Statistics for Business and Economics 6 th Edition
Statistics for Business and Economics 6 th Edition Chapter 11 Hypothesis Testing II Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 11 -1
Chapter Goals After completing this chapter, you should be able to: n Test hypotheses for the difference between two population means n Two means, matched pairs n Independent populations, population variances known n 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 n Use the F table to find critical F values n Complete an F test for the equality of two variances Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 2
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, 6 e © 2007 Pearson Education, Inc. 3
Matched Pairs Tests Means of 2 Related Populations Matched Pairs n n n Paired or matched samples Repeated measures (before/after) Use difference between paired values: d i = x i - yi n Assumptions: n Both Populations Are Normally Distributed Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 4
Test Statistic: Matched Pairs The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: Where D 0 = hypothesized mean difference sd = sample standard dev. of differences n = the sample size (number of pairs) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 5
Decision Rules: Matched Pairs Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H 0: μ x – μ y 0 H 1: μ x – μ y < 0 H 0: μ x – μ y ≤ 0 H 1: μ x – μ y > 0 H 0: μ x – μ y = 0 H 1: μ x – μ y ≠ 0 -t t Reject H 0 if t < -tn-1, Where Reject H 0 if t > tn-1, /2 -t /2 Reject H 0 if t < -tn-1 , /2 or t > tn-1 , /2 has n - 1 d. f. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 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) n Salesperson C. B. T. F. M. H. R. K. M. O. Before (1) 6 20 3 0 4 After (2) 4 6 2 0 0 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Difference, di - 2 -14 - 1 0 - 4 -21 = - 4. 2 7
Matched Pairs: Solution n Has the training made a difference in the number of complaints (at the = 0. 01 level)? H 0: μ x – μ y = 0 H 1: μ x – μ y 0 =. 01 d = - 4. 2 Critical Value = ± 4. 604 d. f. = n - 1 = 4 Test Statistic: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Reject /2 - 4. 604 - 1. 66 4. 604 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. 8
Difference Between Two Means Population means, independent samples n Goal: Form a confidence interval for the difference between two population means, μx – μy Different data sources n Unrelated n Independent n Sample selected from one population has no effect on the sample selected from the other population Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 9
Difference Between Two Means (continued) Population means, independent samples σx 2 and σy 2 known Test statistic is a z value σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal Test statistic is a a value from the Student’s t distribution Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 10
σx 2 and σy 2 Known Population means, independent samples σx 2 and σy 2 known σx 2 and σy 2 unknown Assumptions: * § Samples are randomly and independently drawn § both population distributions are normal § Population variances are known Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 11
σx 2 and σy 2 Known (continued) When σx 2 and σy 2 are known and both populations are normal, the variance of X – Y is Population means, independent samples σx 2 and σy 2 known σx 2 and σy 2 unknown * …and the random variable has a standard normal distribution Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 12
Test Statistic, σx 2 and σy 2 Known Population means, independent samples σx 2 and σy 2 known The test statistic for μx – μy is: * σx 2 and σy 2 unknown Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 13
Hypothesis Tests for Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H 0: μ x μ y H 1: μ x < μ y H 0: μ x ≤ μ y H 1: μ x > μ y 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 H 0: μ x – μ y ≤ 0 H 1: μ x – μ y > 0 H 0: μ x – μ y = 0 H 1: μ x – μ y ≠ 0 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 14
Decision Rules Two Population Means, Independent Samples, Variances Known Lower-tail test: Upper-tail test: Two-tail test: H 0: μ x – μ y 0 H 1: μ x – μ y < 0 H 0: μ x – μ y ≤ 0 H 1: μ x – μ y > 0 H 0: μ x – μ y = 0 H 1: μ x – μ y ≠ 0 -z Reject H 0 if z < -z z Reject H 0 if z > z Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 15
σx 2 and σy 2 Unknown, Assumed Equal Assumptions: Population means, independent samples § Samples are randomly and independently drawn σx 2 and σy 2 known § Populations are normally distributed σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal * § Population variances are unknown but assumed equal σx 2 and σy 2 assumed unequal Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 16
σx 2 and σy 2 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 σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * § use a t value with (nx + ny – 2) degrees of freedom Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 17
Test Statistic, σx 2 and σy 2 Unknown, Equal The test statistic for μx – μy is: σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal * σx 2 and σy 2 assumed unequal Where t has (n 1 + n 2 – 2) d. f. , and Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 18
σx 2 and σy 2 Unknown, Assumed Unequal Assumptions: Population means, independent samples § Samples are randomly and independently drawn σx 2 and σy 2 known § Populations are normally distributed σx 2 and σy 2 unknown § Population variances are unknown and assumed unequal σx 2 and σy 2 assumed unequal * Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 19
σx 2 and σy 2 Unknown, Assumed Unequal (continued) Forming interval estimates: Population means, independent samples § The population variances are assumed unequal, so a pooled variance is not appropriate σx 2 and σy 2 known § use a t value with degrees of freedom, where σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 20
Test Statistic, σx 2 and σy 2 Unknown, Unequal The test statistic for μx – μy is: σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * Where t has degrees of freedom: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 21
Decision Rules Two Population Means, Independent Samples, Variances Unknown Lower-tail test: Upper-tail test: Two-tail test: H 0: μ x – μ y 0 H 1: μ x – μ y < 0 H 0: μ x – μ y ≤ 0 H 1: μ x – μ y > 0 H 0: μ x – μ y = 0 H 1: μ x – μ y ≠ 0 -t Reject H 0 if t < -tn-1, t Reject H 0 if t > tn-1, /2 -t /2 Reject H 0 if t < -tn-1 , /2 or t > tn-1 , /2 Where t has n - 1 d. f. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 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 ( = 0. 05)? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 23
Calculating the Test Statistic The test statistic is: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 24
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 = 21 + 25 - 2 = 44 Critical Values: t = ± 2. 0154 Test Statistic: Reject H 0 . 025 -2. 0154 Reject H 0 . 025 0 2. 0154 t 2. 040 Decision: Reject H 0 at = 0. 05 Conclusion: There is evidence of a difference in means. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 25
Two Population Proportions Population proportions Goal: Test hypotheses for the difference between two population proportions, Px – Py Assumptions: Both sample sizes are large, n. P(1 – P) > 9 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 26
Two Population Proportions (continued) Population proportions n The random variable is approximately normally distributed Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 27
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, 6 e © 2007 Pearson Education, Inc. 28
Decision Rules: Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H 0: p x – p y 0 H 1: p x – p y < 0 H 0: p x – p y ≤ 0 H 1: p x – p y > 0 H 0: p x – p y = 0 H 1: p x – p y ≠ 0 -z Reject H 0 if z < -z z Reject H 0 if z > z Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 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? n n 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, 6 e © 2007 Pearson Education, Inc. 30
Example: Two Population Proportions (continued) The hypothesis test is: n H 0: PM – PW = 0 (the two proportions are equal) H 1: PM – PW ≠ 0 (there is a significant difference between proportions) n The sample proportions are: n Men: = 36/72 =. 50 n Women: = 31/50 =. 62 § The estimate for the common overall proportion is: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 31
Example: Two Population Proportions (continued) The test statistic for PM – PW = 0 is: Reject H 0 . 025 -1. 96 -1. 31 1. 96 Decision: Do not reject H 0 Critical Values = ± 1. 96 For =. 05 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes. 32
Hypothesis Tests of one Population Variance § Goal: Test hypotheses about the population variance, σ2 § If the population is normally distributed, follows a chi-square distribution with (n – 1) degrees of freedom Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 33
Confidence Intervals for the Population Variance (continued) Population Variance The test statistic for hypothesis tests about one population variance is Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 34
Decision Rules: Variance Population variance Lower-tail test: Upper-tail test: Two-tail test: H 0: σ 2 σ 02 H 1: σ 2 < σ 02 H 0: σ 2 ≤ σ 02 H 1: σ 2 > σ 02 H 0: σ 2 = σ 02 H 1: σ 2 ≠ σ 02 Reject H 0 if /2 Reject H 0 if or Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 35
Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic § Goal: Test hypotheses about two population variances H 0: σx 2 σy 2 H 1: σx 2 < σy 2 H 0: σx 2 ≤ σy 2 H 1: σx 2 > σy 2 H 0: σx 2 = σy 2 H 1: σx 2 ≠ σy 2 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, 6 e © 2007 Pearson Education, Inc. 36
Hypothesis Tests for Two Variances (continued) Tests for Two Population Variances F test statistic The random variable 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, 6 e © 2007 Pearson Education, Inc. 37
Test Statistic Tests for Two Population Variances The critical value for a hypothesis test about two population variances is F test statistic where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 38
Decision Rules: Two Variances Use sx 2 to denote the larger variance. H 0: σx 2 = σy 2 H 1: σx 2 ≠ σy 2 H 0: σx 2 ≤ σy 2 H 1: σx 2 > σy 2 /2 0 Do not reject H 0 Reject H 0 F rejection region for a twotail test is: n where sx 2 is the larger of the two sample variances Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 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 between the NYSE at the = 0. 10 level? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. variances & NASDAQ 40
F Test: Example Solution n n Form the hypothesis test: H 0: σx 2 = σy 2 (there is no difference between variances) H 1: σx 2 ≠ σy 2 (there is a difference between variances) Find the F critical values for =. 10/2: Degrees of Freedom: n Numerator (NYSE has the larger standard deviation): n n nx – 1 = 21 – 1 = 20 d. f. Denominator: n ny – 1 = 25 – 1 = 24 d. f. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 41
F Test: Example Solution (continued) n The test statistic is: H 0 : σx 2 = σy 2 H 1 : σx 2 ≠ σy 2 /2 =. 05 n n F = 1. 256 is not in the rejection region, so we do not reject H 0 Do not reject H 0 Reject H 0 F Conclusion: There is not sufficient evidence of a difference in variances at =. 10 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 42
Two-Sample Tests in EXCEL For paired samples (t test): n Tools | data analysis… | t-test: paired two sample for means For independent samples: n Independent sample Z test with variances known: n Tools | data analysis | z-test: two sample for means For variances… n F test for two variances: n Tools | data analysis | F-test: two sample for variances Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 43
Two-Sample Tests in PHStatistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 44
Sample PHStat Output Input Output Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 45
Sample PHStat Output (continued) Input Output Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 46
Chapter Summary n Compared two dependent samples (paired samples) n n Compared two independent samples n n n 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 n Performed z-test for two population proportions Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 47
Chapter Summary (continued) n n n 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, 6 e © 2007 Pearson Education, Inc. 48
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