Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-1 Introduction to Statistics Chapter 10 Estimation and Hypothesis Testing for Two Population Parameters

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-2 Chapter Goals After completing this chapter, you should be able to: Test hypotheses or form interval estimates for two independent population means Standard deviations known Standard deviations unknown two means from paired samples the difference between two population proportions

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-3 Estimation for Two Populations Estimating two population values Population means, independent samples Paired samples Population proportions Group 1 vs. independent Group 2 Same group before vs. after treatment Proportion 1 vs. Proportion 2 Examples:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-4 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Goal: 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 *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-5 Independent Samples Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population Use the difference between 2 sample means Use z test or pooled variance t test *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-6 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 known Assumptions:  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are  30  Population standard deviations are known *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-7 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 …and the standard error of x 1 – x 2 is When σ 1 and σ 2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z-value… (continued) σ 1 and σ 2 known *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-8 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: σ 1 and σ 2 known (continued) *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-9 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, large samples Assumptions:  Samples are randomly and independently drawn  both sample sizes are  30  Population standard deviations are unknown *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-10 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, large samples Forming interval estimates:  use sample standard deviation s to estimate σ  the test statistic is a z value (continued) *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-11 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: σ 1 and σ 2 unknown, large samples (continued) *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-12 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples Assumptions:  populations are normally distributed  the populations have equal variances  samples are independent *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-13 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples Forming interval estimates:  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom (continued) *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-14 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples The pooled standard deviation is (continued) *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-15 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: σ 1 and σ 2 unknown, small samples (continued) Where t  /2 has (n 1 + n 2 – 2) d.f., and *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-16 Paired Samples 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 Paired samples d = x 1 - x 2

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-17 Paired Differences The i th paired difference is d i, where Paired samples d i = x 1i - x 2i The point estimate for the population mean paired difference is d : The sample standard deviation is n is the number of pairs in the paired sample

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-18 Paired Differences The confidence interval for d is Paired samples Where t  /2 has n - 1 d.f. and s d is: (continued) n is the number of pairs in the paired sample

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-19 Hypothesis Tests for the Difference Between Two Means Testing Hypotheses about μ 1 – μ 2 Use the same situations discussed already: Standard deviations known or unknown Sample sizes  30 or not  30

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-20 Hypothesis Tests for Two Population Proportions Lower tail test: H 0 : μ 1  μ 2 H A : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 ≤ μ 2 H A : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 = μ 2 H A : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-21 Hypothesis tests for μ 1 – μ 2 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Use a z test statistic Use s to estimate unknown σ, approximate with a z test statistic Use s to estimate unknown σ, use a t test statistic and pooled standard deviation

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-22 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The test statistic for μ 1 – μ 2 is: σ 1 and σ 2 known *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-23 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, large samples The test statistic for μ 1 – μ 2 is: *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-24 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples Where t  /2 has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 – μ 2 is: *

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-25 Two Population Means, Independent Samples Lower tail test: H 0 : μ 1 – μ 2  0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 – μ 2 ≤ 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z  /2  or z > z  /2 Hypothesis tests for μ 1 – μ 2

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-26 Pooled s p t Test: Example You’re 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 equal variances, is there a difference in average yield (  = 0.05)?

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-27 Calculating the Test Statistic The test statistic is:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-28 Solution H 0 : μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 ) H A : 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

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-29 The test statistic for d is Paired samples Where t  /2 has n - 1 d.f. and s d is: n is the number of pairs in the paired sample Hypothesis Testing for Paired Samples

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-30 Lower tail test: H 0 : μ d  0 H A : μ d < 0 Upper tail test: H 0 : μ d ≤ 0 H A : μ d > 0 Two-tailed test: H 0 : μ d = 0 H A : μ d ≠ 0 Paired Samples Hypothesis Testing for Paired Samples  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t  Reject H 0 if t > t  Reject H 0 if t < -t   or t > t  Where t has n - 1 d.f. (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-31 Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: Paired Samples 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

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-32  Has the training made a difference in the number of complaints (at the 0.01 level)? - 4.2d = H 0 : μ d = 0 H A :  μ d  0 Test Statistic: Critical Value = ± 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 Samples: Solution Reject  /  =.01

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-33 Two Population Proportions Goal: Form a confidence interval for or test a hypothesis about the difference between two population proportions, p 1 – p 2 The point estimate for the difference is p 1 – p 2 Population proportions Assumptions: n 1 p 1  5, n 1 (1-p 1 )  5 n 2 p 2  5, n 2 (1-p 2 )  5

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-34 Confidence Interval for Two Population Proportions Population proportions The confidence interval for p 1 – p 2 is:

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-35 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1  p 2 H A : p 1 < p 2 i.e., H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 ≤ p 2 H A : p 1 > p 2 i.e., H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 = p 2 H A : p 1 ≠ p 2 i.e., H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-36 Two Population Proportions Population proportions The pooled estimate for the overall proportion is: where x 1 and x 2 are the numbers from samples 1 and 2 with the characteristic of interest Since we begin by assuming the null hypothesis is true, we assume p 1 = p 2 and pool the two p estimates

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-37 Two Population Proportions Population proportions The test statistic for p 1 – p 2 is: (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-38 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z   or z > z 

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-39 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

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-40 The hypothesis test is: H 0 : p 1 – p 2 = 0 (the two proportions are equal) H A : p 1 – p 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: Example: Two population Proportions (continued)

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-41 The test statistic for p 1 – p 2 is: 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

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-42 Chapter Summary Compared two independent samples Formed confidence intervals for the differences between two means Performed Z test for the differences in two means Performed t test for the differences in two means Compared two related samples (paired samples) Formed confidence intervals for the paired difference Performed paired sample t tests for the mean difference Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions