PowerPoint Presentation: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 7- 1 Chapter 7 Sampling and Sampling Distributions Basic Business Statistics 11 th Edition
Learning Objectives: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 2 Learning Objectives In this chapter, you learn: To distinguish between different sampling methods The concept of the sampling distribution To compute probabilities related to the sample mean and the sample proportion The importance of the Central Limit Theorem
Why Sample?: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 3 Why Sample? Selecting a sample is less time-consuming than selecting every item in the population (census). Selecting a sample is less costly than selecting every item in the population. An analysis of a sample is less cumbersome and more practical than an analysis of the entire population.
A Sampling Process Begins With A Sampling Frame: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 4 A Sampling Process Begins With A Sampling Frame The sampling frame is a listing of items that make up the population Frames are data sources such as population lists, directories, or maps Inaccurate or biased results can result if a frame excludes certain portions of the population Using different frames to generate data can lead to dissimilar conclusions
Types of Samples: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 5 Types of Samples Samples Non-Probability Samples Judgment Probability Samples Simple Random Systematic Stratified Cluster Convenience
Types of Samples: Nonprobability Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 6 Types of Samples: Nonprobability Sample In a nonprobability sample, items included are chosen without regard to their probability of occurrence. In convenience sampling , items are selected based only on the fact that they are easy, inexpensive, or convenient to sample. In a judgment sample, you get the opinions of pre-selected experts in the subject matter.
Types of Samples: Probability Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 7 Types of Samples: Probability Sample In a probability sample , items in the sample are chosen on the basis of known probabilities. Probability Samples Simple Random Systematic Stratified Cluster
Probability Sample: Simple Random Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 8 Probability Sample: Simple Random Sample Every individual or item from the frame has an equal chance of being selected Selection may be with replacement (selected individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame). Samples obtained from table of random numbers or computer random number generators.
Selecting a Simple Random Sample Using A Random Number Table: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 9 Selecting a Simple Random Sample Using A Random Number Table Sampling Frame For Population With 850 Items Item Name Item # Bev R. 001 Ulan X. 002 . . . . . . . . Joann P. 849 Paul F. 850 Portion Of A Random Number Table 49280 88924 35779 00283 81163 07275 11100 02340 12860 74697 96644 89439 09893 23997 20048 49420 88872 08401 The First 5 Items in a simple random sample Item # 492 Item # 808 Item # 892 -- does not exist so ignore Item # 435 Item # 779 Item # 002
Probability Sample: Systematic Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 10 Decide on sample size: n Divide frame of N individuals into groups of k individuals: k = N / n Randomly select one individual from the 1 st group Select every k th individual thereafter Probability Sample: Systematic Sample N = 40 n = 4 k = 10 First Group
PowerPoint Presentation: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 11
Probability Sample: Stratified Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 12 Probability Sample: Stratified Sample Divide population into two or more subgroups (called strata ) according to some common characteristic A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes Samples from subgroups are combined into one This is a common technique when sampling population of voters, stratifying across racial or socio-economic lines. Population Divided into 4 strata
Probability Sample Cluster Sample: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 13 Probability Sample Cluster Sample Population is divided into several “clusters,” each representative of the population A simple random sample of clusters is selected All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique A common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled. Population divided into 16 clusters. Randomly selected clusters for sample
Probability Sample: Comparing Sampling Methods: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 14 Probability Sample: Comparing Sampling Methods Simple random sample and Systematic sample Simple to use May not be a good representation of the population’s underlying characteristics Stratified sample Ensures representation of individuals across the entire population Cluster sample More cost effective Less efficient (need larger sample to acquire the same level of precision)
Evaluating Survey Worthiness: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 15 Evaluating Survey Worthiness What is the purpose of the survey? Is the survey based on a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit good responses Sampling error – always exists
Types of Survey Errors: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 16 Types of Survey Errors Coverage error or selection bias Exists if some groups are excluded from the frame and have no chance of being selected Non response error or bias People who do not respond may be different from those who do respond Sampling error Variation from sample to sample will always exist Measurement error Due to weaknesses in question design, respondent error, and interviewer’s effects on the respondent (“Hawthorne effect”)
Types of Survey Errors: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 17 Types of Survey Errors Coverage error Non response error Sampling error Measurement error Excluded from frame Follow up on nonresponses Random differences from sample to sample Bad or leading question (continued)
Sampling Distributions: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 18 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population. For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students.
Developing a Sampling Distribution: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 19 Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) A B C D
Developing a Sampling Distribution: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 20 .3 .2 .1 0 18 20 22 24 A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution
Now consider all possible samples of size n=2: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 21 16 possible samples (sampling with replacement) Now consider all possible samples of size n=2 (continued) Developing a Sampling Distribution 16 Sample Means 1 st Obs 2 nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24
Sampling Distribution of All Sample Means: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 22 Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0 .1 .2 .3 P(X) X Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) (no longer uniform) _
Summary Measures of this Sampling Distribution:: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 23 Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued)
Comparing the Population Distribution to the Sample Means Distribution: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 24 Comparing the Population Distribution to the Sample Means Distribution 18 19 20 21 22 23 24 0 .1 .2 .3 P(X) X 18 20 22 24 A B C D 0 .1 .2 .3 Population N = 4 P(X) X _ Sample Means Distribution n = 2 _
Sample Mean Sampling Distribution: Standard Error of the Mean: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 25 Sample Mean Sampling Distribution: Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) Note that the standard error of the mean decreases as the sample size increases
Sample Mean Sampling Distribution: If the Population is Normal: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 26 Sample Mean Sampling Distribution: If the Population is Normal If a population is normal with mean μ and standard deviation σ , the sampling distribution of is also normally distributed with and
Z-value for Sampling Distribution of the Mean: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 27 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of : where: = sample mean = population mean = population standard deviation n = sample size
Sampling Distribution Properties: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 28 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution Properties (i.e. is unbiased )
Sampling Distribution Properties: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 29 Sampling Distribution Properties As n increases, decreases Larger sample size Smaller sample size (continued)
Determining An Interval Including A Fixed Proportion of the Sample Means: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 30 Determining An Interval Including A Fixed Proportion of the Sample Means Find a symmetrically distributed interval around µ that will include 95% of the sample means when µ = 368, σ = 15, and n = 25. Since the interval contains 95% of the sample means 5% of the sample means will be outside the interval Since the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit. From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96.
Determining An Interval Including A Fixed Proportion of the Sample Means: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 31 Determining An Interval Including A Fixed Proportion of the Sample Means Calculating the lower limit of the interval Calculating the upper limit of the interval 95% of all sample means of sample size 25 are between 362.12 and 373.88 (continued)
Sample Mean Sampling Distribution: If the Population is not Normal: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 32 Sample Mean Sampling Distribution: If the Population is not Normal We can apply the Central Limit Theorem : Even if the population is not normal , …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution: and
Central Limit Theorem: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 33 n ↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population
Sample Mean Sampling Distribution: If the Population is not Normal: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 34 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation Larger sample size Smaller sample size Sample Mean Sampling Distribution: If the Population is not Normal (continued) Sampling distribution properties:
How Large is Large Enough?: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 35 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 For normal population distributions, the sampling distribution of the mean is always normally distributed
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 36 Example Suppose a population has mean μ = 8 and standard deviation σ = 3 . Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 37 Example Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation (continued)
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 38 Example Solution (continued): (continued) Z 7.8 8.2 -0.4 0.4 Sampling Distribution Standard Normal Distribution .1554 +.1554 Population Distribution ? ? ? ? ? ? ? ? ? ? ? ? Sample Standardize X
Population Proportions: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 39 Population Proportions π = the proportion of the population having some characteristic Sample proportion ( p ) provides an estimate of π : 0 ≤ p ≤ 1 p is approximately distributed as a normal distribution when n is large (assuming sampling with replacement from a finite population or without replacement from an infinite population)
Sampling Distribution of p: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 40 Sampling Distribution of p Approximated by a normal distribution if: where and (where π = population proportion) Sampling Distribution P( p s ) .3 .2 .1 0 0 . 2 .4 .6 8 1 p
Z-Value for Proportions: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 41 Z-Value for Proportions Standardize p to a Z value with the formula:
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 42 Example If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45? i.e.: if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ?
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 43 Example if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? (continued) Find : Convert to standardized normal:
Example: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 44 Example Z 0.45 1.44 0.4251 Standardize Sampling Distribution Standardized Normal Distribution if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? (continued) Use standardized normal table: P(0 ≤ Z ≤ 1.44) = 0.4251 0.40 0 p
Chapter Summary: Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7- 45 Chapter Summary Discussed probability and nonprobability samples Described four common probability samples Examined survey worthiness and types of survey errors Introduced sampling distributions Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion Calculated probabilities using sampling distributions