Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-1 Chapter 10 Two-Sample Tests and One-Way ANOVA Business Statistics, A First Course 6 th Edition

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-2 Learning Objectives In this chapter, you learn hypothesis testing procedures to test:  The means of two independent populations  The proportions of two independent populations  The means of more than two populations

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-3 Chapter Overview One-Way Analysis of Variance (ANOVA) F-test Tukey-Kramer test Two-Sample Tests Population Means, Independent Samples Population Proportions

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-4 Tests For Equal Means Two-Sample Tests Categorical Variable has 2 categories If you conclude H 1, be sure to report sample means so that reader will know the direction of difference H 1 : Not all means are equal If you conclude H 1, be sure to carry out the Tukey-Kramer test to let readers know which means are significantly bigger or smaller C-Sample Tests Categorical Variable has c categories P-value from Tools - Data Analysis - ANOVA Single Factor Assuming normality and equal variability

Business Statistics, A First Course (4e) © 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 Single-Factor ANOVA to obtain p-value * σ 1 and σ 2 known σ 1 and σ 2 unknown, assumed equal

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-6 Population means, independent samples 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

Types of t Test  Separate variance t Test  If SD 1 > 2 * SD 2  F test for equality of two variances  Pooled Variance t Test  Paired t test  Repeated measures  Paired together according to some characteristic Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-7

F Test for equality of two variances  H 0 : σ 1 2 = σ 2 2  H 1 : σ 1 2 ≠ σ 2 2  The data set contains 480 ceramic strength measurements for two batches of material.  No of observation – 240  SD1 =  SD2 =  Do two samples come from populations with equal variances?  Does a new process, treatment, or test reduce the variability of the current process? Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-8

Purpose of two sample t Test  Is process 1 equivalent to process 2?  Is the new process better than the current process?  Is the new process better than the current process by at least some pre-determined threshold amount? Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 10-9

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap Single Factor ANOVA 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)?

Z test for difference between two proportions Sample Space Hotel Waikiki West Waikiki EastTotals Choose Again?Yes No Totals Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap  H 0 : P 1 = P 2 H 1 : P 1 ≠ P 2

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap Single Factor Analysis of Variance Analysis of Variance (ANOVA) F-test Tukey-Kramer test

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap One-Way Analysis of Variance  Evaluate the difference among the means of two or more groups Examples: Accident rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five brands of tires  Assumptions  Populations are normally distributed  Populations have equal variances  Samples are randomly and independently drawn

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap Hypotheses of One-Way ANOVA   All population means are equal  i.e., no treatment effect (no variation in means among groups)   At least one population mean is different  i.e., there is a treatment effect  Does not mean that all population means are different (some pairs may be the same)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap One-Way ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap One-Way ANOVA At least one mean is different: The Null Hypothesis is NOT true or (continued)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap One-Way ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the 0.05 significance level, is there a difference in mean distance? Club 1 Club 2 Club

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap One-Way ANOVA Example: Scatter Diagram Distance Club 1 Club 2 Club Club 1 2 3

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap SUMMARY GroupsCountSumAverageVariance Club Club Club ANOVA Source of Variation SSdfMSFP-valueF crit Between Groups E Within Groups Total One-Way ANOVA Excel Output EXCEL: tools | data analysis | ANOVA: single factor

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap The Tukey-Kramer Procedure  Tells which population means are significantly different  e.g.: μ 1 = μ 2  μ 3  Done after rejection of equal means in ANOVA  Allows pair-wise comparisons  Compare absolute mean differences with critical range x μ 1 = μ 2 μ 3

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap Tukey-Kramer Critical Range where: Q U = Value from Studentized Range Distribution with c and n - c degrees of freedom for the desired level of  (see appendix E.8 table) MSW = Mean Square Within n j and n j’ = Sample sizes from groups j and j’

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap The Tukey-Kramer Procedure: Example 1. Compute absolute mean differences: Club 1 Club 2 Club Find the Q U value from the table in appendix E.6 with c = 3 (numerator) and (n – c) = (15 – 3) = 12 (denominator) degrees of freedom for the desired level of  (  = 0.05 used here):

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap The Tukey-Kramer Procedure: Example 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. Thus, with 95% confidence we can conclude that the mean distance for club 1 is greater than club 2 and 3, and club 2 is greater than club Compute Critical Range: 4. Compare: (continued)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap Chapter Summary  Described single factor analysis of variance  The logic of ANOVA  ANOVA assumptions  F test for difference in c means  The Tukey-Kramer procedure for multiple comparisons