Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-1 Chapter 10 Two-Sample Tests Basic Business Statistics 10 th Edition

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-2 Learning Objectives In this chapter, you learn:  How to use hypothesis testing for comparing the difference between  The means of two independent populations  The means of two related populations  The proportions of two independent populations  The variances of two independent populations

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-3 Two-Sample Tests Population Means, Independent Samples Means, Related Samples Population Variances Population 1 vs. independent Population 2 Same population before vs. after treatment Variance 1 vs. Variance 2 Examples: Population Proportions Proportion 1 vs. Proportion 2

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-4 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known Goal: Test hypothesis 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 * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-5 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, a pooled-variance t test, or a separate-variance t test * σ 1 and σ 2 known σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-6 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known * Use a Z test statistic Use S p to estimate unknown σ, use a t test statistic and pooled standard deviation σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal Use S 1 and S 2 to estimate unknown σ 1 and σ 2, use a separate-variance t test

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-7 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, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-8 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, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 10-9 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 * (continued) σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 zz 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown, Assumed Equal 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, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known (continued) * Forming interval estimates:  The population variances are assumed equal, so use the two sample variances and pool them to estimate the common σ 2  the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known The pooled variance is (continued) * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 – μ 2 is: * (continued) σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known The confidence interval for μ 1 – μ 2 is: Where * Confidence Interval, σ 1 and σ 2 Unknown σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 Sample mean Sample std dev Assuming both populations are approximately normal with equal variances, is there a difference in average yield (  = 0.05)?

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Calculating the Test Statistic The test statistic is:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 = = 44 Critical Values: t = ± Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a difference in means. t Reject H

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 Unknown, Not Assumed Equal Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but cannot be assumed to be equal * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known (continued) * Forming the test statistic:  The population variances are not assumed equal, so include the two sample variances in the computation of the t-test statistic  the test statistic is a t value with v degrees of freedom (see next slide) σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Not Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known The number of degrees of freedom is the integer portion of: (continued) * σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Not Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Population means, independent samples σ 1 and σ 2 known The test statistic for μ 1 – μ 2 is: * (continued) σ 1 and σ 2 unknown, assumed equal σ 1 and σ 2 unknown, not assumed equal σ 1 and σ 2 Unknown, Not Assumed Equal

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Related Populations 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 i = X 1i - X 2i

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Confidence Interval, σ D Known The confidence interval for μ D is Paired samples Where n = the sample size (number of pairs in the paired sample)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  Use a paired t test, 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)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap The confidence interval for μ D is Paired samples where Confidence Interval, σ D Unknown

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 tt 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.

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  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: Paired t Test Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, D i C.B T.F M.H R.K M.O D =  DiDi n = -4.2

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  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 = ± d.f. = n - 1 = 4 Reject  / 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 t Test: Solution Reject  /  =.01

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Two Population Proportions Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π 1 – π 2 The point estimate for the difference is Population proportions Assumptions: n 1 π 1  5, n 1 (1- π 1 )  5 n 2 π 2  5, n 2 (1- π 2 )  5

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 π 1 = π 2 and pool the two sample estimates

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Two Population Proportions Population proportions The test statistic for p 1 – p 2 is a Z statistic: (continued) where

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Confidence Interval for Two Population Proportions Population proportions The confidence interval for π 1 – π 2 is:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : π 1  π 2 H 1 : π 1 < π 2 i.e., H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 ≤ π 2 H 1 : π 1 > π 2 i.e., H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 = π 2 H 1 : π 1 ≠ π 2 i.e., H 0 : π 1 – π 2 = 0 H 1 : π 1 – π 2 ≠ 0

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Hypothesis Tests for Two Population Proportions Population proportions Lower-tail test: H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 – π 2 = 0 H 1 : π 1 – π 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if Z < -Z  Reject H 0 if Z > Z  Reject H 0 if Z < -Z   or Z > Z  (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  The hypothesis test is: H 0 : π 1 – π 2 = 0 (the two proportions are equal) H 1 : π 1 – π 2 ≠ 0 (there is a significant difference between proportions)  The sample proportions are:  Men: p 1 = 36/72 =.50  Women: p 2 = 31/50 =.62  The pooled estimate for the overall proportion is: Example: Two population Proportions (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap The test statistic for π 1 – π 2 is: Example: Two population Proportions (continued) 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 *

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Hypothesis Tests for Variances Tests for Two Population Variances F test statistic The F test statistic is: = Variance of Sample 1 n = numerator degrees of freedom n = denominator degrees of freedom = Variance of Sample 2 * (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  The F critical value is found from the F table  There 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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 Mean Std dev Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.05 level?

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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  = 0.05:  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 = FU:FU:FL:FL:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap  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 = 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 =0.43  /2 =.025 Reject H 0 F

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Two-Sample Tests in EXCEL For independent samples:  Independent sample Z test with variances known:  Tools | data analysis | z-test: two sample for means  Pooled variance t test:  Tools | data analysis | t-test: two sample assuming equal variances  Separate-variance t test:  Tools | data analysis | t-test: two sample assuming unequal variances 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap Chapter Summary  Compared two independent samples  Performed Z test for the difference in two means  Performed pooled variance t test for the difference in two means  Performed separate-variance t test for difference in two means  Formed confidence intervals for the difference 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 mean difference

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 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)