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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-1 Chapter 9 Two-Sample Tests Statistics for Managers Using Microsoft ® Excel 4 th Edition
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-2 Chapter Goals After completing this chapter, you should be able to: Test hypotheses for the difference between two independent population means ( standard deviations known or unknown) Test two means from related samples for the mean difference Complete a Z test for the difference between two proportions Use the F table to find critical F values Complete an F test for the difference between two variances
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-3 Two Sample Tests Population Means, Independent Samples Means, Related 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
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-4 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown Goal: Test hypotheses or 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 *
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-5 Independent Samples Population means, independent samples Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population Use the difference between 2 sample means Use Z test or pooled variance t test * σ 1 and σ 2 known σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-6 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown * Use a Z test statistic Use S to estimate unknown σ, use a t test statistic and pooled standard deviation
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-7 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Known Assumptions: Samples are randomly and independently drawn population distributions are normal or both sample sizes are 30 Population standard deviations are known * σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-8 Population means, independent samples σ 1 and σ 2 known …and the standard error of X 1 – X 2 is When σ 1 and σ 2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a Z-value… (continued) σ 1 and σ 2 Known * σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-9 Population means, independent samples σ 1 and σ 2 known The test statistic for μ 1 – μ 2 is: σ 1 and σ 2 Known * σ 1 and σ 2 unknown (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-10 Hypothesis Tests for Two Population Means Lower-tail test: H 0 : μ 1 μ 2 H 1 : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2 0 H 1 : μ 1 – μ 2 < 0 Upper-tail test: H 0 : μ 1 ≤ μ 2 H 1 : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tail test: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-11 Two Population Means, Independent Samples Lower-tail test: H 0 : μ 1 – μ 2 0 H 1 : μ 1 – μ 2 < 0 Upper-tail test: H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tail test: H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 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 Hypothesis tests for μ 1 – μ 2
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-12 Population means, independent samples σ 1 and σ 2 known The confidence interval for μ 1 – μ 2 is: Confidence Interval, σ 1 and σ 2 Known * σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-13 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown Assumptions: Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 Population variances are unknown but assumed equal * σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-14 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown (continued) * σ 1 and σ 2 unknown Forming interval estimates: The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-15 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown The pooled standard deviation is (continued) * σ 1 and σ 2 unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-16 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 – μ 2 is: * σ 1 and σ 2 unknown (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-17 Population means, independent samples σ 1 and σ 2 known The confidence interval for μ 1 – μ 2 is: Where * σ 1 and σ 2 unknown Confidence Interval, σ 1 and σ 2 Unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-18 Pooled S p 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)?
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-19 Calculating the Test Statistic The test statistic is:
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-20 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: 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
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-21 Related Samples Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Or, if Not Normal, use large samples Related samples D = X 1 - X 2
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-22 Mean Difference, σ D Known The i th paired difference is D i, where Related samples D i = X 1i - X 2i The point estimate for the population mean paired difference is D : Suppose the population standard deviation of the difference scores, σ D, is known n is the number of pairs in the paired sample
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-23 The test statistic for the mean difference is a Z value: Paired samples Mean Difference, σ D Known (continued) Where μ D = hypothesized mean difference σ D = population standard dev. of differences n = the sample size (number of pairs)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-24 Confidence Interval, σ D Known The confidence interval for D is Paired samples Where n = the sample size (number of pairs in the paired sample)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-25 If σ D is unknown, we can estimate the unknown population standard deviation with a sample standard deviation: Related samples The sample standard deviation is Mean Difference, σ D Unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-26 The test statistic for D is now a t statistic, with n-1 d.f.: Paired samples Where t has n - 1 d.f. and S D is: Mean Difference, σ D Unknown (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-27 The confidence interval for D is Paired samples where Confidence Interval, σ D Unknown
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-28 Lower-tail test: H 0 : μ D 0 H 1 : μ D < 0 Upper-tail test: H 0 : μ D ≤ 0 H 1 : μ D > 0 Two-tail test: H 0 : μ D = 0 H 1 : μ D ≠ 0 Paired Samples Hypothesis Testing for Mean Difference, σ D Unknown /2 -t -t /2 tt t /2 Reject H 0 if t < -t Reject H 0 if t > t Reject H 0 if t < -t or t > t Where t has n - 1 d.f.
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-29 Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: Paired Samples 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
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-30 Has the training made a difference in the number of complaints (at the 0.01 level)? - 4.2D = H 0 : μ D = 0 H 1 : μ D 0 Test Statistic: Critical Value = ± 4.604 d.f. = n - 1 = 4 Reject /2 - 4.604 4.604 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. Paired Samples: Solution Reject /2 - 1.66 =.01
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-31 Two Population Proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, p 1 – p 2 The point estimate for the difference is Population proportions Assumptions: n 1 p 1 5, n 1 (1-p 1 ) 5 n 2 p 2 5, n 2 (1-p 2 ) 5
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-32 Two Population Proportions Population proportions 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 Since we begin by assuming the null hypothesis is true, we assume p 1 = p 2 and pool the two p s estimates
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-33 Two Population Proportions Population proportions The test statistic for p 1 – p 2 is a Z statistic: (continued) where
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-34 Confidence Interval for Two Population Proportions Population proportions The confidence interval for p 1 – p 2 is:
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-35 Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : p 1 p 2 H 1 : p 1 < p 2 i.e., H 0 : p 1 – p 2 0 H 1 : p 1 – p 2 < 0 Upper-tail test: H 0 : p 1 ≤ p 2 H 1 : p 1 > p 2 i.e., H 0 : p 1 – p 2 ≤ 0 H 1 : p 1 – p 2 > 0 Two-tail test: H 0 : p 1 = p 2 H 1 : p 1 ≠ p 2 i.e., H 0 : p 1 – p 2 = 0 H 1 : p 1 – p 2 ≠ 0
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-36 Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : p 1 – p 2 0 H 1 : p 1 – p 2 < 0 Upper-tail test: H 0 : p 1 – p 2 ≤ 0 H 1 : p 1 – p 2 > 0 Two-tail test: H 0 : p 1 – p 2 = 0 H 1 : p 1 – p 2 ≠ 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 (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-37 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
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-38 The hypothesis test is: H 0 : p 1 – p 2 = 0 (the two proportions are equal) H 1 : p 1 – p 2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p s1 = 36/72 =.50 Women: p s2 = 31/50 =.62 The pooled estimate for the overall proportion is: Example: Two population Proportions (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-39 The test statistic for p 1 – p 2 is: Example: Two population Proportions (continued).025 -1.961.96.025 -1.31 Decision: Do not reject H 0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. Reject H 0 Critical Values = ±1.96 For =.05
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-40 Hypothesis Tests for Variances Tests for Two Population Variances F test statistic H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 Two-tail test Lower-tail test Upper-tail test H 0 : σ 1 2 σ 2 2 H 1 : σ 1 2 < σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 *
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-41 Hypothesis Tests for Variances Tests for Two Population Variances F test statistic The F test statistic is: = Variance of Sample 1 n 1 - 1 = numerator degrees of freedom n 2 - 1 = denominator degrees of freedom = Variance of Sample 2 * (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-42 The F critical value is found from the F table The are two appropriate degrees of freedom: numerator and denominator In the F table, numerator degrees of freedom determine the column denominator degrees of freedom determine the row The F Distribution where df 1 = n 1 – 1 ; df 2 = n 2 – 1
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-43 F0 Finding the Rejection Region rejection region for a two-tail test is: FLFL Reject H 0 Do not reject H 0 F 0 FUFU Reject H 0 Do not reject H 0 F0 /2 Reject H 0 Do not reject H 0 FUFU H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 H 0 : σ 1 2 σ 2 2 H 1 : σ 1 2 < σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 FLFL /2 Reject H 0 Reject H 0 if F < F L Reject H 0 if F > F U
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-44 Finding the Rejection Region F0 /2 Reject H 0 Do not reject H 0 FUFU H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 FLFL /2 Reject H 0 (continued) 2. Find F L using the formula: Where F U* is from the F table with n 2 – 1 numerator and n 1 – 1 denominator degrees of freedom (i.e., switch the d.f. from F U ) 1. Find F U from the F table for n 1 – 1 numerator and n 2 – 1 denominator degrees of freedom To find the critical F values:
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-45 F Test: An Example 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 = 0.05 level?
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-46 F Test: Example Solution Form the hypothesis test: H 0 : σ 2 1 – σ 2 2 = 0 ( there is no difference between variances) H 1 : σ 2 1 – σ 2 2 ≠ 0 ( there is a difference between variances) Numerator: n 1 – 1 = 21 – 1 = 20 d.f. Denominator: n 2 – 1 = 25 – 1 = 24 d.f. F U = F.025, 20, 24 = 2.33 Find the F critical values for =.05: Numerator: n 2 – 1 = 25 – 1 = 24 d.f. Denominator: n 1 – 1 = 21 – 1 = 20 d.f. F L = 1/F.025, 24, 20 = 1/2.41 =.41 FU:FU:FL:FL:
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-47 The test statistic is: 0 /2 =.025 F U =2.33 Reject H 0 Do not reject H 0 H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 F Test: Example Solution 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 =.05 F L =0.41 /2 =.025 Reject H 0 F
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-48 Two-Sample Tests in EXCEL For independent samples: Independent sample Z test with variances known: Tools | data analysis | z-test: two sample for means For paired samples (t test): Tools | data analysis… | t-test: paired two sample for means For variances… F test for two variances: Tools | data analysis | F-test: two sample for variances
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-49 Two-Sample Tests in PHStat
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-50 Sample PHStat Output Input Output
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-51 Sample PHStat Output Input Output (continued)
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-52 Chapter Summary Compared two independent samples Performed Z test for the differences in two means Performed pooled variance t test for the differences in two means Formed confidence intervals for the differences between two means Compared two related samples (paired samples) Performed paired sample Z and t tests for the mean difference Formed confidence intervals for the paired difference
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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-53 Chapter Summary Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions Performed F tests for the difference between two population variances Used the F table to find F critical values (continued)
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