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Chapter 9:

Chapter 9 Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses

Learning Objectives:

Learning Objectives 1. Solve Hypothesis Testing Problems for Two Populations Mean Proportion Variance 2. Distinguish Independent & Related Populations 3. Explain the F Distribution

Thinking Challenge:

Thinking Challenge Who Gets Higher Grades: Males or Females? Which Programs Are Faster to Learn: Windows or DOS? How Would You Try to Answer These Questions?

Two Population Tests:

Two Population Tests

Testing Two Means:

Testing Two Means Independent Sampling & Paired Difference Experiments

Two Population Tests:

Two Population Tests

Independent & Related Populations:

Independent & Related Populations Independent Related

Independent & Related Populations:

Independent & Related Populations 1. Different Data Sources Unrelated Independent Independent Related

Independent & Related Populations:

Independent & Related Populations 1. Different Data Sources Unrelated Independent 1. Same Data Source Paired or Matched Repeated Measures (Before/After) Independent Related

Independent & Related Populations:

Independent & Related Populations 1. Different Data Sources Unrelated Independent 2. Use Difference Between the 2 Sample Means X1 -X2 1. Same Data Source Paired or Matched Repeated Measures (Before/After) Independent Related

Independent & Related Populations:

Two Independent Populations Examples:

Two Independent Populations Examples 1. An economist wishes to determine whether there is a difference in mean family income for households in 2 socioeconomic groups. 2. An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools.

Two Related Populations Examples:

Two Related Populations Examples 1. Nike wants to see if there is a difference in durability of 2 sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair. 2. An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before & after taking a GMAT review course.

Thinking Challenge:

Thinking Challenge 1. Miles per gallon ratings of cars before & after mounting radial tires 2. The life expectancy of light bulbs made in 2 different factories 3. Difference in hardness between 2 metals: one contains an alloy, one doesn’t 4. Tread life of two different motorcycle tires: one on the front, the other on the back Are They Independent or Paired?

Testing 2 Independent Means:

Testing 2 Independent Means

Two Population Tests:

Two Population Tests

Two Independent PopulationsHypotheses for Means:

Two Independent Populations Hypotheses for Means

Two Independent PopulationsHypotheses for Means:

Two Independent Populations Hypotheses for Means

Two Independent PopulationsHypotheses for Means:

Two Independent Populations Hypotheses for Means

Sampling Distribution:

Sampling Distribution

Sampling Distribution:

Sampling Distribution Population 1  1  1

Sampling Distribution:

Sampling Distribution Population 1 Population 2  1  1  2

Sampling Distribution:

Sampling Distribution Select simple random sample, size n 1 . Compute  X 1 Population 1 Population 2  1  1  2

Sampling Distribution:

Sampling Distribution Select simple random sample, size n 1 . Compute  X 1 Select simple random sample, size n 2 . Compute  X 2 Population 1 Population 2  1  1  2

Sampling Distribution:

Sampling Distribution Select simple random sample, size n 1 . Compute  X 1 Select simple random sample, size n 2 . Compute  X 2 Compute  X 1 -  X 2 for every pair of samples Population 1 Population 2  1  1  2

Sampling Distribution:

Sampling Distribution Select simple random sample, size n 1 . Compute  X 1 Select simple random sample, size n 2 . Compute  X 2 Compute  X 1 -  X 2 for every pair of samples Astronomical number of  X 1 -  X 2 values Population 1 Population 2  1  1  2

Sampling Distribution:

Sampling Distribution

Large-Sample Z Test for 2 Independent Means:

Large-Sample Z Test for 2 Independent Means

Two Population Tests:

Two Population Tests

Large-Sample Z Test for 2 Independent Means:

Large-Sample Z Test for 2 Independent Means

Large-Sample Z Test for 2 Independent Means:

Large-Sample Z Test for 2 Independent Means 1. Assumptions Independent, Random Samples Populations Are Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n1  30 & n2  30 )

A Useful Fact About Normal Variables:

A Useful Fact About Normal Variables If Y1 and Y2 are independent and normally distributed, then Y1+Y2 is normally distributed Mean SD So is Y1-Y2 Mean SD

Sample Distribution for Difference Between Means :

Sample Distribution for Difference Between Means

Large-Sample Z Test for 2 Independent Means:

Large-Sample Z Test Example:

Large-Sample Z Test Example You’re a financial analyst for Charles Schwab. You want to find out if there is a difference in dividend yield between stocks listed on NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 121 125 Mean 3.27 2.53 Std Dev 1.30 1.16 Is there a difference in average yield ( = .05)? © 1984-1994 T/Maker Co.

Large-Sample Z Test Solution:

Large-Sample Z Test Solution

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: Ha:   n1 = , n2 = Critical Value(s): Test Statistic: Decision: Conclusion:

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   n1 = , n2 = Critical Value(s): Test Statistic: Decision: Conclusion:

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 n1 = 121, n2 = 125 Critical Value(s): Test Statistic: Decision: Conclusion:

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 n1 = 121, n2 = 125 Critical Value(s): Test Statistic: Decision: Conclusion: z 0 1.96 -1.96 Reject H 0 Reject H 0 .025

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 n1 = 121, n2 = 125 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = .05 z 0 1.96 -1.96 .025 Reject H 0 Reject H 0 .025 z . . . . .      3 27 2 53 1 698 121 1 353 4 69 125

Large-Sample Z Test Solution:

Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 n1 = 121, n2 = 125 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = .05 There is Evidence of a Difference in Means z 0 1.96 -1.96 .025 Reject H 0 Reject H 0 .025 z . . . . .      3 27 2 53 1 698 121 1 353 4 69 125

Large-Sample Z Test Thinking Challenge:

You’re an economist for the Department of Education. You want to find out if there is a difference in spending per pupil between urban & rural high schools. You collect the following: Urban Rural Number 35 35 Mean $ 6,012 $ 5,832 Std Dev $ 602 $ 497 Is there any difference in population means ( = .10)? Large-Sample Z Test Thinking Challenge

Large-Sample Z Test Solution*:

Large-Sample Z Test Solution* H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .10 n1 = 35, n2 = 35 Critical Value(s): Test Statistic: Decision: Conclusion: Do Not Reject at  = .10 There is No Evidence of a Difference in Means z 0 1.645 -1.645 .05 Reject H 0 Reject H 0 .05 z .       6012 5832 0 602 35 497 35 1 36 2 2

Continued: Confidence Interval:

Continued: Confidence Interval What is the 90% confidence interval for the difference between the means?

Small-Sample t Test for 2 Independent Means:

Small-Sample t Test for 2 Independent Means

Two Population Tests:

Two Population Tests

Small-Sample t Test for 2 Independent Means:

Small-Sample t Test for 2 Independent Means 1. Tests Means of 2 Independent Populations Having Equal Variances 2. Assumptions Independent, Random Samples Both Populations Are Normally Distributed Population Variances Are Unknown But Assumed Equal

Sampling Distribution:

Sampling Distribution

Exercise:

Exercise Verify that as ns get big, expression on previous slide converges to that for large sample normal distribution

Small-Sample t Test Test Statistic:

Small-Sample t Test Test Statistic Hypothesized difference

Small-Sample t Test Example:

Small-Sample t Test Example You’re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Mean 3.27 2.53 Std Dev 1.30 1.16 Assuming normal populations, is there a difference in average yield ( = .05)? © 1984-1994 T/Maker Co.

Small-Sample t Test Solution:

Small-Sample t Test Solution

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: Ha:   df  Critical Value(s): Test Statistic: Decision: Conclusion:

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   df  Critical Value(s): Test Statistic: Decision: Conclusion:

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision: Conclusion:

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision: Conclusion:

Small-Sample t Test Solution:

Small-Sample t Test Solution

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision: Conclusion:

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = .05

Small-Sample t Test Solution:

Small-Sample t Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = .05 There is evidence of a difference in means

Small-Sample t Test Thinking Challenge:

You’re a research analyst for General Motors. Assuming equal variances, is there a difference in the average miles per gallon (mpg) of two car models ( = .05)? You collect the following: Sedan Van Number 15 11 Mean 22.00 20.27 Std Dev 4.77 3.64 Small-Sample t Test Thinking Challenge

Small-Sample t Test Solution*:

Small-Sample t Test Solution* H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  15 + 11 - 2 = 24 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  = .05 There is no evidence of a difference in means

Small-Sample t Test Solution*:

Small-Sample t Test Solution*

Paired-Sample t Test:

Paired-Sample t Test Paired Difference Experiments

Two Population Tests:

Two Population Tests

Paired-Sample t Test for Mean Difference:

Paired-Sample t Test for Mean Difference 1. Tests Means of 2 Related Populations Paired or Matched Repeated Measures (Before/After) 2. Eliminates Variation Among Subjects 3. Assumptions The Paired Differences Are Normally Distributed If Not Normal, Mean Can Be Approximated by Normal Distribution

Paired-Sample t Test Hypotheses:

Paired-Sample t Test Hypotheses Note: Di = X1i - X2i for ith observation Research Questions Hypothesis No Difference Any Difference Pop 1  Pop 2 Pop 1 < Pop 2 Pop 1  Pop 2 Pop 1 > Pop 2 H 0  D = 0  D  0  D  0 H 1  D  0  D < 0  D > 0

Paired-Sample t Test Data Collection Table:

Paired-Sample t Test Data Collection Table Observation Group 1 Group 2 Difference 1 x 11 x 21 D 1 = x 11 -x 21 2 x 12 x 22 D 2 = x 12 -x 22     i x 1i x 2i D i = x 1i - x 2i     n x 1n x 2n D n = x 1n - x 2n

Paired-Sample t Test Test Statistic:

Paired-Sample t Test Test Statistic Sample Mean Sample Standard Deviation t xD S n df n x D n S (Di - xD)2 n 0 D i i n D i n            1 1 1 1 D D D D D D

Paired-Sample t TestExample:

Paired-Sample t Test Example You work in Human Resources. You want to see if a training program is effective. You collect the following test score data: Name Before (1) After (2) Sam 85 94 Tamika 94 87 Brian 78 79 Mike 87 88 At the .10 level, was the training effective?

Computation Table:

Computation Table Observation Before After Difference Sam 85 94 -9 Tamika 94 87 7 Brian 78 79 -1 Mike 87 88 -1 Total - 4

Null HypothesisSolution:

Null Hypothesis Solution 1. Was the training effective? 2. Effective means ‘After’ > ‘Before’. 3. Statistically, this means A > B. 4. Rearranging terms gives 0 B - A. 5. Defining D = B - A & substituting into (4) gives 0 D or D . 6. The alternative hypothesis is Ha: D 0.

Paired-Sample t Test Solution:

Paired-Sample t Test Solution

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: Ha:  = df = Critical Value(s): Test Statistic: Decision: Conclusion:

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = df = Critical Value(s): Test Statistic: Decision: Conclusion:

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = .10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion:

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = .10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion: t 0 -1.6377 .10 Reject

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = .10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion: t 0 -1.6377 .10 Reject t x S n 0 D        D 1 0 6 53 4 306 . . D D

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = .10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion: Do Not Reject at  = .10 t 0 -1.6377 .10 Reject t x S n 0 D        D 1 0 6 53 4 306 . . D D

Paired-Sample t Test Solution:

Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  = .10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion: Do Not Reject at  = .10 There Is No Evidence Training Was Effective t 0 -1.6377 .10 Reject t x S n 0 D        D 1 0 6 53 4 306 . . D D

Paired-Sample t Test Thinking Challenge:

Paired-Sample t Test Thinking Challenge You’re a marketing research analyst. You want to compare a client’s calculator to a competitor’s. You sample 8 retail stores. At the .01 level, does your client’s calculator sell for less than their competitor’s (1) (2) Store Client Competitor 1 $ 10 $ 11 2 8 11 3 7 10 4 9 12 5 11 11 6 10 13 7 9 12 8 8 10

Paired-Sample t Test Solution*:

Paired-Sample t Test Solution* H0: D = 0 (D = 1 - 2) Ha: D < 0  = .01 df = 8 - 1 = 7 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = .01 There Is Evidence Client’s Brand (1) Sells for Less t 0 -2.998 .01 Reject t S n D        2 25 0 1 16 8 5 486 . . . x 0 D D D

Z Test for Differences in Two Proportions:

Z Test for Differences in Two Proportions

Two Population Tests:

Two Population Tests

Z Test for Difference in Two Proportions:

Z Test for Difference in Two Proportions

Z Test for Difference in Two Proportions:

Z Test for Difference in Two Proportions 1. Assumptions Populations Are Independent Populations Follow Binomial Distribution Normal Approximation Can Be Used (for each) Does Not Contain 0 or n

Sample Distribution for Difference Between Proportions :

Sample Distribution for Difference Between Proportions

Z Test for Difference in Two Proportions:

Z Test for Difference in Two Proportions 1. Assumptions Populations Are Independent Populations Follow Binomial Distribution Normal Approximation Can Be Used Does Not Contain 0 or n Z-Test Statistic for Two Proportions