10-1 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Chapter 10 Two-Sample Tests Statistics for Managers using Microsoft Excel 6 th Edition

10-2 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 The variances of two independent populations by testing the ratio of the two variances

10-3 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Two-Sample Tests Population Means, Independent Samples Population Means, Related Samples Population Variances Group 1 vs. Group 2 Same group before vs. after treatment Variance 1 vs. Variance 2 Examples: Population Proportions Proportion 1 vs. Proportion 2 DCOVA

10-4 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-5 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-6 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-7 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-8 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-9 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-10 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-11 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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)? DCOVA

10-12 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 ) DCOVA

10-13 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-14 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-15 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-16 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-17 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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) DCOVA

10-18 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall The test statistic for μ D is: Paired samples Where t STAT has n - 1 d.f. The Paired Difference Test: Finding t STAT DCOVA

10-19 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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. DCOVA

10-20 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall The confidence interval for μ D is Paired samples where The Paired Difference Confidence Interval DCOVA

10-21 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-22 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-23 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-24 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-25 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Two Population Proportions Population proportions The test statistic for π 1 – π 2 is a Z statistic: (continued) where DCOVA

10-26 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-27 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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) DCOVA

10-28 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 35 of 50 women indicated they would vote Yes Test at the.05 level of significance DCOVA

10-29 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 = 0.50 Women: p 2 = 35/50 = 0.70  The pooled estimate for the overall proportion is: Hypothesis Test Example: Two population Proportions (continued) DCOVA

10-30 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

10-31 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Confidence Interval for Two Population Proportions Population proportions The confidence interval for π 1 – π 2 is: DCOVA

10-32 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Testing for the Ratio Of Two Population Variances Tests for Two Population Variances F test statistic H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 * HypothesesF STAT S 1 2 / S 2 2 S 1 2 = Variance of sample 1 (the larger sample variance) n 1 = sample size of sample 1 S 2 2 = Variance of sample 2 (the smaller sample variance) n 2 = sample size of sample 2 n 1 –1 = numerator degrees of freedom n 2 – 1 = denominator degrees of freedom Where: DCOVA

10-33 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall The F critical value is found from the F table There are two degrees of freedom required: numerator and denominator The larger sample variance is always the numerator When In the F table, numerator degrees of freedom determine the column denominator degrees of freedom determine the row The F Distribution df 1 = n 1 – 1 ; df 2 = n 2 – 1 DCOVA

10-34 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Finding the Rejection Region H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 F 0  FαFα Reject H 0 Do not reject H 0 Reject H 0 if F STAT > F α F 0  /2 Reject H 0 Do not reject H 0 F α/2 Reject H 0 if F STAT > F α/2 DCOVA

10-35 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall F Test: An Example You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data : NYSE NASDAQ Number 2125 Mean Std dev Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.05 level? DCOVA

10-36 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall F Test: Example Solution Form the hypothesis test: H 0 : σ 2 1 = σ 2 2 ( there is no difference between variances) H 1 : σ 2 1 ≠ σ 2 2 ( there is a difference between variances) Find the F critical value for  = 0.05: Numerator d.f. = n 1 – 1 = 21 –1 =20 Denominator d.f. = n2 – 1 = 25 –1 = 24 F α/2 = F.025, 20, 24 = 2.33 DCOVA

10-37 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall The test statistic is: 0  /2 =.025 F =2.33 Reject H 0 Do not reject H 0 H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 F Test: Example Solution F STAT = is not in the rejection region, so we do not reject H 0 (continued) Conclusion: There is not sufficient evidence of a difference in variances at  =.05 F DCOVA

10-38 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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

10-39 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Chapter Summary Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions Performed F test for the ratio of two population variances (continued)

10-40 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 the prior written permission of the publisher. Printed in the United States of America.