Chap 11-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 11 Hypothesis Testing II Statistics for Business and Economics 6 th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-2 Chapter Goals After completing this chapter, you should be able to: Test hypotheses for the difference between two population means Two means, matched pairs Independent populations, population variances known Independent populations, population variances unknown but equal Complete a hypothesis test for the difference between two proportions (large samples) Use the chi-square distribution for tests of the variance of a normal distribution Use the F table to find critical F values Complete an F test for the equality of two variances

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-3 Two Sample Tests Population Means, Independent Samples Population Means, Matched Pairs Population Variances Group 1 vs. independent Group 2 Same group before vs. after treatment Variance 1 vs. Variance 2 Examples: Population Proportions Proportion 1 vs. Proportion 2 (Note similarities to Chapter 9)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-4 Matched Pairs Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Assumptions: Both Populations Are Normally Distributed Matched Pairs d i = x i - y i

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-5 The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: Test Statistic: Matched Pairs Where D 0 = hypothesized mean difference s d = sample standard dev. of differences n = the sample size (number of pairs) Matched Pairs

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-6 Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0 Paired Samples Decision Rules: Matched Pairs  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t n-1,  Reject H 0 if t > t n-1,  Reject H 0 if t < -t n-1,    or t > t n-1,  Where has n - 1 d.f.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-7 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: Matched Pairs 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-8 Has the training made a difference in the number of complaints (at the  = 0.01 level)? - 4.2d = H 0 : μ x – μ y = 0 H 1 : μ x – μ y  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. Matched Pairs: Solution Reject  /  =.01

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-9 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μ x – μ y Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Difference Between Two Means Population means, independent samples Test statistic is a z value Test statistic is a a value from the Student’s t distribution σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples σ x 2 and σ y 2 Known Assumptions:  Samples are randomly and independently drawn  both population distributions are normal  Population variances are known * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples …and the random variable has a standard normal distribution When σ x 2 and σ y 2 are known and both populations are normal, the variance of X – Y is (continued) * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 Known

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples Test Statistic, σ x 2 and σ y 2 Known * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown The test statistic for μ x – μ y is:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Hypothesis Tests for Two Population Means Lower-tail test: H 0 : μ x  μ y H 1 : μ x < μ y i.e., H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x ≤ μ y H 1 : μ x > μ y i.e., H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x = μ y H 1 : μ x ≠ μ y i.e., H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0 Two Population Means, Independent Samples

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two Population Means, Independent Samples, Variances Known Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 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 Decision Rules

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Equal Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed  Population variances are unknown but assumed equal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples (continued) Forming interval estimates:  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  use a t value with (n x + n y – 2) degrees of freedom * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal σ x 2 and σ y 2 Unknown, Assumed Equal

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap * Test Statistic, σ x 2 and σ y 2 Unknown, Equal σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ x – μ y is:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed  Population variances are unknown and assumed unequal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal (continued) Forming interval estimates:  The population variances are assumed unequal, so a pooled variance is not appropriate  use a t value with degrees of freedom, where σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 assumed unequal

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap * Test Statistic, σ x 2 and σ y 2 Unknown, Unequal σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Where t has degrees of freedom: The test statistic for μ x – μ y is:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0 Decision Rules  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t n-1,  Reject H 0 if t > t n-1,  Reject H 0 if t < -t n-1,    or t > t n-1,  Where t has n - 1 d.f. Two Population Means, Independent Samples, Variances Unknown

Statistics for Business and Economics, 6e © 2007 Pearson Education, 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 average yield (  = 0.05)?

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Calculating the Test Statistic The test statistic is:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two Population Proportions Goal: Test hypotheses for the difference between two population proportions, P x – P y Population proportions Assumptions: Both sample sizes are large, nP(1 – P) > 9

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two Population Proportions Population proportions (continued) The random variable is approximately normally distributed

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Test Statistic for Two Population Proportions Population proportions The test statistic for H 0 : P x – P y = 0 is a z value: Where

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Rules: Proportions Population proportions Lower-tail test: H 0 : p x – p y  0 H 1 : p x – p y < 0 Upper-tail test: H 0 : p x – p y ≤ 0 H 1 : p x – p y > 0 Two-tail test: H 0 : p x – p y = 0 H 1 : p x – p y ≠ 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 

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The hypothesis test is: H 0 : P M – P W = 0 (the two proportions are equal) H 1 : P M – P W ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: = 36/72 =.50 Women: = 31/50 =.62  The estimate for the common overall proportion is: Example: Two Population Proportions (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The test statistic for P M – P W = 0 is: Example: Two Population Proportions (continued) Decision: Do not reject H 0 Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes. Reject H 0 Critical Values = ±1.96 For  =.05

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Population Variance follows a chi-square distribution with (n – 1) degrees of freedom  Goal: Test hypotheses about the population variance, σ 2  If the population is normally distributed, Hypothesis Tests of one Population Variance

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Confidence Intervals for the Population Variance Population Variance The test statistic for hypothesis tests about one population variance is (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Rules: Variance Population variance Lower-tail test: H 0 : σ 2  σ 0 2 H 1 : σ 2 < σ 0 2 Upper-tail test: H 0 : σ 2 ≤ σ 0 2 H 1 : σ 2 > σ 0 2 Two-tail test: H 0 : σ 2 = σ 0 2 H 1 : σ 2 ≠ σ 0 2  /2  Reject H 0 if or

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 Two-tail test Lower-tail test Upper-tail test H 0 : σ x 2  σ y 2 H 1 : σ x 2 < σ y 2 H 0 : σ x 2 ≤ σ y 2 H 1 : σ x 2 > σ y 2  Goal: Test hypotheses about two population variances The two populations are assumed to be independent and normally distributed

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic The random variable Has an F distribution with (n x – 1) numerator degrees of freedom and (n y – 1) denominator degrees of freedom Denote an F value with 1 numerator and 2 denominator degrees of freedom by (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Test Statistic Tests for Two Population Variances F test statistic The critical value for a hypothesis test about two population variances is where F has (n x – 1) numerator degrees of freedom and (n y – 1) denominator degrees of freedom

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Rules: Two Variances rejection region for a two- tail test is: F 0  Reject H 0 Do not reject H 0 F0  /2 Reject H 0 Do not reject H 0 H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 H 0 : σ x 2 ≤ σ y 2 H 1 : σ x 2 > σ y 2 Use s x 2 to denote the larger variance. where s x 2 is the larger of the two sample variances

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Example: F Test 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  = 0.10 level?

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap F Test: Example Solution Form the hypothesis test: H 0 : σ x 2 = σ y 2 ( there is no difference between variances) H 1 : σ x 2 ≠ σ y 2 ( there is a difference between variances) Degrees of Freedom: Numerator (NYSE has the larger standard deviation): n x – 1 = 21 – 1 = 20 d.f. Denominator: n y – 1 = 25 – 1 = 24 d.f. Find the F critical values for  =.10/2:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The test statistic is:  /2 =.05 Reject H 0 Do not reject H 0 H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 F Test: Example Solution F = 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  =.10 F

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Sample Tests in EXCEL For paired samples (t test): Tools | data analysis… | t-test: paired two sample for means For independent samples: Independent sample Z test with variances known: Tools | data analysis | z-test: two sample for means For variances… F test for two variances: Tools | data analysis | F-test: two sample for variances

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Sample Tests in PHStat

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sample PHStat Output Input Output

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sample PHStat Output Input Output (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Chapter Summary Compared two dependent samples (paired samples) Performed paired sample t test for the mean difference Compared two independent samples Performed z test for the differences in two means Performed pooled variance t test for the differences in two means Compared two population proportions Performed z-test for two population proportions

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Chapter Summary Used the chi-square test for a single population variance Performed F tests for the difference between two population variances Used the F table to find F critical values (continued)