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Chap09 2 sample test

This chapter discusses two-sample tests, including tests for the difference between two independent population means, the difference between two related (paired) sample means, the difference between two population proportions, and the difference between two variances. It provides the formulas and procedures for conducting Z tests, t tests, and F tests for these comparisons in situations where the population standard deviations are both known and unknown. The goal is to test hypotheses about differences between parameters of two populations or to construct confidence intervals for these differences.

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Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 9 Two-Sample Tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-1
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 Statistics for Managers Using the difference between two variances Microsoft Excel, 4e © 2004 Chap 9-2 Prentice-Hall, Inc. 
Two Sample Tests Two Sample Tests Population Means, Independent Samples Means, Related Samples Population Proportions Population Variances Examples: Group 1 vs. independent Group 2 Statistics for Same group before vs. after treatment Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 Chap 9-3
Difference Between Two Means Population means, independent samples * σ1 and σ2 known Goal: Test hypotheses or form a confidence interval for the difference between two population means, μ1 – μ2 σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. The point estimate for the difference is X1 – X2 Chap 9-4
Independent Samples Population means, independent samples  * Different data sources  Unrelated  Independent  σ1 and σ2 known σ1 and σ2 unknown   Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 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 Chap 9-5
Difference Between Two Means Population means, independent samples * σ1 and σ2 known Use a Z test statistic σ1 and σ2 unknown Use S to estimate unknown σ , use a t test statistic and pooled standard deviation Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-6
σ1 and σ2 Known Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown Assumptions: *  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are ≥ 30  Population standard Statistics for Managers Using deviations are known Microsoft Excel, 4e © 2004 Chap 9-7 Prentice-Hall, Inc.
σ1 and σ2 Known (continued) Population means, independent samples σ1 and σ2 known 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 X1 – X2 is σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. σ X1 − X2 2 1 2 σ σ2 = + n1 n2 Chap 9-8
σ1 and σ2 Known (continued) Population means, independent samples σ1 and σ2 known The test statistic for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (X Z= 1 ) − X 2 − ( μ1 − μ2 ) 2 1 2 σ σ2 + n1 n2 Chap 9-9
Hypothesis Tests for Two Population Means Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 ≥ μ2 H1: μ1 < μ2 H0: μ1 ≤ μ2 H1: μ1 > μ2 H0: μ1 = μ2 H1: μ1 ≠ μ2 i.e., i.e., i.e., H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-10
Hypothesis tests for μ1 – μ2 Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-11
Confidence Interval, σ1 and σ2 Known Population means, independent samples σ1 and σ2 known The confidence interval for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ( ) 2 1 2 σ σ2 X1 − X 2 ± Z + n1 n2 Chap 9-12
σ1 and σ2 Unknown Assumptions: Population means, independent samples  Samples are randomly and independently drawn σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but assumed equal Chap 9-13
σ1 and σ2 Unknown (continued) Forming interval estimates: Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown *  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 Statistics for Managers Using with (n1 + n2 – 2) degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-14 Prentice-Hall, Inc.
σ1 and σ2 Unknown (continued) Population means, independent samples The pooled standard deviation is σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sp = ( n1 − 1) S12 + ( n2 − 1) S2 2 (n1 − 1) + (n2 − 1) Chap 9-15
σ1 and σ2 Unknown (continued) The test statistic for μ1 – μ2 is: Population means, independent samples (X t= 1 σ1 and σ2 known σ1 and σ2 unknown ) − X 2 − ( μ1 − μ2 ) 1 1  S  +  n n  2   1 2 p * Where t has (n1 + n2 – 2) d.f., Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. and S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-16
Confidence Interval, σ1 and σ2 Unknown Population means, independent samples The confidence interval for μ1 – μ2 is: σ1 and σ2 known σ1 and σ2 unknown (X 1 * ) − X 2 ± t n1 +n2 -2 1 1  S  +  n n  2   1 2 p Where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-17
Pooled Sp 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 Number 21 Sample mean 3.27 Sample std dev 1.30 NASDAQ 25 2.53 1.16 Assuming both populations are approximately normal with equal variances, is there for Managers average Statistics a difference in Using yield (α = 0.05)? Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-18
Calculating the Test Statistic The test statistic is: (X t= 1 ) − X 2 − ( μ1 − μ2 ) 1 1 S  +  n n  2   1 = 2 p ( 3.27 − 2.53 ) − 0 1   1 1.5021 +   21 25  ( n1 − 1)S12 + ( n2 − 1)S2 2 = ( 21 − 1)1.30 2 + ( 25 − 1)1.16 2 S2 = p (n1 − 1) + (n2 − 1) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (21 - 1) + (25 − 1) = 2.040 = 1.5021 Chap 9-19
Solution H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) α = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: 3.27 − 2.53 t= = 2.040 1   1 1.5021  +   21 25  Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Decision: Reject H0 at α = 0.05 Conclusion: There is evidence of a difference in means. Chap 9-20
Related Samples Tests Means of 2 Related Populations Related samples    Paired or matched samples Repeated measures (before/after) Use difference between paired values: D = X1 - X2 Eliminates Variation Among Subjects  Assumptions:  Both Populations Are Normally Distributed  Or, if Not Normal, use large samples Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 9-21 Prentice-Hall, Inc. 
Mean Difference, σD Known The ith paired difference is Di , where Related samples Di = X1i - X2i The point estimate for the population mean paired difference is D : n D= ∑D i= 1 i n Suppose the population standard deviation of the difference scores, σD, is known Statistics for Managers Using n is the number of pairs in the paired sample Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-22
Mean Difference, σD Known (continued) Paired samples The test statistic for the mean difference is a Z value: D − μD Z= σD n Where μD = hypothesized mean difference σD = population standard dev. of differences Managers Using n = the sample size (number of pairs) Statistics for Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-23
Confidence Interval, σD Known Paired samples The confidence interval for D is σD D±Z n Where n = the sample size (number of pairs in the paired sample) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-24
Mean Difference, σD Unknown Related samples If σD is unknown, we can estimate the unknown population standard deviation with a sample standard deviation: The sample standard deviation is Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. n SD = (Di − D)2 ∑ i=1 n −1 Chap 9-25
Mean Difference, σD Unknown (continued) Paired samples The test statistic for D is now a t statistic, with n-1 d.f.: D − μD t= SD n n Where t has n - 1 d.f. and Statistics for Managers Using SD is: Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D i=1 i − D) 2 n−1 Chap 9-26
Confidence Interval, σD Unknown Paired samples The confidence interval for D is SD D ± t n−1 n n where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D − D) i=1 2 i n −1 Chap 9-27
Hypothesis Testing for Mean Difference, σD Unknown Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μD ≥ 0 H1: μD < 0 H0: μD ≤ 0 H1: μD > 0 H0: μD = 0 H1: μD ≠ 0 α α -tα Reject H if t < -t tα Reject H if t > t α α Statistics 0 Managers Using 0 for Microsoft Excel, 4e © Where t has n - 1 d.f. 2004 Prentice-Hall, Inc. α /2 α /2 -tα/2 tα/2 Reject H0 if t < -tα/2 or t > tα/2 Chap 9-28
Paired Samples 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 4 6 2 0 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. - 2 -14 - 1 0 - 4 -21 D = Σ Di n = -4.2 SD = ∑ (D − D) i n −1 = 5.67 Chap 9-29 2
Paired Samples: Solution  Has the training made a difference in the number of complaints (at the 0.01 level)? H0: μD = 0 H1: μD ≠ 0 α = .01 D = - 4.2 Critical Value = ± 4.604 d.f. = n - 1 = 4 Test Statistic: D − for Managers Statistics μD = − 4.2 − 0 Using t= = − 1.66 SD / n 5.67/ 2004 Microsoft Excel, 4e © 5 Prentice-Hall, Inc. Reject Reject α/2 α/2 - 4.604 4.604 - 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. Chap 9-30
Two Population Proportions Population proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, p1 – p2 Assumptions: n1p1 ≥ 5 , n1(1-p1) ≥ 5 n2p2 ≥ 5 , n2(1-p2) ≥ 5 The point estimate for Statistics for Managers Using is the difference Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ps1 − ps2 Chap 9-31
Two Population Proportions Population proportions Since we begin by assuming the null hypothesis is true, we assume p1 = p2 and pool the two ps estimates The pooled estimate for the overall proportion is: X1 + X 2 p= n1 + n2 where X1 and X2 are the numbers from Statistics for Managers Using 1 and 2 with the characteristic of samples Microsoft Excel, 4e © 2004 interest Prentice-Hall, Inc. Chap 9-32
Two Population Proportions (continued) Population proportions The test statistic for p1 – p2 is a Z statistic: (p Z= s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 X Statistics for ManagerspUsing+ X 2 , p s = X1 , p s = X 2 where = 1 n1 n2 Microsoft Excel, 4e © 2004n1 + n2 Chap 9-33 Prentice-Hall, Inc. 1 2
Confidence Interval for Two Population Proportions Population proportions (p s1 The confidence interval for p1 – p2 is: ) − p s2 ± Z Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. p s1 (1 − p s1 ) n1 + p s2 (1 − p s2 ) n2 Chap 9-34
Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 ≥ p2 H1: p1 < p2 H0: p1 ≤ p2 H1: p1 > p2 H0: p1 = p2 H1: p1 ≠ p2 i.e., i.e., i.e., H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-35
Hypothesis Tests for Two Population Proportions (continued) Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-36
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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-37
Example: Two population Proportions (continued) The hypothesis test is:  H0: p1 – p2 = 0 (the two proportions are equal) H1: p1 – p2 ≠ 0 (there is a significant difference between proportions)  The sample proportions are:  Men:  Women: ps1 = 36/72 = .50 ps2 = 31/50 = .62  The pooled estimate for the overall proportion is: X1 + X 2 36 + 31 67 p= = = = .549 Statistics for Managers Using + 50 122 n1 + n2 72 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-38
Example: Two population Proportions (continued) Reject H0 The test statistic for p1 – p2 is: (p z= = s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 ( .50 − .62) − ( 0) 1   1 .549 (1 − .549)  +  72 50   Reject H0 .025 .025 -1.96 -1.31 = − 1.31 Critical Values = ±1.96 StatisticsFor Managers Using for α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1.96 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. 9-39 Chap
Hypothesis Tests for Variances Tests for Two Population Variances F test statistic * H0: σ12 = σ22 H1: σ12 ≠ σ22 Two-tail test H0: σ12 ≥ σ22 H1: σ12 < σ22 Lower-tail test H0: σ12 ≤ σ22 H1: σ12 > σ22 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Upper-tail test Chap 9-40
Hypothesis Tests for Variances (continued) Tests for Two Population Variances F test statistic The F test statistic is: * 2 1 2 2 S F= S 2 S1 = Variance of Sample 1 n1 - 1 = numerator degrees of freedom S 2 = Variance of Sample 2 2 Statistics for Managers Using n2 - 1 = denominator degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-41 Prentice-Hall, Inc.
The F Distribution  The F critical value is found from the F table  The are two appropriate degrees of freedom: numerator and denominator 2 S1 F= 2 S2  where df1 = n1 – 1 ; df2 = n2 – 1 In the F table,  numerator degrees of freedom determine the column  denominator degrees of freedom determine the row Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-42
Finding the Rejection Region H0: σ12 ≥ σ22 H1: σ12 < σ22 α α/2 0 Reject H0 FL F Do not reject H0 Reject H0 H0: σ12 ≤ σ22 H1: σ12 > σ22 α 0 Do StatisticsnotH Managers Using F for F Reject H reject U Microsoft Excel, 4e © 2004 Reject H if F > F Prentice-Hall, Inc. 0 0 U α/2 0 Reject H0 if F < FL 0 H0: σ12 = σ22 H1: σ12 ≠ σ22 FL Do not reject H0 FU rejection region for a two-tail test is:  Reject H0 2 S1 F = 2 > FU S2 2 S1 F = 2 < FL S2 Chap 9-43 F
Finding the Rejection Region (continued) α/2 H0: σ12 = σ22 H1: σ12 ≠ σ22 α/2 0 Reject H0 To find the critical F values: 1. Find FU from the F table for n1 – 1 numerator and n2 – 1 denominator degrees of Managers Statistics forfreedom Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. FL Do not reject H0 FU Reject H0 F 1 2. Find FL using the formula: FL = FU* Where FU* is from the F table with n2 – 1 numerator and n1 – 1 denominator degrees of freedom (i.e., switch the d.f. from FU) Chap 9-44
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 21 25 Mean 3.27 2.53 Std dev 1.30 1.16 Is there a difference in the variances between the NYSE Statistics for ManagersαUsing level? & NASDAQ at the = 0.05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-45
F Test: Example Solution  Form the hypothesis test: H0: σ21 – σ22 = 0 (there is no difference between variances) H1: σ21 – σ22 ≠ 0  (there is a difference between variances) Find the F critical values for α = .05: FU:  Numerator:   n1 – 1 = 21 – 1 = 20 d.f. Denominator:  FL:  Numerator: n2 – 1 = 25 – 1 = 24 d.f. Statistics UforF.025, 20, 24 = 2.33 F = Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.   n2 – 1 = 25 – 1 = 24 d.f. Denominator:  n1 – 1 = 21 – 1 = 20 d.f. FL = 1/F.025, 24, 20 = 1/2.41 = .41 Chap 9-46
F Test: Example Solution (continued)  The test statistic is: 2 1 2 2 H0: σ12 = σ22 H1: σ12 ≠ σ22 2 S 1.30 F= = = 1.256 2 S 1.16 α/2 = .025 0 α/2 = .025 Reject H0  F = 1.256 is not in the rejection region, so we do not reject H0 Do not reject H0 FL=0.41 Conclusion: There is not sufficient evidence of a difference in variances Statistics for Managers Using at α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 FU=2.33  Chap 9-47 F
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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-48
Two-Sample Tests in PHStat Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-49
Sample PHStat Output Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-50
Sample PHStat Output (continued) Input Output Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-51
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 Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-52
Chapter Summary (continued)  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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-53

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Chap09 2 sample test

  • 1.
    Statistics for Managers UsingMicrosoft® Excel 4th Edition Chapter 9 Two-Sample Tests Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-1
  • 2.
    Chapter Goals After completingthis 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 Statistics for Managers Using the difference between two variances Microsoft Excel, 4e © 2004 Chap 9-2 Prentice-Hall, Inc. 
  • 3.
    Two Sample Tests TwoSample Tests Population Means, Independent Samples Means, Related Samples Population Proportions Population Variances Examples: Group 1 vs. independent Group 2 Statistics for Same group before vs. after treatment Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 Chap 9-3
  • 4.
    Difference Between TwoMeans Population means, independent samples * σ1 and σ2 known Goal: Test hypotheses or form a confidence interval for the difference between two population means, μ1 – μ2 σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. The point estimate for the difference is X1 – X2 Chap 9-4
  • 5.
    Independent Samples Population means, independent samples  * Differentdata sources  Unrelated  Independent  σ1 and σ2 known σ1 and σ2 unknown   Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 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 Chap 9-5
  • 6.
    Difference Between TwoMeans Population means, independent samples * σ1 and σ2 known Use a Z test statistic σ1 and σ2 unknown Use S to estimate unknown σ , use a t test statistic and pooled standard deviation Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-6
  • 7.
    σ1 and σ2Known Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown Assumptions: *  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are ≥ 30  Population standard Statistics for Managers Using deviations are known Microsoft Excel, 4e © 2004 Chap 9-7 Prentice-Hall, Inc.
  • 8.
    σ1 and σ2Known (continued) Population means, independent samples σ1 and σ2 known 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 X1 – X2 is σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. σ X1 − X2 2 1 2 σ σ2 = + n1 n2 Chap 9-8
  • 9.
    σ1 and σ2Known (continued) Population means, independent samples σ1 and σ2 known The test statistic for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (X Z= 1 ) − X 2 − ( μ1 − μ2 ) 2 1 2 σ σ2 + n1 n2 Chap 9-9
  • 10.
    Hypothesis Tests for TwoPopulation Means Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 ≥ μ2 H1: μ1 < μ2 H0: μ1 ≤ μ2 H1: μ1 > μ2 H0: μ1 = μ2 H1: μ1 ≠ μ2 i.e., i.e., i.e., H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-10
  • 11.
    Hypothesis tests forμ1 – μ2 Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 ≥ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-11
  • 12.
    Confidence Interval, σ1 andσ2 Known Population means, independent samples σ1 and σ2 known The confidence interval for μ1 – μ2 is: * σ1 and σ2 unknown Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ( ) 2 1 2 σ σ2 X1 − X 2 ± Z + n1 n2 Chap 9-12
  • 13.
    σ1 and σ2Unknown Assumptions: Population means, independent samples  Samples are randomly and independently drawn σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but assumed equal Chap 9-13
  • 14.
    σ1 and σ2Unknown (continued) Forming interval estimates: Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown *  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 Statistics for Managers Using with (n1 + n2 – 2) degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-14 Prentice-Hall, Inc.
  • 15.
    σ1 and σ2Unknown (continued) Population means, independent samples The pooled standard deviation is σ1 and σ2 known σ1 and σ2 unknown * Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Sp = ( n1 − 1) S12 + ( n2 − 1) S2 2 (n1 − 1) + (n2 − 1) Chap 9-15
  • 16.
    σ1 and σ2Unknown (continued) The test statistic for μ1 – μ2 is: Population means, independent samples (X t= 1 σ1 and σ2 known σ1 and σ2 unknown ) − X 2 − ( μ1 − μ2 ) 1 1  S  +  n n  2   1 2 p * Where t has (n1 + n2 – 2) d.f., Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. and S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-16
  • 17.
    Confidence Interval, σ1 andσ2 Unknown Population means, independent samples The confidence interval for μ1 – μ2 is: σ1 and σ2 known σ1 and σ2 unknown (X 1 * ) − X 2 ± t n1 +n2 -2 1 1  S  +  n n  2   1 2 p Where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. S 2 p ( n1 − 1) S12 + ( n2 − 1) S2 2 = (n1 − 1) + (n2 − 1) Chap 9-17
  • 18.
    Pooled Sp tTest: 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 Number 21 Sample mean 3.27 Sample std dev 1.30 NASDAQ 25 2.53 1.16 Assuming both populations are approximately normal with equal variances, is there for Managers average Statistics a difference in Using yield (α = 0.05)? Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-18
  • 19.
    Calculating the TestStatistic The test statistic is: (X t= 1 ) − X 2 − ( μ1 − μ2 ) 1 1 S  +  n n  2   1 = 2 p ( 3.27 − 2.53 ) − 0 1   1 1.5021 +   21 25  ( n1 − 1)S12 + ( n2 − 1)S2 2 = ( 21 − 1)1.30 2 + ( 25 − 1)1.16 2 S2 = p (n1 − 1) + (n2 − 1) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. (21 - 1) + (25 − 1) = 2.040 = 1.5021 Chap 9-19
  • 20.
    Solution H0: μ1 -μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) α = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: 3.27 − 2.53 t= = 2.040 1   1 1.5021  +   21 25  Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Decision: Reject H0 at α = 0.05 Conclusion: There is evidence of a difference in means. Chap 9-20
  • 21.
    Related Samples Tests Meansof 2 Related Populations Related samples    Paired or matched samples Repeated measures (before/after) Use difference between paired values: D = X1 - X2 Eliminates Variation Among Subjects  Assumptions:  Both Populations Are Normally Distributed  Or, if Not Normal, use large samples Statistics for Managers Using Microsoft Excel, 4e © 2004 Chap 9-21 Prentice-Hall, Inc. 
  • 22.
    Mean Difference, σDKnown The ith paired difference is Di , where Related samples Di = X1i - X2i The point estimate for the population mean paired difference is D : n D= ∑D i= 1 i n Suppose the population standard deviation of the difference scores, σD, is known Statistics for Managers Using n is the number of pairs in the paired sample Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-22
  • 23.
    Mean Difference, σDKnown (continued) Paired samples The test statistic for the mean difference is a Z value: D − μD Z= σD n Where μD = hypothesized mean difference σD = population standard dev. of differences Managers Using n = the sample size (number of pairs) Statistics for Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-23
  • 24.
    Confidence Interval, σDKnown Paired samples The confidence interval for D is σD D±Z n Where n = the sample size (number of pairs in the paired sample) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-24
  • 25.
    Mean Difference, σDUnknown Related samples If σD is unknown, we can estimate the unknown population standard deviation with a sample standard deviation: The sample standard deviation is Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. n SD = (Di − D)2 ∑ i=1 n −1 Chap 9-25
  • 26.
    Mean Difference, σDUnknown (continued) Paired samples The test statistic for D is now a t statistic, with n-1 d.f.: D − μD t= SD n n Where t has n - 1 d.f. and Statistics for Managers Using SD is: Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D i=1 i − D) 2 n−1 Chap 9-26
  • 27.
    Confidence Interval, σDUnknown Paired samples The confidence interval for D is SD D ± t n−1 n n where Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. SD = ∑ (D − D) i=1 2 i n −1 Chap 9-27
  • 28.
    Hypothesis Testing for MeanDifference, σD Unknown Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μD ≥ 0 H1: μD < 0 H0: μD ≤ 0 H1: μD > 0 H0: μD = 0 H1: μD ≠ 0 α α -tα Reject H if t < -t tα Reject H if t > t α α Statistics 0 Managers Using 0 for Microsoft Excel, 4e © Where t has n - 1 d.f. 2004 Prentice-Hall, Inc. α /2 α /2 -tα/2 tα/2 Reject H0 if t < -tα/2 or t > tα/2 Chap 9-28
  • 29.
    Paired Samples Example Assumeyou 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 4 6 2 0 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. - 2 -14 - 1 0 - 4 -21 D = Σ Di n = -4.2 SD = ∑ (D − D) i n −1 = 5.67 Chap 9-29 2
  • 30.
    Paired Samples: Solution Has the training made a difference in the number of complaints (at the 0.01 level)? H0: μD = 0 H1: μD ≠ 0 α = .01 D = - 4.2 Critical Value = ± 4.604 d.f. = n - 1 = 4 Test Statistic: D − for Managers Statistics μD = − 4.2 − 0 Using t= = − 1.66 SD / n 5.67/ 2004 Microsoft Excel, 4e © 5 Prentice-Hall, Inc. Reject Reject α/2 α/2 - 4.604 4.604 - 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. Chap 9-30
  • 31.
    Two Population Proportions Population proportions Goal:test a hypothesis or form a confidence interval for the difference between two population proportions, p1 – p2 Assumptions: n1p1 ≥ 5 , n1(1-p1) ≥ 5 n2p2 ≥ 5 , n2(1-p2) ≥ 5 The point estimate for Statistics for Managers Using is the difference Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. ps1 − ps2 Chap 9-31
  • 32.
    Two Population Proportions Population proportions Sincewe begin by assuming the null hypothesis is true, we assume p1 = p2 and pool the two ps estimates The pooled estimate for the overall proportion is: X1 + X 2 p= n1 + n2 where X1 and X2 are the numbers from Statistics for Managers Using 1 and 2 with the characteristic of samples Microsoft Excel, 4e © 2004 interest Prentice-Hall, Inc. Chap 9-32
  • 33.
    Two Population Proportions (continued) Population proportions Thetest statistic for p1 – p2 is a Z statistic: (p Z= s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 X Statistics for ManagerspUsing+ X 2 , p s = X1 , p s = X 2 where = 1 n1 n2 Microsoft Excel, 4e © 2004n1 + n2 Chap 9-33 Prentice-Hall, Inc. 1 2
  • 34.
    Confidence Interval for TwoPopulation Proportions Population proportions (p s1 The confidence interval for p1 – p2 is: ) − p s2 ± Z Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. p s1 (1 − p s1 ) n1 + p s2 (1 − p s2 ) n2 Chap 9-34
  • 35.
    Hypothesis Tests for TwoPopulation Proportions Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 ≥ p2 H1: p1 < p2 H0: p1 ≤ p2 H1: p1 > p2 H0: p1 = p2 H1: p1 ≠ p2 i.e., i.e., i.e., H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-35
  • 36.
    Hypothesis Tests for TwoPopulation Proportions (continued) Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: p1 – p2 ≥ 0 H1: p1 – p2 < 0 H0: p1 – p2 ≤ 0 H1: p1 – p2 > 0 H0: p1 – p2 = 0 H1: p1 – p2 ≠ 0 α α -zα Reject H0 if Z < -Zα zα Reject H0 if Z > Zα Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. α /2 α /2 -zα/2 zα/2 Reject H0 if Z < -Zα/2 or Z > Zα/2 Chap 9-36
  • 37.
    Example: Two population Proportions Isthere 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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-37
  • 38.
    Example: Two population Proportions (continued) Thehypothesis test is:  H0: p1 – p2 = 0 (the two proportions are equal) H1: p1 – p2 ≠ 0 (there is a significant difference between proportions)  The sample proportions are:  Men:  Women: ps1 = 36/72 = .50 ps2 = 31/50 = .62  The pooled estimate for the overall proportion is: X1 + X 2 36 + 31 67 p= = = = .549 Statistics for Managers Using + 50 122 n1 + n2 72 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-38
  • 39.
    Example: Two population Proportions (continued) RejectH0 The test statistic for p1 – p2 is: (p z= = s1 ) − p s2 − ( p1 − p 2 ) 1 1 p (1 − p)  +  n n  2   1 ( .50 − .62) − ( 0) 1   1 .549 (1 − .549)  +  72 50   Reject H0 .025 .025 -1.96 -1.31 = − 1.31 Critical Values = ±1.96 StatisticsFor Managers Using for α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1.96 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. 9-39 Chap
  • 40.
    Hypothesis Tests forVariances Tests for Two Population Variances F test statistic * H0: σ12 = σ22 H1: σ12 ≠ σ22 Two-tail test H0: σ12 ≥ σ22 H1: σ12 < σ22 Lower-tail test H0: σ12 ≤ σ22 H1: σ12 > σ22 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Upper-tail test Chap 9-40
  • 41.
    Hypothesis Tests forVariances (continued) Tests for Two Population Variances F test statistic The F test statistic is: * 2 1 2 2 S F= S 2 S1 = Variance of Sample 1 n1 - 1 = numerator degrees of freedom S 2 = Variance of Sample 2 2 Statistics for Managers Using n2 - 1 = denominator degrees of freedom Microsoft Excel, 4e © 2004 Chap 9-41 Prentice-Hall, Inc.
  • 42.
    The F Distribution  TheF critical value is found from the F table  The are two appropriate degrees of freedom: numerator and denominator 2 S1 F= 2 S2  where df1 = n1 – 1 ; df2 = n2 – 1 In the F table,  numerator degrees of freedom determine the column  denominator degrees of freedom determine the row Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-42
  • 43.
    Finding the RejectionRegion H0: σ12 ≥ σ22 H1: σ12 < σ22 α α/2 0 Reject H0 FL F Do not reject H0 Reject H0 H0: σ12 ≤ σ22 H1: σ12 > σ22 α 0 Do StatisticsnotH Managers Using F for F Reject H reject U Microsoft Excel, 4e © 2004 Reject H if F > F Prentice-Hall, Inc. 0 0 U α/2 0 Reject H0 if F < FL 0 H0: σ12 = σ22 H1: σ12 ≠ σ22 FL Do not reject H0 FU rejection region for a two-tail test is:  Reject H0 2 S1 F = 2 > FU S2 2 S1 F = 2 < FL S2 Chap 9-43 F
  • 44.
    Finding the RejectionRegion (continued) α/2 H0: σ12 = σ22 H1: σ12 ≠ σ22 α/2 0 Reject H0 To find the critical F values: 1. Find FU from the F table for n1 – 1 numerator and n2 – 1 denominator degrees of Managers Statistics forfreedom Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. FL Do not reject H0 FU Reject H0 F 1 2. Find FL using the formula: FL = FU* Where FU* is from the F table with n2 – 1 numerator and n1 – 1 denominator degrees of freedom (i.e., switch the d.f. from FU) Chap 9-44
  • 45.
    F Test: AnExample 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 Statistics for ManagersαUsing level? & NASDAQ at the = 0.05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-45
  • 46.
    F Test: ExampleSolution  Form the hypothesis test: H0: σ21 – σ22 = 0 (there is no difference between variances) H1: σ21 – σ22 ≠ 0  (there is a difference between variances) Find the F critical values for α = .05: FU:  Numerator:   n1 – 1 = 21 – 1 = 20 d.f. Denominator:  FL:  Numerator: n2 – 1 = 25 – 1 = 24 d.f. Statistics UforF.025, 20, 24 = 2.33 F = Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.   n2 – 1 = 25 – 1 = 24 d.f. Denominator:  n1 – 1 = 21 – 1 = 20 d.f. FL = 1/F.025, 24, 20 = 1/2.41 = .41 Chap 9-46
  • 47.
    F Test: ExampleSolution (continued)  The test statistic is: 2 1 2 2 H0: σ12 = σ22 H1: σ12 ≠ σ22 2 S 1.30 F= = = 1.256 2 S 1.16 α/2 = .025 0 α/2 = .025 Reject H0  F = 1.256 is not in the rejection region, so we do not reject H0 Do not reject H0 FL=0.41 Conclusion: There is not sufficient evidence of a difference in variances Statistics for Managers Using at α = .05 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Reject H0 FU=2.33  Chap 9-47 F
  • 48.
    Two-Sample Tests inEXCEL 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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-48
  • 49.
    Two-Sample Tests inPHStat Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-49
  • 50.
    Sample PHStat Output Input Output Statisticsfor Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-50
  • 51.
    Sample PHStat Output (continued) Input Output Statisticsfor Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-51
  • 52.
    Chapter Summary  Compared twoindependent 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 Statistics for Managers Using  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-52
  • 53.
    Chapter Summary (continued)  Compared twopopulation 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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 9-53