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Statistics for Business and Economics 7th Edition Chapter 10 Hypothesis Testing: Additional Topics Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-1 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-2 Two Sample Tests Two Sample Tests Population Means, Dependent Samples Population Means, Independent Samples Population Proportions Population Variances Examples: Same group before vs after treatment Group vs independent Group Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Proportion vs Proportion Variance vs Variance Ch 10-3 10.1 Dependent Samples Tests Means of Related Populations Dependent Samples Paired or matched samples Repeated measures (before/after) Use difference between paired values: di = x i - y i Assumptions: Both Populations Are Normally Distributed Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-4 Test Statistic: Dependent Samples The test statistic for the mean difference is a t value, with n – degrees of freedom: Dependent Samples where d D0 t sd n d d n i x y D0 = hypothesized mean difference sd = sample standard dev of differences n = the sample size (number of pairs) Copyright © 2010 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: H0: μx – μy H1: μx – μy < H0: μx – μy ≤ H1: μx – μy > H0: μx – μy = H1: μx – μy ≠ -t t Reject H0 if t < -tn-1, Where Reject H0 if t > tn-1, t d D0 sd n /2 -t/2 /2 t/2 Reject H0 if t < -tn-1 , or t > tn-1 , has n - d.f Copyright © 2010 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) Salesperson C.B T.F M.H R.K M.O Before (1) 20 After (2) 0 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Difference, di - -14 - - -21 = - 4.2 Sd (d d ) i n 5.67 Ch 10-7 Matched Pairs: Solution Has the training made a difference in the number of complaints (at the = 0.05 level)? H0: μx – μy = H1: μx – μy = 05 d = - 4.2 Critical Value = ± 2.776 d.f = n − = Test Statistic: d D0 4.2 t 1.66 sd / n 5.67/ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Reject Reject /2 /2 - 2.776 2.776 - 1.66 Decision: Do not reject H0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints Ch 10-8 10.2 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μx – μy 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 Ch 10-9 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Test statistic is a a value from the Student’s t distribution Ch 10-10 Test Statistic for Two Population Proportions The test statistic for H0: Px – Py = Population proportions is a z value: z pˆ x pˆ y pˆ (1 pˆ ) pˆ (1 pˆ ) nx ny Where Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall pˆ n xpˆ x n ypˆ y nx ny Ch 10-28 Decision Rules: Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: Px – Py H1: Px – Py < H0: Px – Py ≤ H1: Px – Py > H0: Px – Py = H1: Px – Py ≠ -z Reject H0 if z < -z z Reject H0 if z > z Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall /2 -z/2 /2 z/2 Reject H0 if z < -z or z > z Ch 10-29 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-30 Example: Two Population Proportions (continued) The hypothesis test is: H0: PM – PW = (the two proportions are equal) H1: PM – PW ≠ (there is a significant difference between proportions) The sample proportions are: Men: pˆ M = 36/72 = 50 pˆ W = 31/50 = 62 Women: The estimate for the common overall proportion is: ˆ M n W pˆ W 72(36/72) 50(31/50) 67 n p M pˆ .549 nM n W 72 50 122 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-31 Example: Two Population Proportions (continued) The test statistic for PM – PW = is: pˆ pˆ W z pˆ (1 pˆ ) pˆ (1 pˆ ) 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 = 05 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall significant evidence of a difference between men and women in proportions who will vote yes Ch 10-32 10.4 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-33 Hypothesis Tests for Two Variances (continued) Tests for Two Population Variances F test statistic The random variable x y s /σ F s /σ x 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 Fν1,ν Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-34 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 x y where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-35 Decision Rules: Two Variances Use sx2 to denote the larger variance H0: σx2 = σy2 H1: σx2 ≠ σy2 H0: σx2 ≤ σy2 H1: σx2 > σy2 /2 Do not reject H0 Reject H0 Fn x 1,ny 1,α F Reject H0 if F Fn x 1,ny 1,α Do not reject H0 F Reject H0 Fnx 1,ny 1,α / rejection region for a twotail test is: Reject H0 if F Fnx 1,n y 1,α / where sx2 is the larger of the two sample variances Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-36 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 3.272.53 Std dev 1.301.16 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 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 = 10/2: Degrees of Freedom: Fn x 1, ny 1, α / Numerator (NYSE has the larger F20 , 24 , 0.10/2 standard deviation): 2.03 nx – = 21 – = 20 d.f Denominator: ny – = 25 – = 24 d.f Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-38 F Test: Example Solution (continued) The test statistic is: H0 : σx = σy H1: σx2 ≠ σy2 s2x 1.30 F 1.256 s y 1.16 F = 1.256 is not in the rejection region, so we not reject H0 /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 = 10 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-39 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 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 10-41 Chapter Summary (continued) 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 Ch 10-42 ... Same group before vs after treatment Group vs independent Group Copyright © 2 010 Pearson Education, Inc Publishing as Prentice Hall Proportion vs Proportion Variance vs Variance Ch 10- 3 10. 1 Dependent... significant change in the number of complaints Ch 10- 8 10. 2 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population... unknown The test statistic for μx – μy is: x y D0 z 2 σy σx nx ny Copyright © 2 010 Pearson Education, Inc Publishing as Prentice Hall Ch 10- 13 Hypothesis Tests for Two Population Means
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