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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7 th Edition Chapter 10 Hypothesis Testing: Additional Topics Ch. 10-1

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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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-2

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Two Sample Tests Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Two Sample Tests Population Means, Independent Samples Population Means, Dependent Samples 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 Ch. 10-3

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Dependent Samples 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Dependent Samples d i = x i - y i Ch. 10-4 10.1

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Test Statistic: Dependent Samples The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: where Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall D 0 = hypothesized mean difference s d = sample standard dev. of differences n = the sample size (number of pairs) Dependent Samples Ch. 10-5

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Decision Rules: Matched Pairs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Matched or Paired Samples /2 -t -t /2 tt 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. Ch. 10-6

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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: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Matched Pairs Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, d i C.B. 6 4 - 2 T.F. 20 6 -14 M.H. 3 2 - 1 R.K. 0 0 0 M.O. 4 0 - 4 -21 d = didi n = - 4.2 Ch. 10-7

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Has the training made a difference in the number of complaints (at the = 0.05 level)? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall - 4.2d = H 0 : μ x – μ y = 0 H 1 : μ x – μ y 0 Test Statistic: Critical Value = ± 2.776 d.f. = n − 1 = 4 Reject /2 - 2.776 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. Matched Pairs: Solution Reject /2 - 1.66 =.05 Ch. 10-8

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Difference Between Two Means Different populations Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population Normally distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μ x – μ y Ch. 10-9 10.2

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Difference Between Two Means Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 10-10

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σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples 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 Ch. 10-11

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σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 10-12

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Test Statistic, σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples * 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: Ch. 10-13

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Hypothesis Tests for Two Population Means Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 10-14

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Decision Rules Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 zz 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 Ch. 10-15

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σ x 2 and σ y 2 Unknown, Assumed Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples 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 Ch. 10-16

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σ x 2 and σ y 2 Unknown, Assumed Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples (continued) 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 Ch. 10-17

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Test Statistic, σ x 2 and σ y 2 Unknown, Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall * 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: Ch. 10-18

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Decision Rules Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 -t -t /2 tt t /2 Reject H 0 if t < -t (n1+n2 – 2), Reject H 0 if t > t (n1+n2 – 2), Reject H 0 if t < -t (n1+n2 – 2), or t > t (n1+n2 – 2), Two Population Means, Independent Samples, Variances Unknown Ch. 10-19

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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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Assuming both populations are approximately normal with equal variances, is there a difference in average yield ( = 0.05)? Ch. 10-20

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Calculating the Test Statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The test statistic is: Ch. 10-21

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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: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Decision: Conclusion: Reject H 0 at = 0.05 There is evidence of a difference in means. t 0 2.0154-2.0154.025 Reject H 0.025 2.040 Ch. 10-22

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σ x 2 and σ y 2 Unknown, Assumed Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples 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 Ch. 10-23

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σ x 2 and σ y 2 Unknown, Assumed Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples (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 Ch. 10-24

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Test Statistic, σ x 2 and σ y 2 Unknown, Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall * 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: Ch. 10-25

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Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) > 5 Ch. 10-26 10.3

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Two Population Proportions The random variable is approximately normally distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population proportions (continued) Ch. 10-27

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Test Statistic for Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population proportions The test statistic for H 0 : P x – P y = 0 is a z value: Where Ch. 10-28

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Decision Rules: Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 zz z /2 Reject H 0 if z < -z Reject H 0 if z > z Reject H 0 if z < -z or z > z Ch. 10-29

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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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-30

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The hypothesis test is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 10-31

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Example: Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The test statistic for P M – P W = 0 is: (continued).025 -1.961.96.025 -1.31 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 Ch. 10-32

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Hypothesis Tests for Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 10-33 10.4

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Hypothesis Tests for Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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) Ch. 10-34

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Test Statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 10-35

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Decision Rules: Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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 Ch. 10-36

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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 Mean3.272.53 Std dev1.301.16 Is there a difference in the variances between the NYSE & NASDAQ at the = 0.10 level? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-37

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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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 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: Ch. 10-38

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F Test: Example Solution The test statistic is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall /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 = 1.256 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 Ch. 10-39

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Two-Sample Tests in EXCEL 2007 For paired samples (t test): Data | data analysis… | t-test: paired two sample for means For independent samples: Independent sample Z test with variances known: Data | data analysis | z-test: two sample for means For variances… F test for two variances: Data | data analysis | F-test: two sample for variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-40

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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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-41

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Chapter Summary Performed F tests for the difference between two population variances Used the F table to find F critical values Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 10-42

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