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Newbold_chap11.ppt
1.
Chap 11-1 Statistics for
Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 11 Hypothesis Testing II Statistics for Business and Economics 6th Edition
2.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-2 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
3.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-3 Two Sample Tests Two Sample Tests Population Means, Independent Samples Population Means, Matched Pairs 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 (Note similarities to Chapter 9)
4.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-4 Matched Pairs 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 Matched Pairs di = xi - yi
5.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-5 The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: n s D d t d 0 Test Statistic: Matched Pairs Where D0 = hypothesized mean difference sd = sample standard dev. of differences n = the sample size (number of pairs) Matched Pairs
6.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-6 Lower-tail test: H0: μx – μy 0 H1: μx – μy < 0 Upper-tail test: H0: μx – μy ≤ 0 H1: μx – μy > 0 Two-tail test: H0: μx – μy = 0 H1: μx – μy ≠ 0 Paired Samples Decision Rules: Matched Pairs a a/2 a/2 a -ta -ta/2 ta ta/2 Reject H0 if t < -tn-1, a Reject H0 if t > tn-1, a Reject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 Where n s D d t d 0 has n - 1 d.f.
7.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-7 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: Matched Pairs Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, di 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 = di n 5.67 1 n ) d (d S 2 i d = - 4.2
8.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-8 Has the training made a difference in the number of complaints (at the a = 0.01 level)? - 4.2 d = 1.66 5 5.67/ 0 4.2 n / s D d t d 0 H0: μx – μy = 0 H1: μx – μy 0 Test Statistic: Critical Value = ± 4.604 d.f. = n - 1 = 4 Reject a/2 - 4.604 4.604 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. Matched Pairs: Solution Reject a/2 - 1.66 a = .01
9.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-9 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
10.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-10 Difference Between Two Means Population means, independent samples Test statistic is a z value Test statistic is a a value from the Student’s t distribution σx 2 and σy 2 assumed equal σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal (continued)
11.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-11 Population means, independent samples σx 2 and σy 2 Known Assumptions: Samples are randomly and independently drawn both population distributions are normal Population variances are known * σx 2 and σy 2 known σx 2 and σy 2 unknown
12.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-12 Population means, independent samples …and the random variable has a standard normal distribution When σx 2 and σy 2 are known and both populations are normal, the variance of X – Y is y 2 y x 2 x 2 Y X n σ n σ σ (continued) * Y 2 y X 2 x Y X n σ n σ ) μ (μ ) y x ( Z σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 Known
13.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-13 Population means, independent samples Test Statistic, σx 2 and σy 2 Known * σx 2 and σy 2 known σx 2 and σy 2 unknown y 2 y x 2 x 0 n σ n σ D y x z The test statistic for μx – μy is:
14.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-14 Hypothesis Tests for Two Population Means Lower-tail test: H0: μx μy H1: μx < μy i.e., H0: μx – μy 0 H1: μx – μy < 0 Upper-tail test: H0: μx ≤ μy H1: μx > μy i.e., H0: μx – μy ≤ 0 H1: μx – μy > 0 Two-tail test: H0: μx = μy H1: μx ≠ μy i.e., H0: μx – μy = 0 H1: μx – μy ≠ 0 Two Population Means, Independent Samples
15.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-15 Two Population Means, Independent Samples, Variances Known Lower-tail test: H0: μx – μy 0 H1: μx – μy < 0 Upper-tail test: H0: μx – μy ≤ 0 H1: μx – μy > 0 Two-tail test: H0: μx – μy = 0 H1: μx – μy ≠ 0 a a/2 a/2 a -za -za/2 za za/2 Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -za/2 or z > za/2 Decision Rules
16.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-16 Population means, independent samples σx 2 and σy 2 Unknown, Assumed Equal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown but assumed equal * σx 2 and σy 2 assumed equal σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal
17.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-17 Population means, independent samples (continued) Forming interval estimates: The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ use a t value with (nx + ny – 2) degrees of freedom * σx 2 and σy 2 assumed equal σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal σx2 and σy 2 Unknown, Assumed Equal
18.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-18 * Test Statistic, σx 2 and σy 2 Unknown, Equal σx 2 and σy 2 assumed equal σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal 2 n n 1)s (n 1)s (n s y x 2 y y 2 x x 2 p Where t has (n1 + n2 – 2) d.f., and y x 2 p y x n 1 n 1 S μ μ t y x The test statistic for μx – μy is:
19.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-19 Population means, independent samples σx 2 and σy 2 Unknown, Assumed Unequal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown and assumed unequal * σx 2 and σy 2 assumed equal σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal
20.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-20 Population means, independent samples σx 2 and σy 2 Unknown, Assumed Unequal (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 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 1) /(n n s 1) /(n n s ) n s ( ) n s ( y 2 y 2 y x 2 x 2 x 2 y 2 y x 2 x v
21.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-21 * Test Statistic, σx 2 and σy 2 Unknown, Unequal σx 2 and σy 2 assumed equal σx 2 and σy 2 unknown σx 2 and σy 2 assumed unequal 1) /(n n s 1) /(n n s ) n s ( ) n s ( y 2 y 2 y x 2 x 2 x 2 y 2 y x 2 x v Where t has degrees of freedom: The test statistic for μx – μy is: Y 2 y X 2 x 0 n σ n σ D ) y x ( t
22.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-22 Lower-tail test: H0: μx – μy 0 H1: μx – μy < 0 Upper-tail test: H0: μx – μy ≤ 0 H1: μx – μy > 0 Two-tail test: H0: μx – μy = 0 H1: μx – μy ≠ 0 Decision Rules a a/2 a/2 a -ta -ta/2 ta ta/2 Reject H0 if t < -tn-1, a Reject H0 if t > tn-1, a Reject H0 if t < -tn-1 , a/2 or t > tn-1 , a/2 Where t has n - 1 d.f. Two Population Means, Independent Samples, Variances Unknown
23.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-23 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)?
24.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-24 Calculating the Test Statistic 1.5021 1) 25 ( 1) - (21 1.16 1 25 1.30 1 21 1) n ( ) 1 (n S 1 n S 1 n S 2 2 2 1 2 2 2 2 1 1 2 p 2.040 25 1 21 1 5021 . 1 0 2.53 3.27 n 1 n 1 S μ μ X X t 2 1 2 p 2 1 2 1 The test statistic is:
25.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-25 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 Test Statistic: Decision: Conclusion: Reject H0 at a = 0.05 There is evidence of a difference in means. t 0 2.0154 -2.0154 .025 Reject H0 Reject H0 .025 2.040 2.040 25 1 21 1 5021 . 1 2.53 3.27 t
26.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-26 Two Population Proportions Goal: Test hypotheses for the difference between two population proportions, Px – Py Population proportions Assumptions: Both sample sizes are large, nP(1 – P) > 9
27.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-27 Two Population Proportions Population proportions (continued) The random variable is approximately normally distributed y y y x x x y x y x n ) p (1 p n ) p (1 p ) p (p ) p p ( Z ˆ ˆ ˆ ˆ ˆ ˆ
28.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-28 Test Statistic for Two Population Proportions Population proportions The test statistic for H0: Px – Py = 0 is a z value: y 0 0 x 0 0 y x n ) p (1 p n ) p (1 p p p z ˆ ˆ ˆ ˆ ˆ ˆ y x y y x x 0 n n p n p n p ˆ ˆ ˆ Where
29.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-29 Decision Rules: Proportions Population proportions Lower-tail test: H0: px – py 0 H1: px – py < 0 Upper-tail test: H0: px – py ≤ 0 H1: px – py > 0 Two-tail test: H0: px – py = 0 H1: px – py ≠ 0 a a/2 a/2 a -za -za/2 za za/2 Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -za/2 or z > za/2
30.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-30 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
31.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-31 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: = 36/72 = .50 Women: = 31/50 = .62 .549 122 67 50 72 50(31/50) 72(36/72) n n p n p n p y x y y x x 0 ˆ ˆ ˆ The estimate for the common overall proportion is: Example: Two Population Proportions (continued) M p̂ W p̂
32.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-32 The test statistic for PM – PW = 0 is: Example: Two Population Proportions (continued) .025 -1.96 1.96 .025 -1.31 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes. 1.31 50 .549) (1 .549 72 .549) (1 .549 .62 .50 n ) p (1 p n ) p (1 p p p z 2 0 0 1 0 0 W M ˆ ˆ ˆ ˆ ˆ ˆ Reject H0 Reject H0 Critical Values = ±1.96 For a = .05
33.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-33 Population Variance 2 2 2 1 n σ 1)s (n follows a chi-square distribution with (n – 1) degrees of freedom Goal: Test hypotheses about the population variance, σ2 If the population is normally distributed, Hypothesis Tests of one Population Variance
34.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-34 Confidence Intervals for the Population Variance Population Variance The test statistic for hypothesis tests about one population variance is 2 0 2 2 1 n σ 1)s (n χ (continued)
35.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-35 Decision Rules: Variance Population variance Lower-tail test: H0: σ2 σ0 2 H1: σ2 < σ0 2 Upper-tail test: H0: σ2 ≤ σ0 2 H1: σ2 > σ0 2 Two-tail test: H0: σ2 = σ0 2 H1: σ2 ≠ σ0 2 a a/2 a/2 a Reject H0 if Reject H0 if Reject H0 if or 2 , 1 n a χ 2 ,1 1 n a χ 2 ,1 1 n 2 / a χ 2 , 1 n 2 / a χ 2 ,1 1 n 2 1 n a χ χ 2 , 1 n 2 1 n a χ χ 2 , 1 n 2 1 n 2 / a χ χ 2 ,1 1 n 2 1 n 2 / a χ χ
36.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-36 Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic H0: σx 2 = σy 2 H1: σx 2 ≠ σy 2 Two-tail test Lower-tail test Upper-tail test H0: σx 2 σy 2 H1: σx 2 < σy 2 H0: σx 2 ≤ σy 2 H1: σx 2 > σy 2 Goal: Test hypotheses about two population variances The two populations are assumed to be independent and normally distributed
37.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-37 Hypothesis Tests for Two Variances Tests for Two Population Variances F test statistic 2 y 2 y 2 x 2 x /σ s /σ s F 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 (continued)
38.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-38 Test Statistic Tests for Two Population Variances F test statistic 2 y 2 x s s F The critical value for a hypothesis test about two population variances is where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom
39.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-39 Decision Rules: Two Variances rejection region for a two- tail test is: F 0 a Reject H0 Do not reject H0 F 0 a/2 Reject H0 Do not reject H0 H0: σx 2 = σy 2 H1: σx 2 ≠ σy 2 H0: σx 2 ≤ σy 2 H1: σx 2 > σy 2 Use sx 2 to denote the larger variance. α 1, n 1, n y x F 2 / α 1, n 1, n 0 y x F F if H Reject 2 / α 1, n 1, n y x F where sx 2 is the larger of the two sample variances α 1, n 1, n 0 y x F F if H Reject
40.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-40 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?
41.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-41 F Test: Example Solution Form the hypothesis test: H0: σx 2 = σy 2 (there is no difference between variances) H1: σx 2 ≠ σy 2 (there is a difference between variances) Degrees of Freedom: Numerator (NYSE has the larger standard deviation): nx – 1 = 21 – 1 = 20 d.f. Denominator: ny – 1 = 25 – 1 = 24 d.f. Find the F critical values for a = .10/2: 2.03 F F 0.10/2 , 24 , 20 , 1 n , 1 n y x 2 / α
42.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-42 The test statistic is: 1.256 1.16 1.30 s s F 2 2 2 y 2 x a/2 = .05 Reject H0 Do not reject H0 H0: σx 2 = σy 2 H1: σx 2 ≠ σy 2 F Test: Example Solution F = 1.256 is not in the rejection region, so we do not reject H0 (continued) Conclusion: There is not sufficient evidence of a difference in variances at a = .10 F 2.03 F 0.10/2 , 24 , 20
43.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-43 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
44.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-44 Two-Sample Tests in PHStat
45.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-45 Sample PHStat Output Input Output
46.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-46 Sample PHStat Output Input Output (continued)
47.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-47 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
48.
Statistics for Business
and Economics, 6e © 2007 Pearson Education, Inc. Chap 11-48 Chapter Summary 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 (continued)
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