as small or smaller than the one for this sample would occur if the airline’s claims about the population of baggage weight is accurate Comment on the results. 7-38. ACNielsen is a New York–based corporation and a member of the modern marketing research industry. One of the items that ACNielsen tracks is the expenditure on over-the-counter OTC cough medicines. ACNielsen recently indicated that consumers spent 620 million on OTC cough medicines in the United States. The article also indicated that nearly 30 million visits for coughs were made to doctors’ offices in the United States. a. Determine the average cost of OTC cough medicines per doctor’s office visit based on 30 million purchases. b. Assuming that the average cost indicated in part a is the true average cost of OTC cough medicines per doctor’s visit and the standard deviation is 10 determine the probability that the average cost for a random selection of 30 individuals will result in an average expenditure of more than 25 in OTC cough medicines. c. Determine the 90th percentile for the average cost of OTC cough medicines for a sample of 36 individuals all of whom have visited a doctor’s office for cough symptoms. Computer Database Exercises 7-39. One of the top-selling video games continues to be Mad- den NFL 12. While prices vary widely depending on store or website the suggested retail price for this video game is 59.95. The file titled Madden contains a ran- dom sample of the retail prices paid for Madden NFL 12. a. Calculate the sample mean and standard deviation of retail prices paid for Madden NFL 12. b. To determine if the average retail price has fallen assume the population mean is 59.95 calculate the probability that a sample of size 200 would result in a sample mean no larger than the one calculated in part a. Assume that the sample standard deviation is representative of the population standard deviation. c. In part b you used 59.95 as the population mean. Calculate the probability required in part b assuming that the population mean is 59.50. d. On the basis of your calculations in parts b and c does it seem likely that the average retail price for Madden NFL 12 has decreased Explain. meet the quality standard for strength. Boards that are too heavy are strong but are difficult for customers to handle. The company knows that there will be variation in the boards’ weight. Product specifications call for the weight per sheet to average 10 pounds with a standard deviation of 1.75 pounds. During each shift Sands employees select and weigh a random sample of 25 boards. The boards are thought to have a normally distributed weight distribution. If the average of the sample slips below 9.60 pounds an adjustment is made to the process to add more moisture and resins to increase the weight and Sands hopes the strength. a. Assuming that the process is operating correctly according to specifications what is the probability that a sample will indicate that an adjustment is needed b. Assume the population mean weight per sheet slips to 9 pounds. Determine the probability that the sample will indicate an adjustment is not needed. c. Assuming that 10 pounds is the mean weight what should the cutoff be if the company wants no more than a 5 chance that a sample of 25 boards will have an average weight less than 9.6 lbs 7-35. The branch manager for United Savings and Loan in Seaside Virginia has worked with her employees in an effort to reduce the waiting time for customers at the bank. Recently she and the team concluded that average waiting time is now down to 3.5 minutes with a standard deviation equal to 1.0 minute. However before making a statement at a managers’ meeting this branch manager wanted to double-check that the process was working as thought. To make this check she randomly sampled 25 customers and recorded the time they had to wait. She discovered that mean wait time for this sample of customers was 4.2 minutes. Based on the team’s claims about waiting time what is the probability that a sample mean for n 25 people would be as large or larger than 4.2 minutes What should the manager conclude based on these data 7-36. Mileage ratings for cars and trucks generally come with a qualifier stating actual mileage will depend on driving conditions and habits. Ford is stating the Ecoboost F-150 pickup truck will get 19 miles per gallon with combined town and country driving. Assume the mean stated by Ford is the actual average and the distribution has a standard deviation of 3 mpg. a. Given the above mean and standard deviation what is the probability that 100 drivers will get more than 19.2 miles per gallon average b. Suppose 1000 drivers were randomly selected. What is the probability the average obtained by these drivers would exceed 19.2 mpg 7-37. Airlines have recently toughened their standards for the weight of checked baggage limiting the weight of a bag to 50 pounds on domestic U.S. flights. Heavier bags will be carried but at an additional fee. Suppose that one major airline has stated in an internal memo to employees that the mean weight for bags checked last year on the www.downloadslide.com

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Introduction to Sampling Distributions b. Use Excel to select a simple random sample of 30 executive compensation amounts. Compute the sample mean for this sample. Find the probability of getting a sample mean as extreme or more extreme than the one you got. Hint: Use the finite population correction factor because the sample is large relative to the size of the population. 7-43. The data file called CEO Compensation contains data for the most highly paid CEOs in the nation. Separate the values into those from 2010 and those from 2011. Treat the values found for 2010 data as population values. Assume the value found for the mean of the 2011 data is a sample from the 2010 population. What is the probability of finding a value this large or larger 7-44. The file Salaries contains the annual salary for all faculty at a small state college in the Midwest. Assume that these faculty salaries represent the population of interest. a. Compute the population mean and population standard deviation. b. Develop a frequency distribution of these data using 10 classes. Do the population data appear to be normally distributed c. What is the probability that a random sample of 16 faculty selected from the population would have a sample mean annual salary greater than or equal to 56650 d. Suppose the following 25 faculty were randomly sampled from the population and used to estimate the population mean annual salary: Faculty ID Number 137 040 054 005 064 134 013 199 168 027 095 065 193 059 192 084 176 029 143 182 009 033 152 068 044 What would the sampling error be e. Referring to part d what is the probability of obtaining a sample mean smaller than the one obtained from this sample 7-40. Acee Bottling and Distributing bottles and markets Pepsi-Cola products in southwestern Montana. The average fill volume for Pepsi cans is supposed to be 12 ounces. The filling machine has a known standard deviation of 0.05 ounces. Each week the company selects a simple random sample of 60 cans and carefully measures the volume in each can. The results of the latest sample are shown in the file called Acee Bottling. Based on the data in the sample what would you conclude about whether the filling process is working as expected Base your answer on the probability of observing the sample mean you compute for these sample data. 7-41. Bruce Leichtman is president of Leichtman Research Group Inc. LRG which specializes in research and consulting on broadband media and entertainment industries. In a recent survey the company determined the cost of extra high-definition HD gear needed to watch television in HD. The costs ranged from 5 a month for a set-top box to 200 for a new satellite. The file titled HDCosts contains a sample of the cost of the extras whose purchase was required to watch television in HD. Assume that the population average cost is 150 and the standard deviation is 50. a. Create a box and whisker plot and use it and the sample average to determine if the population from which this sample was obtained could be normally distributed. b. Determine the probability that the mean of a random sample of size 150 costs for HD extras would be more than 5 away from the mean of the sample described above. c. Given your response to part a do you believe the results obtained in part b are valid Explain. 7-42. CEO pay has been a controversial subject for several years and even came up in the 2012 presidential election. The file CEO Compensation contains data on the 100 top paid CEOs in either 2010 or 2011 depending on when the information was made available. a. Treating the data in the file as the population of interest compute the population mean and standard deviation for CEO compensation. END EXERCISES 7-2 3 Sampling Distribution of a Proportion Working with Proportions In many instances the objective of sampling is to estimate a population proportion. For instance an accountant may be interested in determining the proportion of accounts payable balances that are correct. A production supervisor may wish to determine the percentage of product that is defect- free. A marketing research department might want to know the proportion of potential customers who will purchase a particular product. In all these instances the decision makers could select a sample compute the sample proportion and make their decision based on the sample results. Sample proportions are subject to sampling error just as are sample means. The concept of sampling distributions provides us a way to assess the potential magnitude of the sampling error for proportions in given situations. www.downloadslide.com

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Introduction to Sampling Distributions BUSINESS APPLICATION SAMPLING DISTRIBUTIONS FOR PROPORTIONS FLORIDA POWER AND LIGHT Customer service managers at Florida Power and Light surveyed every customer who had new power service installed in a suburb of Tampa during the month of September last year. The key question in the survey was “Are you satisfied with the service received” The population size was 80 customers. The number of customers who answered “Yes” to the question was 72. The value of interest in this example is the population proportion. Equation 6 is used to compute a population proportion. Population Proportion The fraction of values in a population that have a specific attribute. Population Proportion X N p 6 where: p Population proportion X Number of items in the population having the attribute of interest N Population size The proportion of customers in the population who are satisfied with the service by Flor- ida Power and Light is 72 80 p 090 . Therefore 90 of the population responded “Yes” to the survey question. This is the param- eter. It is a measurement taken from the population. It is the “true value.” Now suppose that Florida Power and Light wishes to do a follow-up survey for a simple random sample of n 20 of the same 80 customers. What fraction of this sample will be people who had previously responded “Yes” to the satisfaction question The answer depends on which sample is selected. There are many 3.5353 10 18 to be precise possible random samples of 20 that could be selected from 80 people. How- ever the company will select only one of these possible samples. At one extreme suppose the 20 people selected for the sample included all 8 who answered “No” to the satisfaction question and 12 others who answered “Yes.” The sample proportion is computed using Equation 7. Sample Proportion The fraction of items in a sample that have the attribute of interest. Sample Proportion p x n 5 7 where: p Sample proportion x Number of items in the sample with the attribute of interest n Sample size For the Florida Power and Light example the sample proportion of “Yes” responses is 12 20 060 . p55 The sample proportion of “Yes” responses is 0.60 whereas the population proportion is 0.90. The difference between the sample value and the population value is sampling error. Equation 8 is used to compute the sampling error involving a single proportion. www.downloadslide.com

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Introduction to Sampling Distributions Then for this extreme situation we get Sampling error 0.60 - 0.90- 0.30 If a sample on the other extreme had been selected and all 20 people came from the original list of 72 who had responded “Yes” the sample proportion would be 20 20 100 . 55 p For this sample the sampling error is Sampling error 1.00 - 0.90 0.10 Thus the range of sampling error in this example is from - 0.30 to 0.10. As with any sam- pling situation you can expect some sampling error. The sample proportion will probably not equal the population proportion because the sample selected will not be a perfect replica of the population. EXAMPLE 4 SAMPLING ERROR FOR A PROPORTION Hewlett-Packard–Compaq Merger In 2002 a proxy fight took place between the management of Hewlett-Packard HP and Walter Hewlett the son of one of HP’s founders over whether the merger between HP and Compaq should be approved. Each outstanding share of common stock was allocated one vote. After the vote in March 2002 the initial tally showed that the proportion of shares in the approval col- umn was 0.51.