Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 10-1 Chapter 2c Two-Sample Tests

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-2 Learning Objectives In this chapter, you learn: How to use hypothesis testing for comparing the difference between The means of two independent populations The means of two related populations The proportions of two independent populations

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-3 Two-Sample Tests Population Means, Independent Samples Population Means, Related Samples Group 1 vs. Group 2 Same group before vs. after treatment Examples: Population Proportions Proportion 1 vs. Proportion 2

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-4 Difference Between Two Means Population means, independent samples Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ 1 – μ 2 The point estimate for the difference is X 1 – X 2 * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-5 Difference Between Two Means: Independent Samples Population means, independent samples * Use S p to estimate unknown σ. Use a Pooled-Variance t test. σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal Use S 1 and S 2 to estimate unknown σ 1 and σ 2. Use a Separate-variance t test Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-6 Hypothesis Tests for Two Population Means Lower-tail test: H 0 : μ 1  μ 2 H 1 : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 H 1 : μ 1 – μ 2 < 0 Upper-tail test: H 0 : μ 1 ≤ μ 2 H 1 : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tail test: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-7 Two Population Means, Independent Samples Lower-tail test: H 0 : μ 1 – μ 2  0 H 1 : μ 1 – μ 2 < 0 Upper-tail test: H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tail test: H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0  /2  -t  -t  /2 tt t  /2 Reject H 0 if t STAT < -t  Reject H 0 if t STAT > t  Reject H 0 if t STAT < -t  /2  or t STAT > t  /2 Hypothesis tests for μ 1 – μ 2

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-8 Population means, independent samples Hypothesis tests for µ 1 - µ 2 with σ 1 and σ 2 unknown and assumed equal Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but assumed equal * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 10-9 Population means, independent samples The pooled variance is: The test statistic is: Where t STAT has d.f. = (n 1 + n 2 – 2) (continued) * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal Hypothesis tests for µ 1 - µ 2 with σ 1 and σ 2 unknown and assumed equal

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Population means, independent samples The confidence interval for μ 1 – μ 2 is: Where t α/2 has d.f. = n 1 + n 2 – 2 * Confidence interval for µ 1 - µ 2 with σ 1 and σ 2 unknown and assumed equal σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Pooled-Variance t Test Example You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number Sample mean Sample std dev Assuming both populations are approximately normal with equal variances, is there a difference in mean yield (  = 0.05)?

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Pooled-Variance t Test Example: Calculating the Test Statistic The test statistic is: (continued) H0: μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 ) H1: μ 1 - μ 2 ≠ 0 i.e. (μ 1 ≠ μ 2 )

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Pooled-Variance t Test Example: Hypothesis Test Solution H 0 : μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 ) H 1 : μ 1 - μ 2 ≠ 0 i.e. (μ 1 ≠ μ 2 )  = 0.05 df = = 44 Critical Values: t = ± Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a difference in means. t Reject H

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Pooled-Variance t Test Example: Confidence Interval for µ 1 - µ 2 Since we rejected H 0 can we be 95% confident that µ NYSE > µ NASDAQ ? 95% Confidence Interval for µ NYSE - µ NASDAQ Since 0 is less than the entire interval, we can be 95% confident that µ NYSE > µ NASDAQ

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Population means, independent samples Hypothesis tests for µ 1 - µ 2 with σ 1 and σ 2 unknown, not assumed equal Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown and cannot be assumed to be equal * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Population means, independent samples (continued) * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal Hypothesis tests for µ 1 - µ 2 with σ 1 and σ 2 unknown and not assumed equal The test statistic is: t STAT has d.f. ν =

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Related Populations The Paired Difference Test Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Or, if not Normal, use large samples Related samples D i = X 1i - X 2i

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Related Populations The Paired Difference Test The i th paired difference is D i, where Related samples D i = X 1i - X 2i The point estimate for the paired difference population mean μ D is D : n is the number of pairs in the paired sample The sample standard deviation is S D (continued)

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap The test statistic for μ D is: Paired samples Where t STAT has n - 1 d.f. The Paired Difference Test: Finding t STAT

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Lower-tail test: H 0 : μ D  0 H 1 : μ D < 0 Upper-tail test: H 0 : μ D ≤ 0 H 1 : μ D > 0 Two-tail test: H 0 : μ D = 0 H 1 : μ D ≠ 0 Paired Samples The Paired Difference Test: Possible Hypotheses  /2  -t  -t  /2 tt t  /2 Reject H 0 if t STAT < -t  Reject H 0 if t STAT > t  Reject H 0 if t STAT < -t   or t STAT > t  Where t STAT has n - 1 d.f.

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap The confidence interval for μ D is Paired samples where The Paired Difference Confidence Interval

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data: Paired Difference Test: Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, D i C.B T.F M.H R.K M.O D =  DiDi n = -4.2

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Has the training made a difference in the number of complaints (at the 0.01 level)? - 4.2D = H 0 : μ D = 0 H 1 :  μ D  0 Test Statistic: t = ± d.f. = n - 1 = 4 Reject  / Decision: Do not reject H 0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints. Paired Difference Test: Solution Reject  /  =.01

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Two Population Proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π 1 – π 2 The point estimate for the difference is Population proportions Assumptions: n 1 π 1  5, n 1 (1- π 1 )  5 n 2 π 2  5, n 2 (1- π 2 )  5

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Two Population Proportions Population proportions The pooled estimate for the overall proportion is: where X 1 and X 2 are the number of items of interest in samples 1 and 2 In the null hypothesis we assume the null hypothesis is true, so we assume π 1 = π 2 and pool the two sample estimates

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Two Population Proportions Population proportions The test statistic for π 1 – π 2 is a Z statistic: (continued) where

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : π 1  π 2 H 1 : π 1 < π 2 i.e., H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 ≤ π 2 H 1 : π 1 > π 2 i.e., H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 = π 2 H 1 : π 1 ≠ π 2 i.e., H 0 : π 1 – π 2 = 0 H 1 : π 1 – π 2 ≠ 0

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 – π 2 = 0 H 1 : π 1 – π 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if Z STAT < -Z  Reject H 0 if Z STAT > Z  Reject H 0 if Z STAT < -Z   or Z STAT > Z  (continued)

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Hypothesis Test Example: Two population Proportions Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes Test at the.05 level of significance

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap The hypothesis test is: H 0 : π 1 – π 2 = 0 (the two proportions are equal) H 1 : π 1 – π 2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p 1 = 36/72 =.50 Women: p 2 = 31/50 =.62  The pooled estimate for the overall proportion is: Hypothesis Test Example: Two population Proportions (continued)

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap The test statistic for π 1 – π 2 is: Hypothesis Test Example: Two population Proportions (continued) Decision: Do not reject H 0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. Reject H 0 Critical Values = ±1.96 For  =.05

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Confidence Interval for Two Population Proportions Population proportions The confidence interval for π 1 – π 2 is:

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Chapter Summary Compared two independent samples Performed pooled-variance t test for the difference in two means Performed separate-variance t test for difference in two means Formed confidence intervals for the difference between two means Compared two related samples (paired samples) Performed paired t test for the mean difference Formed confidence intervals for the mean difference

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap Chapter Summary Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions (continued)