Business Statistics A DecisionMaking Approach 8 th Edition
Business Statistics: A Decision-Making Approach 8 th Edition Chapter 10 Estimation and Hypothesis Testing for Two Population Parameters Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 10 -1
Chapter Goals After completing this chapter, you should be able to: n Understand the logic of hypothesis testing n Test hypotheses and form interval estimates n Two independent population means n n Standard deviations known Standard deviations unknown Two means from paired samples The difference between two population proportions Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 2
Estimation for Two Populations Estimating two population values Population means, independent samples Group 1 vs. Group 2 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall Paired samples Population proportions Same group before vs. after Proportion 1 vs. Proportion 2 3
Difference Between Two Means Population means, independent samples * σ1 and σ2 known σ1 and σ2 unknown but assumed equal σ1 and σ2 unknown, not assumed equal Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall Goal: Form a confidence interval for the difference between two population means, μ 1 – μ 2 The point estimate for the difference is x 1 – x 2 4
Independent Samples n n Different data sources n Unrelated n Independent n Sample selected from one population has no effect on the sample selected from the other population Use the difference between 2 sample means Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 5
σ1 and σ2 known Assumptions: § Samples are randomly and independently drawn § population distributions are normal or both sample sizes are 30 § Population standard deviations are known Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 6
σ1 and σ2 known (continued) When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z value……and the standard error of x 1 – x 2 is Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 7
Confidence Interval: σ1 and σ2 known The confidence interval for μ 1 – μ 2 is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 8
General Steps 1. Define the population parameter of interest and select independent samples from each population 2. Specify the confidence interval 3. Compute the point estimate 4. Determine the standard error 5. Determine the critical value 6. Develop the confidence interval estimate Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 9
σ1 and σ2 unknown, large samples Assumptions: Population means, independent samples § Samples are randomly and independently drawn § Populations follow the normal distribution σ1 and σ2 known σ1 and σ2 unknown but assumed equal * § The two standard deviations are equal σ1 and σ2 unknown, not assumed equal Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 10
σ1 and σ2 unknown, large samples (continued) Forming interval estimates: § The population standard deviations are assumed equal, so use the two sample standard deviations and pool them to estimate σ (called pooled standard deviation) § the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11
σ1 and σ2 unknown, large samples (continued) The pooled standard deviation is: The confidence interval for μ 1 – μ 2 is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 12
σ1 and σ2 unknown, small samples Population means, independent samples Assumptions: § populations are normally distributed σ1 and σ2 known § there is a reason to believe that the populations do not have equal variances σ1 and σ2 unknown but assumed equal σ1 and σ2 unknown, not assumed equal Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall * § samples are independent 13
σ1 and σ2 unknown, small samples (continued) Forming interval estimates: § The population variances are not assumed equal, so we do not pool them § the test statistic is a t value with degrees of freedom given by: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 14
σ1 and σ2 unknown, small samples (continued) The confidence interval for μ 1 – μ 2 is: Where t /2 has d. f. given by Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 15
Hypothesis Tests for the Difference Between Two Means n Testing Hypotheses about μ 1 – μ 2 n Use the same situations discussed already: n Standard deviations known n Standard deviations unknown Assumed equal n Assumed not equal (small samples) n Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 16
Hypothesis Tests for Two Population Means, Independent Samples Lower tail test: Upper tail test: Two-tailed test: H 0: μ 1 μ 2 HA : μ 1 < μ 2 H 0: μ 1 ≤ μ 2 HA : μ 1 > μ 2 H 0: μ 1 = μ 2 HA : μ 1 ≠ μ 2 i. e. , H 0: μ 1 – μ 2 0 HA : μ 1 – μ 2 < 0 H 0: μ 1 – μ 2 ≤ 0 HA : μ 1 – μ 2 > 0 H 0: μ 1 – μ 2 = 0 HA : μ 1 – μ 2 ≠ 0 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 17
Hypothesis tests for μ 1 – μ 2 Population means, independent samples σ1 and σ2 known Use a z test statistic σ1 and σ2 unknown but assumed equal Use sp to estimate unknown σ , use a t test statistic with n 1 + n 2 – 2 d. f. σ1 and σ2 unknown, not assumed equal Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall Use s 1 and s 2 to estimate unknown σ1 and σ2 , use a t test statistic and calculate the required degrees of freedom 18
σ1 and σ2 known The test statistic for μ 1 – μ 2 is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 19
Steps: Hypothesis Test for Two Population Means 1. 2. 3. 4. Specify parameter of interest Formulate hypotheses Specify the significance level ( ) Construct the rejection region and develop the decision rule 5. Compute the test statistics and/or the p-value 6. Reach a decision 7. Draw a conclusion See Chapter 9 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 20
σ1 and σ2 unknown, large samples The test statistic for μ 1 – μ 2 is: Where t has (n 1 + n 2 – 2) d. f. , and Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 21
σ1 and σ2 unknown, small samples The test statistic for μ 1 – μ 2 is: Where t has d. f. given by Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 22
Hypothesis tests for μ 1 – μ 2 Two Population Means, Independent Samples Lower tail test: Upper tail test: Two-tailed test: H 0: μ 1 – μ 2 0 HA : μ 1 – μ 2 < 0 H 0: μ 1 – μ 2 ≤ 0 HA : μ 1 – μ 2 > 0 H 0: μ 1 – μ 2 = 0 HA : μ 1 – μ 2 ≠ 0 Example: σ1 and σ2 known: -z Reject H 0 if z < -z z Reject H 0 if z > z Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 23
Example σ1 and σ2 unknown, assumed equal You’re 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 equal variances, is there a difference in average yield ( = 0. 05)? Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 24
Calculating the Test Statistic The test statistic is: Where: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 25
Solution H 0: μ 1 - μ 2 = 0 i. e. (μ 1 = μ 2) HA: 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 0. 025 -2. 0154 Reject H 0 0. 025 0 2. 0154 t 2. 040 Decision: Reject H 0 at = 0. 05 Conclusion: There is evidence that the means are different. Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 26
Paired Samples Tests Means of 2 Related Populations n Paired or matched samples n n Repeated measures (before/after) n n e. g. , two different value appraisals on the same home e. g. , weight before and after a diet program Use difference between paired values: d = x 1 - x 2 n n Eliminates Variation Among Subjects Assumptions: n Both Populations Are Normally Distributed n Or, if Not Normal, use large samples Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 27
Paired Differences The ith paired difference is di , where di = x 1 i - x 2 i The point estimate for the population mean paired difference is d : The sample standard deviation is the number of pairs in the paired sample Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 28
Paired Differences (continued) The confidence interval for d is Where t has n - 1 d. f. and sd is: n is the number of pairs in the paired sample Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 29
Hypothesis Testing for Paired Samples The test statistic for d is n is the number of pairs in the paired sample Where t has n - 1 d. f. Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall and sd is: 30
Hypothesis Testing for Paired Samples (continued) Paired Samples Lower tail test: Upper tail test: Two-tailed test: H 0: μ d 0 HA : μd < 0 H 0: μ d ≤ 0 HA : μd > 0 H 0: μ d = 0 HA : μd ≠ 0 -t Reject H 0 if t < -t t Reject H 0 if t > t Where t has n - 1 d. f. Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall /2 -t /2 Reject H 0 if t < -t /2 or t > t /2 31
Paired Sample Example Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, di C. B. T. F. M. H. R. K. M. O. 6 20 3 0 4 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 4 6 2 0 0 - 2 -14 - 1 0 - 4 -21 d = di n = -4. 2 32
Paired Sample: Solution Has the training made a difference in the number of complaints (at the 0. 05 level)? H 0: μ d = 0 HA: μd 0 = 0. 05 d = - 4. 2 Critical Value = ± 2. 7765 d. f. = n - 1 = 4 Test Statistic: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall Reject /2 - 2. 7765 - 1. 66 2. 7765 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. 33
Two Population Proportions Goal: Form a confidence interval for or test a hypothesis about the difference between two population proportions, π1 – π2 Assumptions: n 1π1 5 , n 1(1 -π1) 5 n 2π2 5 , n 2(1 -π2) 5 The point estimate for the difference is Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall p 1 – p 2 34
Confidence Interval for Two Population Proportions The confidence interval for π1 – π2 is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 35
Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: Upper tail test: Two-tailed test: H 0: π 1 π 2 HA : π 1 < π 2 H 0: π 1 ≤ π 2 HA : π 1 > π 2 H 0: π 1 = π 2 HA : π 1 ≠ π 2 i. e. , H 0: π 1 – π 2 0 HA : π 1 – π 2 < 0 H 0: π 1 – π 2 ≤ 0 HA : π 1 – π 2 > 0 H 0: π 1 – π 2 = 0 HA : π 1 – π 2 ≠ 0 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 36
Two Population Proportions Since we begin by assuming the null hypothesis is true, we assume π1 = π2 and pool the two p estimates The pooled estimate for the overall proportion is: Where x 1 and x 2 are the numbers from samples 1 and 2 with the characteristic of interest Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 37
Two Population Proportions (continued) The test statistic for π1 – π2 is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 38
Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: Upper tail test: Two-tailed test: H 0: π 1 – π 2 0 HA : π 1 – π 2 < 0 H 0: π 1 – π 2 ≤ 0 HA : π 1 – π 2 > 0 H 0: π 1 – π 2 = 0 HA : π 1 – π 2 ≠ 0 -z Reject H 0 if z < -z z Reject H 0 if z > z Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall /2 -z /2 Reject H 0 if z < -z /2 or z > z /2 39
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 0. 05 level of significance Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 40
Example: Two population Proportions (continued) n The hypothesis test is: H 0: π1 – π2 = 0 (the two proportions are equal) HA: π1 – π2 ≠ 0 (there is a significant difference between proportions) n The sample proportions are: Men: p 1 = 36/72 = 0. 50 Women: p 2 = 31/50 = 0. 62 § The pooled estimate for the overall proportion is: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 41
Example: Two population Proportions (continued) Reject H 0 The test statistic for π1 – π2 is: 0. 025 -1. 96 -1. 31 Reject H 0 0. 025 1. 96 Decision: Do not reject H 0 Critical Values = ± 1. 96 For = 0. 05 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall Conclusion: There is not significant evidence of a difference in the proportion who will vote yes between men and women. 42
Two Sample Tests in EXCEL For independent samples: n Independent sample z test with variances known: n n Data | data analysis | z-Test: Two Sample for Means Independent sample z test with variance unknown: n Data | data analysis | t-Test: Two Sample Assuming Equal Variances n Data | data analysis | t-Test: Two Sample Assuming Unequal Variances For paired samples (t test): n Data | data analysis… | t-Test: Paired Two Sample for Means Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 43
Two Sample Tests in PHStat Options Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 44
Two Sample Tests in PHStat Input Output Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 45
Two Sample Tests in PHStat Input Output Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 46
Chapter Summary n Compared two independent samples n n Compared two related samples (paired samples) n n n Formed confidence intervals for the differences between two means Performed z test for the differences in two means Performed t test for the differences in two means Formed confidence intervals for the paired difference Performed paired sample t tests for the mean difference Compared two population proportions n n Formed confidence intervals for the difference between two population proportions Performed z test for two population proportions Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 47
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