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Statistics for Business and Economics 6th Edition Chapter 11 Hypothesis Testing II Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-1 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 11-2 Two Sample Tests Two Sample Tests Population Means, Matched Pairs Population Means, Independent Samples Population Proportions Population Variances Examples: Same group before vs. after treatment Group 1 vs. independent Group 2 Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 (Note similarities to Chapter 9) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-3 Matched Pairs Tests Means of 2 Related Populations Matched Pairs Paired or matched samples Repeated measures (before/after) Use difference between paired values: di = xi - yi Assumptions: Both Populations Are Normally Distributed Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-4 Test Statistic: Matched Pairs Matched Pairs The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: d D0 t sd n Where D0 = hypothesized mean difference sd = sample standard dev. of differences n = the sample size (number of pairs) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-5 Decision Rules: Matched Pairs Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μx – μy 0 H1: μx – μy < 0 H0: μx – μy ≤ 0 H1: μx – μy > 0 H0: μx – μy = 0 H1: μx – μy ≠ 0 a a -ta ta Reject H0 if t < -tn-1, a Where Reject H0 if t > tn-1, a t d D0 sd n a/2 -ta/2 a/2 ta/2 Reject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 has n - 1 d.f. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-6 Matched Pairs Example 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: di d = n Number of Complaints: (2) - (1) Salesperson C.B. T.F. M.H. R.K. M.O. Before (1) After (2) Difference, di 6 20 3 0 4 4 6 2 0 0 - 2 -14 - 1 0 - 4 -21 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. = - 4.2 Sd 2 (d d ) i n 1 5.67 Chap 11-7 Matched Pairs: Solution Has the training made a difference in the number of complaints (at the a = 0.01 level)? H0: μx – μy = 0 H1: μx – μy 0 a = .01 d = - 4.2 Critical Value = ± 4.604 d.f. = n - 1 = 4 Reject Reject a/2 a/2 - 4.604 4.604 - 1.66 Decision: Do not reject H0 (t stat is not in the reject region) Test Statistic: d D0 4.2 0 t 1.66 sd / n 5.67/ 5 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Conclusion: There is not a significant change in the number of complaints. Chap 11-8 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μx – μy 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 11-9 Difference Between Two Means (continued) Population means, independent samples σx2 and σy2 known Test statistic is a z value σx2 and σy2 unknown σx2 and σy2 assumed equal σx2 and σy2 assumed unequal Test statistic is a a value from the Student’s t distribution Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-10 σx2 and σy2 Known Population means, independent samples σx2 and σy2 known σx2 and σy2 unknown Assumptions: * Samples are randomly and independently drawn both population distributions are normal Population variances are known Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-11 σx2 and σy2 Known (continued) When σx2 and σy2 are known and both populations are normal, the variance of X – Y is Population means, independent samples 2 σx2 and σy2 known σx2 and σy2 unknown σ 2X Y * 2 σy σx nx ny …and the random variable Z (x y) (μX μY ) 2 σ 2x σ y nX nY has a standard normal distribution Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-12 Test Statistic, σx2 and σy2 Known Population means, independent samples σx2 and σy2 known The test statistic for μx – μy is: * σx2 and σy2 unknown Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. x y D0 z 2 σy σx nx ny 2 Chap 11-13 Hypothesis Tests for Two Population Means Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μx μy H1: μx < μy H0: μx ≤ μy H1: μx > μy H0: μx = μy H1: μx ≠ μy i.e., i.e., i.e., H0: μx – μy 0 H1: μx – μy < 0 H0: μx – μy ≤ 0 H1: μx – μy > 0 H0: μx – μy = 0 H1: μx – μy ≠ 0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-14 Decision Rules Two Population Means, Independent Samples, Variances Known Lower-tail test: Upper-tail test: Two-tail test: H0: μx – μy 0 H1: μx – μy < 0 H0: μx – μy ≤ 0 H1: μx – μy > 0 H0: μx – μy = 0 H1: μx – μy ≠ 0 a a -za Reject H0 if z < -za za Reject H0 if z > za Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. a/2 -za/2 a/2 za/2 Reject H0 if z < -za/2 or z > za/2 Chap 11-15 σx2 and σy2 Unknown, Assumed Equal Assumptions: Population means, independent samples Samples are randomly and independently drawn σx2 and σy2 known Populations are normally distributed σx2 and σy2 unknown σx2 and σy2 assumed equal * Population variances are unknown but assumed equal σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-16 σx2 and σy2 Unknown, Assumed Equal (continued) Forming interval estimates: Population means, independent samples The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ σx2 and σy2 known σx2 and σy2 unknown σx2 and σy2 assumed equal σx2 and σy2 assumed unequal * use a t value with (nx + ny – 2) degrees of freedom Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-17 Test Statistic, σx2 and σy2 Unknown, Equal The test statistic for μx – μy is: σx2 and σy2 unknown σx2 and σy2 assumed equal * t σx2 and σy2 assumed unequal x y μx μy 1 1 S n n y x 2 p Where t has (n1 + n2 – 2) d.f., and sp2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. (n x 1)s 2x (n y 1)s 2y nx ny 2 Chap 11-18 σx2 and σy2 Unknown, Assumed Unequal Assumptions: Population means, independent samples Samples are randomly and independently drawn σx2 and σy2 known Populations are normally distributed σx2 and σy2 unknown Population variances are unknown and assumed unequal σx2 and σy2 assumed equal σx2 and σy2 assumed unequal * Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-19 σx2 and σy2 Unknown, Assumed Unequal (continued) Forming interval estimates: Population means, independent samples The population variances are assumed unequal, so a pooled variance is not appropriate σx2 and σy2 known use a t value with degrees of freedom, where σx2 and σy2 unknown 2 σx2 and σy2 assumed equal σx2 and σy2 assumed unequal * Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. s2x s2y ( ) ( ) n y n x v 2 2 2 2 sy sx /(n y 1) /(n x 1) n y nx Chap 11-20 Test Statistic, σx2 and σy2 Unknown, Unequal σx2 and σy2 unknown The test statistic for μx – μy is: σx2 and σy2 assumed equal σx2 and σy2 assumed unequal * t (x y) D0 Where t has degrees of freedom: Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. σ σ nX nY 2 y 2 x v s2x s2y ( ) ( ) n y n x 2 2 s2 s2x /(n x 1) y /(n y 1) n nx y 2 Chap 11-21 Decision Rules Two Population Means, Independent Samples, Variances Unknown Lower-tail test: Upper-tail test: Two-tail test: H0: μx – μy 0 H1: μx – μy < 0 H0: μx – μy ≤ 0 H1: μx – μy > 0 H0: μx – μy = 0 H1: μx – μy ≠ 0 a a -ta Reject H0 if t < -tn-1, a ta Reject H0 if t > tn-1, a a/2 -ta/2 a/2 ta/2 Reject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 Where t has n - 1 d.f. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-22 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 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 average yield (a = 0.05)? Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-23 Calculating the Test Statistic The test statistic is: X X μ μ t 1 2 1 2 1 1 S n1 n2 2 p 3.27 2.53 0 1 1 1.5021 21 25 2 2 2 2 n 1 S n 1 S 21 1 1.30 25 1 1.16 1 2 2 S2 1 p (n1 1) (n2 1) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. (21 - 1) (25 1) 2.040 1.5021 Chap 11-24 Solution 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 Reject H0 .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Test Statistic: Decision: 3.27 2.53 t 2.040 Reject H0 at a = 0.05 1 1 Conclusion: 1.5021 21 25 There is evidence of a difference in means. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-25 Two Population Proportions Population proportions Goal: Test hypotheses for the difference between two population proportions, Px – Py Assumptions: Both sample sizes are large, nP(1 – P) > 9 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-26 Two Population Proportions (continued) Population proportions The random variable Z (pˆ x pˆ y ) (p x p y ) pˆ x (1 pˆ x ) pˆ y (1 pˆ y ) nx ny is approximately normally distributed Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-27 Test Statistic for Two Population Proportions The test statistic for H0: Px – Py = 0 is a z value: Population proportions z pˆ x pˆ y pˆ 0 (1 pˆ 0 ) pˆ 0 (1 pˆ 0 ) nx ny Where Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. pˆ 0 nxpˆ x nypˆ y nx ny Chap 11-28 Decision Rules: Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: px – py 0 H1: px – py < 0 H0: px – py ≤ 0 H1: px – py > 0 H0: px – py = 0 H1: px – py ≠ 0 a a -za Reject H0 if z < -za za Reject H0 if z > za Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. a/2 -za/2 a/2 za/2 Reject H0 if z < -za/2 or z > za/2 Chap 11-29 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 11-30 Example: Two Population Proportions (continued) The hypothesis test is: H0: PM – PW = 0 (the two proportions are equal) H1: PM – PW ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p̂M = 36/72 = .50 p̂ W = 31/50 = .62 Women: The estimate for the common overall proportion is: pˆ 0 nxpˆ x nypˆ y nx ny 72(36/72) 50(31/50) 67 .549 72 50 122 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-31 Example: Two Population Proportions (continued) The test statistic for PM – PW = 0 is: z pˆ pˆ W pˆ 0 (1 pˆ 0 ) pˆ 0 (1 pˆ 0 ) n1 n2 Reject H0 Reject H0 .025 .025 M -1.96 -1.31 1.96 .50 .62 .549 (1 .549) .549 (1 .549) Decision: Do not reject H0 72 50 Conclusion: There is not 1.31 Critical Values = ±1.96 For a = .05 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. significant evidence of a difference between men and women in proportions who will vote yes. Chap 11-32 Hypothesis Tests of one Population Variance Population Variance Goal: Test hypotheses about the population variance, σ2 If the population is normally distributed, 2 n1 (n 1)s σ2 2 follows a chi-square distribution with (n – 1) degrees of freedom Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-33 Confidence Intervals for the Population Variance (continued) Population Variance The test statistic for hypothesis tests about one population variance is χ 2 n 1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. (n 1)s σ 02 2 Chap 11-34 Decision Rules: Variance Population variance Lower-tail test: Upper-tail test: Two-tail test: H0: σ2 σ02 H1: σ2 < σ02 H0: σ2 ≤ σ02 H1: σ2 > σ02 H0: σ2 = σ02 H1: σ2 ≠ σ02 a a χ n21,a χn21,1a Reject H0 if χ 2 n 1 χ 2 n 1,1a Reject H0 if χ n21 χn21,a Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. a/2 a/2 χ n21,1a / 2 χn21,a / 2 Reject H0 if χ n21 χ n21,a / 2 or χn21 χn21,1a / 2 Chap 11-35 Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic Goal: Test hypotheses about two population variances H0: σx2 σy2 H1: σx2 < σy2 H0: σx2 ≤ σy2 H1: σx2 > σy2 H0: σx2 = σy2 H1: σx2 ≠ σy2 Lower-tail test Upper-tail test Two-tail test The two populations are assumed to be independent and normally distributed Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-36 Hypothesis Tests for Two Variances (continued) Tests for Two Population Variances F test statistic The random variable 2 x 2 y s /σ F s /σ 2 x 2 y Has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom Denote an F value with 1 numerator and 2 denominator degrees of freedom by Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-37 Test Statistic Tests for Two Population Variances The critical value for a hypothesis test about two population variances is s F s F test statistic 2 x 2 y where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-38 Decision Rules: Two Variances Use sx2 to denote the larger variance. H0: σx2 = σy2 H1: σx2 ≠ σy2 H0: σx2 ≤ σy2 H1: σx2 > σy2 a/2 a 0 Do not reject H0 Reject H0 Fnx 1,ny 1,α F Reject H0 if F Fnx 1,ny 1,α 0 Do not reject H0 F Reject H0 Fnx 1,ny 1,α / 2 rejection region for a twotail test is: Reject H0 if F Fnx 1,ny 1,α / 2 where sx2 is the larger of the two sample variances Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-39 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 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 = 0.10 level? Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-40 F Test: Example Solution Form the hypothesis test: H0: σx2 = σy2 (there is no difference between variances) H1: σx2 ≠ σy2 (there is a difference between variances) Find the F critical values for a = .10/2: Degrees of Freedom: Numerator (NYSE has the larger standard deviation): nx – 1 = 21 – 1 = 20 d.f. Fnx 1, ny 1, α / 2 F20 , 24 , 0.10/2 2.03 Denominator: ny – 1 = 25 – 1 = 24 d.f. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-41 F Test: Example Solution (continued) The test statistic is: H0: σx2 = σy2 H1: σx2 ≠ σy2 s 2x 1.30 2 F 2 1.256 2 s y 1.16 F = 1.256 is not in the rejection region, so we do not reject H0 a/2 = .05 Do not reject H0 Reject H0 F F20 , 24 , 0.10/2 2.03 Conclusion: There is not sufficient evidence of a difference in variances at a = .10 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-42 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 11-43 Two-Sample Tests in PHStat Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-44 Sample PHStat Output Input Output Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-45 Sample PHStat Output (continued) Input Output Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-46 Chapter Summary Compared two dependent samples (paired samples) Compared two independent samples Performed paired sample t test for the mean difference 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 11-47 Chapter Summary (continued) 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-48