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Statistics for Managers using Microsoft Excel 6th Global Edition Chapter 10 Two-Sample Tests Copyright ©2011 Pearson Education 10-1 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 Copyright ©2011 Pearson Education 10-2 Two-Sample Tests DCOVA Two-Sample Tests Population Means, Independent Samples Population Means, Related Samples Population Proportions Population Variances Examples: Group 1 vs. Group 2 Copyright ©2011 Pearson Education Same group before vs. after treatment Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 10-3 Difference Between Two Means DCOVA Population means, independent samples * σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2011 Pearson Education Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ1 – μ2 The point estimate for the difference is X1 – X2 10-4 Difference Between Two Means: Independent Samples DCOVA Population means, independent samples * Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population σ1 and σ2 unknown, assumed equal Use Sp to estimate unknown σ. Use a Pooled-Variance t test. σ1 and σ2 unknown, not assumed equal Use S1 and S2 to estimate unknown σ1 and σ2. Use a Separate-variance t test Copyright ©2011 Pearson Education 10-5 Hypothesis Tests for Two Population Means DCOVA Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 μ2 H1: μ1 < μ2 H0: μ1 ≤ μ2 H1: μ1 > μ2 H0: μ1 = μ2 H1: μ1 ≠ μ2 i.e., i.e., i.e., H0: μ1 – μ2 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 Copyright ©2011 Pearson Education 10-6 Hypothesis tests for μ1 – μ2 DCOVA Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 a a -ta Reject H0 if tSTAT < -ta Copyright ©2011 Pearson Education ta Reject H0 if tSTAT > ta a/2 -ta/2 a/2 ta/2 Reject H0 if tSTAT < -ta/2 or tSTAT > ta/2 10-7 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal DCOVA Assumptions: Population means, independent samples σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2011 Pearson Education 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 10-8 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal (continued) DCOVA • The pooled variance is: Population means, independent samples σ1 and σ2 unknown, assumed equal Sp 2 * n 1 1S 12 n 2 1S 2 2 (n 1 1) ( n 2 1) • The test statistic is: t STAT X1 X 2 μ1 μ 2 2 Sp σ1 and σ2 unknown, not assumed equal Copyright ©2011 Pearson Education 1 1 n n2 1 • Where tSTAT has d.f. = (n1 + n2 – 2) 10-9 Confidence interval for µ1 - µ2 with σ1 and σ2 unknown and assumed equal DCOVA Population means, independent samples σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2011 Pearson Education * The confidence interval for μ1 – μ2 is: X 1 X 2 t α /2 2 Sp 1 n n 2 1 1 Where tα/2 has d.f. = n1 + n2 – 2 10-10 Pooled-Variance t Test Example DCOVA 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 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming both populations are approximately normal with equal variances, is there a difference in mean yield (a = 0.05)? Copyright ©2011 Pearson Education 10-11 Pooled-Variance t Test Example: Calculating the Test Statistic (continued) H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) DCOVA The test statistic is: X 1 X 2 μ 1 μ 2 t 2 Sp S 2 p 1 1 n n2 1 n 1 1S 12 n 2 1S 2 (n 1 1) ( n 2 1) Copyright ©2011 Pearson Education 3.27 2.53 0 2.040 1 1 1 . 5021 25 21 2 21 11.30 2 25 11.16 (21 - 1) ( 25 1) 2 1.5021 10-12 Pooled-Variance t Test Example: Hypothesis Test Solution DCOVA H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) a = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 t 1 1 1 . 5021 21 25 Copyright ©2011 Pearson Education .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Test Statistic: 3.27 2.53 Reject H0 2.040 Decision: Reject H0 at a = 0.05 Conclusion: There is evidence of a difference in means. 10-13 Pooled-Variance t Test Example: Confidence Interval for µ1 - µ2 DCOVA Since we rejected H0 can we be 95% confident that µNYSE > µNASDAQ? 95% Confidence Interval for µNYSE - µNASDAQ X 1 X 2 t a /2 1 1 0 . 74 2 . 0154 0 . 3628 ( 0 . 009 , 1 . 471 ) S n1 n 2 2 p Since 0 is less than the entire interval, we can be 95% confident that µNYSE > µNASDAQ Copyright ©2011 Pearson Education 10-14 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown, not assumed equal DCOVA Assumptions: Population means, independent samples Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2011 Pearson Education * Population variances are unknown and cannot be assumed to be equal 10-15 Related Populations The Paired Difference Test DCOVA Tests Means of 2 Related Populations Related samples Paired or matched samples Repeated measures (before/after) Use difference between paired values: Di = X1i - X2i Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Or, if not Normal, use large samples Copyright ©2011 Pearson Education 10-16 Related Populations The Paired Difference Test DCOVA (continued) The ith paired difference is Di , where Related samples Di = X1i - X2i The point estimate for the paired difference D population mean μD is D : n D i i 1 n n The sample standard deviation is SD SD (Di D) 2 i 1 n 1 n is the number of pairs in the paired sample Copyright ©2011 Pearson Education 10-17 The Paired Difference Test: Finding tSTAT DCOVA Paired samples The test statistic for μD is: t STAT D μD SD n Copyright ©2011 Pearson Education Where tSTAT has n - 1 d.f. 10-18 The Paired Difference Test: Possible Hypotheses DCOVA Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μD 0 H1: μD < 0 H0: μD ≤ 0 H1: μD > 0 H0: μD = 0 H1: μD ≠ 0 a a -ta Reject H0 if tSTAT < -ta ta Reject H0 if tSTAT > ta Where tSTAT has n - 1 d.f. Copyright ©2011 Pearson Education a/2 -ta/2 a/2 ta/2 Reject H0 if tSTAT < -ta/2 or tSTAT > ta/2 10-19 The Paired Difference Confidence Interval Paired samples DCOVA The confidence interval for μD is D ta / 2 SD n n where Copyright ©2011 Pearson Education SD (D i D) 2 i 1 n 1 10-20 Paired Difference Test: Example DCOVA 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: Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, Di C.B. T.F. M.H. R.K. M.O. 6 20 3 0 4 4 6 2 0 0 - 2 -14 - 1 0 - 4 -21 D = Di n = -4.2 SD (Di D) 2 n 1 5.67 Copyright ©2011 Pearson Education 10-21 Paired Difference Test: DCOVA Solution Has the training made a difference in the number of complaints (at the 0.01 level)? H0: μD = 0 H1: μD 0 a = .01 D = - 4.2 t0.005 = ± 4.604 t STAT SD / n Copyright ©2011 Pearson Education 5.67/ 5 a/2 a/2 - 4.604 4.604 - 1.66 (tstat is not in the reject region) Test Statistic: 4.2 0 Reject Decision: Do not reject H0 d.f. = n - 1 = 4 D μD Reject 1.66 Conclusion: There is not a significant change in the number of complaints. 10-22 Two Population Proportions DCOVA Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π1 – π2 Population proportions Assumptions: n1 π1 5 , n1(1- π1) 5 n2 π2 5 , n2(1- π2) 5 The point estimate for the difference is Copyright ©2011 Pearson Education p1 p 2 10-23 Two Population Proportions DCOVA Population proportions In the null hypothesis we assume the null hypothesis is true, so we assume π1 = π2 and pool the two sample estimates The pooled estimate for the overall proportion is: p X1 X 2 n1 n 2 where X1 and X2 are the number of items of interest in samples 1 and 2 Copyright ©2011 Pearson Education 10-24 Two Population Proportions (continued) DCOVA The test statistic for π1 – π2 is a Z statistic: Population proportions Z STAT where Copyright ©2011 Pearson Education p p1 p 2 π1 π 2 1 1 p (1 p ) n1 n 2 X1 X 2 n1 n 2 , p1 X1 n1 , p2 X2 n2 10-25 Hypothesis Tests for Two Population Proportions DCOVA Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: π1 π2 H1: π1 < π2 H0: π1 ≤ π2 H1: π1 > π2 H0: π1 = π2 H1: π1 ≠ π2 i.e., i.e., i.e., H0: π1 – π2 0 H1: π1 – π2 < 0 H0: π1 – π2 ≤ 0 H1: π1 – π2 > 0 H0: π1 – π2 = 0 H1: π1 – π2 ≠ 0 Copyright ©2011 Pearson Education 10-26 Hypothesis Tests for Two Population Proportions (continued) Population proportions DCOVA Lower-tail test: Upper-tail test: Two-tail test: H0: π1 – π2 0 H1: π1 – π2 < 0 H0: π1 – π2 ≤ 0 H1: π1 – π2 > 0 H0: π1 – π2 = 0 H1: π1 – π2 ≠ 0 a a -za Reject H0 if ZSTAT < -Za Copyright ©2011 Pearson Education za Reject H0 if ZSTAT > Za a/2 -za/2 a/2 za/2 Reject H0 if ZSTAT < -Za/2 or ZSTAT > Za/2 10-27 Hypothesis Test Example: Two population Proportions DCOVA 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 Copyright ©2011 Pearson Education 10-28 Hypothesis Test Example: Two population Proportions (continued) DCOVA The hypothesis test is: H0: π1 – π2 = 0 (the two proportions are equal) H1: π1 – π2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p1 = 36/72 = 0.50 Women: p2 = 35/50 = 0.70 The pooled estimate for the overall proportion is: p Copyright ©2011 Pearson Education X1 X 2 n1 n 2 36 35 72 50 71 .582 122 10-29 Hypothesis Test Example: Two population Proportions DCOVA (continued) The test statistic for π1 – π2 is: z STAT p1 p 2 1 2 1 1 p (1 p ) n1 n 2 .50 .70 0 1 1 .582 (1 .582) 72 50 Critical Values = ±1.96 For a = .05 Copyright ©2011 Pearson Education Reject H0 Reject H0 .025 .025 -1.96 -2.20 2.20 1.96 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. 10-30 Confidence Interval for Two Population Proportions DCOVA Population proportions p1 p 2 Copyright ©2011 Pearson Education The confidence interval for π1 – π2 is: Z a /2 p 1 (1 p 1 ) n1 p 2 (1 p 2 ) n2 10-31 Testing for the Ratio Of Two Population Variances DCOVA Tests for Two Population Variances F test statistic * Hypotheses H0: σ12 = σ22 H1: σ12 ≠ σ22 H0: σ12 ≤ σ22 H1: σ12 > σ22 FSTAT S12 / S22 Where: S12 = Variance of sample 1 (the larger sample variance) n1 = sample size of sample 1 S22 = Variance of sample 2 (the smaller sample variance) n2 = sample size of sample 2 n1 –1 = numerator degrees of freedom n2 – 1 = denominator degrees of freedom Copyright ©2011 Pearson Education 10-32 The F Distribution DCOVA 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 2 When F STAT S1 S 2 2 df1 = n1 – 1 ; df2 = n2 – 1 In the F table, numerator degrees of freedom determine the column denominator degrees of freedom determine the row Copyright ©2011 Pearson Education 10-33 Finding the Rejection Region DCOVA H0: σ12 = σ22 H1: σ12 ≠ σ22 H0: σ12 ≤ σ22 H1: σ12 > σ22 a/2 0 Do not reject H0 Fα/2 Reject H0 Reject H0 if FSTAT > Fα/2 Copyright ©2011 Pearson Education a F 0 Do not reject H0 Fα Reject H0 F Reject H0 if FSTAT > Fα 10-34 F Test: An Example DCOVA 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 21 25 Mean 3.27 2.53 Std dev 1.30 1.16 Is there a difference in the variances between the NYSE & NASDAQ at the a = 0.05 level? Copyright ©2011 Pearson Education 10-35 F Test: Example Solution DCOVA Form the hypothesis test: H0: σ21 = σ22 (there is no difference between variances) H1: σ21 ≠ σ22 (there is a difference between variances) Find the F critical value for a = 0.05: Numerator d.f. = n1 – 1 = 21 –1 =20 Denominator d.f. = n2 – 1 = 25 –1 = 24 Fα/2 = F.025, 20, 24 = 2.33 Copyright ©2011 Pearson Education 10-36 F Test: Example Solution DCOVA (continued) The test statistic is: 2 F STAT S1 S 2 2 1 . 30 2 1 . 16 2 H0: σ12 = σ22 H1: σ12 ≠ σ22 1 . 256 a/2 = .025 0 Do not reject H0 FSTAT = 1.256 is not in the rejection region, so we do not reject H0 Conclusion: There is not sufficient evidence of a difference in variances at a = .05 Copyright ©2011 Pearson Education Reject H0 F F0.025=2.33 10-37 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 Copyright ©2011 Pearson Education 10-38 Chapter Summary (continued) 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 Copyright ©2011 Pearson Education 10-39 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. Copyright ©2011 Pearson Education 10-40