Statistics for Business and Economics 8 th Edition
Statistics for Business and Economics 8 th Edition Chapter 10 Hypothesis Testing: Additional Topics Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 n Independent populations, population variances unknown but equal Complete a hypothesis test for the difference between two proportions (large samples) n Use the F table to find critical F values n Complete an F test for the equality of two variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -2
Two Sample Tests Population Means, Dependent Samples Population Means, Independent Samples Population Proportions Population Variances Examples: Same group before vs. after treatment Group 1 vs. independent Group 2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 Ch. 10 -3
Dependent Samples 10. 1 Dependent Samples Tests of the Difference Between Two Normal Population Means: Dependent Samples Tests Means of 2 Related Populations 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -4
Test Statistic: Dependent Samples Population Means, Dependent Samples The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: For tests of the following form: H 0: μ x – μ y 0 H 0: μ x – μ y ≤ 0 H 0: μ x – μ y = 0 where sd = sample standard dev. of differences n = the sample size (number of pairs) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -5
Decision Rules: Matched Pairs Matched or 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. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Difference, di - 2 -14 - 1 0 - 4 -21 = - 4. 2 Ch. 10 -7
Matched Pairs: Solution n Has the training made a difference in the number of complaints (at the = 0. 05 level)? H 0: μ x – μ y = 0 H 1: μ x – μ y 0 =. 05 d = - 4. 2 Critical Value = ± 2. 776 d. f. = n − 1 = 4 Test Statistic: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Reject /2 - 2. 776 - 1. 66 2. 776 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. Ch. 10 -8
Independent Samples 10. 2 Population means, independent samples n Tests of the Difference Between Two Normal Population Means: Dependent Samples Goal: Form a confidence interval for the difference between two population means, μx – μy Different populations n Unrelated n Independent n n Sample selected from one population has no effect on the sample selected from the other population Normally distributed Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Test statistic is a a value from the Student’s t distribution Ch. 10 -10
σx 2 and σy 2 Known Population means, independent samples σx 2 and σy 2 known Assumptions: * σx 2 and σy 2 unknown § Samples are randomly and independently drawn § both population distributions are normal § Population variances are known Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -16
σx 2 and σy 2 Unknown, Assumed Equal (continued) 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 * § use a t value with (nx + ny – 2) degrees of freedom σx 2 and σy 2 assumed unequal Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -17
Test Statistic, σx 2 and σy 2 Unknown, Equal The test statistic for H 0 : μx – μy = 0 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -18
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 < -t (n 1+n 2 – 2), t Reject H 0 if t > t (n 1+n 2 – 2), Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall /2 -t /2 Reject H 0 if t < -t (n 1+n 2 – 2), /2 or t > t (n 1+n 2 – 2), /2 Ch. 10 -19
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)? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -20
Calculating the Test Statistic 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) The test statistic is: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -21
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 Reject H 0 . 025 -2. 0154 Reject H 0 . 025 0 2. 0154 t 2. 040 Test Statistic: Decision: Reject H 0 at = 0. 05 Conclusion: There is evidence of a difference in means. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -22
Newbold 10. 8 n A screening procedure - measure attitudes toward minorities n n n Independent random samples 151 males 108 female financial analysits for males n n high csores indicate negative attitudes how scores indicate positve attitudes samle mean. 85. 8, std dev: 19. 13 for females n samle mean. 71. 5, std dev: 12. 2
Newbold 10. 8 n n Test the null hypothesis that the two population meand are equal against the alternative that the true mean score is higher for male then for female financial analysts
Solution
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 * Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -26
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 * Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -27
Test Statistic, σx 2 and σy 2 Unknown, Unequal The test statistic for H 0: μx – μy = 0 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: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -28
10. 3 Population proportions Two Population Proportions Tests of the Difference Between Two Population Proportions (Large Samples) Goal: Test hypotheses for the difference between two population proportions, Px – Py Assumptions: Both sample sizes are large, n. P(1 – P) > 5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -29
Two Population Proportions (continued) Population proportions n The random variable has a standard normal distribution Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -30
Test Statistic for Two Population Proportions Population proportions The test statistic for H 0: P x – P y = 0 is a z value: Where Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -31
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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 Ch. 10 -32
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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -33
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: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -34
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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes. Ch. 10 -35
10. 4 Tests for Two Population Variances F test statistic Tests of Equality of Two Variances § 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 0 Do not reject H 0 F Reject H 0 rejection region for a twotail test is: n where sx 2 is the larger of the two sample variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 NYSE 0. 10 level? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall variances between & NASDAQ at the = Ch. 10 -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. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -42
Quiz 1 2013/2014 Spring n A political science professor is interested in comparing the characteritics of students who do and do not vote in national elections. For a random sample of 20 students who claimed to have voted in the last election she found a mean GPA 2. 71 and standard deviation of 0. 64. For an independent random sample of 40 students who did not vote the mean CPA was 2. 79 and standard deviation was 0. 56.
Quiz 1 2013/2014 Spring n n n Test the equity of variances of the two population at a level of 10%. Based on the results of your test apply an appropriate procedure when forming confidence intervals or hypothesis testing about population means. Form a 90% confidence interval for the differences between two population means. Test against two sided alternative, the null hypothesis that the mean populations are equal at a significance level of 10%. What is the critical value and critical reagion of the test. Interpret the result. State your conclusions for confidence interval and hypothesis test clearly.
Solution n n n n Ho: 2 x = 2 y, H 1 2 x 2 y, F = 0. 642/0. 562 = 1. 30 nx-1=19, ny-1=39 =0. 10 F 19, 39, 0. 05 = 2. 39 F < 2, 39 so fail to reject Hö Variances can be assumed to be equal
Solution n n n n Ho: x = y, H 1 x y, =0. 10, t = (2. 71 -2. 79)/sqrt(sp/20+sp/40) s 2 p=[(20 -1)0. 642 + (20 -1) 0. 562](20+40 -2) = 0. 0283 t = -0. 08/0. 168 = -0. 47 t 58, 0. 05 = 1. 67 -1. 67 < -0. 47 <1. 67 Fail to reject Hö cam be assumed equal
Some Comments on Hypothesis Testing n A test with low power can result from: n n n Small sample size Large variances in the underlying populations Poor measurement procedures If sample sizes are large it is possible to find significant differences that are not practically important Researchers should select the appropriate level of significance before computing p-values Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -47
Two-Sample Tests in EXCEL For paired samples (t test): n Data | data analysis | t-test: paired two sample for means For independent samples: n Independent sample z test with variances known: n Data | data analysis | z-test: two sample for means For variances… n F test for two variances: n Data | data analysis | F-test: two sample for variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -48
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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -49
Chapter Summary (continued) n n Performed F tests for the difference between two population variances Used the F table to find F critical values Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10 -50
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