© 2004 Prentice-Hall, Inc.Chap 12-1 Basic Business Statistics (9 th Edition) Chapter 12 Tests for Two or More Samples with Categorical Data

© 2004 Prentice-Hall, Inc. Chap 12-2 Chapter Topics Z Test for Differences in Two Proportions (Independent Samples)  2 Test for Differences in Two Proportions (Independent Samples)  2 Test for Differences in More than Two Proportions (Independent Samples) Marascuilo Procedure  2 Test of Independence

© 2004 Prentice-Hall, Inc. Chap 12-3 Z Test for Differences in Two Proportions What is It Used For? To determine whether there is a difference between 2 population proportions or whether one is larger than the other Assumptions: Independent samples Population follows binomial distribution Sample size large enough: np  5 and n(1-p)  5 for each population

© 2004 Prentice-Hall, Inc. Chap 12-4 Z Test Statistic where X 1 = Number of Successes in Sample 1 X 2 = Number of Successes in Sample 2 Pooled Estimate of the Population Proportion

© 2004 Prentice-Hall, Inc. Chap 12-5 The Hypotheses for the Z Test Research Questions Hypothesi s No Difference Any Difference Prop 1  Prop 2 Prop 1 < Prop 2 Prop 1  Prop 2 Prop 1 > Prop 2 H 0 p 1 - p 2  p 1 -p 2  0p 1 -p 2  0 H 1 p 1 -p 2  0 p 1 -p 2 < 0p 1 - p 2 > 0

© 2004 Prentice-Hall, Inc. Chap 12-6 Z Test for Differences in Two Proportions: Example As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the 0.01 significance level, is there a difference in perceptions?

© 2004 Prentice-Hall, Inc. Chap 12-7 Calculating the Test Statistic

© 2004 Prentice-Hall, Inc. Chap 12-8 Z Test for Differences in Two Proportions: Solution H 0 : p 1 - p 2 = 0 H 1 : p 1 - p 2  0  = 0.01 n 1 = 78 n 2 = 82 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  = There is evidence of a difference in proportions. Z  290. Z Reject H

© 2004 Prentice-Hall, Inc. Chap 12-9 Z Test for Differences in Two Proportions in PHStat PHStat | Two-Sample Tests | Z Test for Differences in Two Proportions … Example Solution in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap Confidence Interval for Differences in Two Proportions The Confidence Interval for Differences in Two Proportions

© 2004 Prentice-Hall, Inc. Chap Confidence Interval for Differences in Two Proportions: Example As personnel director, you want to find out the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. Construct a 99% confidence interval for the difference in two proportions.

© 2004 Prentice-Hall, Inc. Chap Confidence Interval for Differences in Two Proportions: Solution We are 99% confident that the difference between two proportions is somewhere between and

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions: Basic Idea Compares Observed to Expected Frequencies if Null Hypothesis is True The Closer Observed Frequencies are to Expected Frequencies, the More Likely the H 0 is True Measured by squared difference relative to expected frequency Sum of relative squared differences is the test statistic

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions: Contingency Table Evaluation Method Perception12Total Fair Unfair Total Contingency Table (Observed Frequencies) for Comparing Fairness of Performance Evaluation Methods 2 Populations Levels of Variable

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions: Expected Frequencies 112 of 160 Total are “Fair” ( ) 78 Used Evaluation Method 1 Expect (78  112/160) = 54.6 to be “Fair” Evaluation Method Perception12 Total Fair Unfair Total

© 2004 Prentice-Hall, Inc. Chap The  2 Test Statistic

© 2004 Prentice-Hall, Inc. Chap Computation of the  2 Test Statistic f 0 f e (f 0 - f e ) (f 0 - f e ) 2 (f 0 - f e ) 2 / f e Sum = Observed Frequencies Expected Frequencies

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions: Finding the Critical Value  Reject r = 2 (# rows in contingency table) c = 2 (# columns)  =.01 df = (r - 1)(c - 1) = 1  2 Table (Portion) Upper Tail Area DF.995 ….95 … … … … …

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions: Solution H 0 : p 1 - p 2 = 0 H 1 : p 1 - p 2  0 Test Statistic = Decision: Conclusion:  0 Reject  =.01 Reject at  = There is evidence of a difference in proportions. Note: The conclusion obtained using   test is the same as using Z Test. Caution! Each expected frequency should be  5.

© 2004 Prentice-Hall, Inc. Chap  2 Test for Two Proportions in PHStat PHStat | Two-Sample Tests | Chi-Square Test for Differences in Two Proportions … Example Solution in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap  2 Test for More Than Two Proportions Extends the  2 Test to the General Case of c Independent Populations Tests for Equality (=) of Proportions Only Uses Contingency Table Assumptions: Independent random samples “Large” sample sizes All expected frequencies  1

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Hypotheses and Statistic Hypotheses H 0 : p 1 = p 2 =... = p c H 1 : Not all p j are equal Test statistic Degrees of freedom: (r - 1)(c - 1) Observed frequency Expected frequency # Rows # Columns

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Example The University is thinking of switching to a trimester academic calendar. A random sample of 100 undergraduates, 50 graduate students and 50 faculty members were surveyed. OpinionUnderGradFaculty Favor Oppose Totals Test at the.01 level of significance to determine if there is evidence of a difference in attitude between the groups.

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Example (continued) 1. Set Hypotheses: H 0 : p 1 = p 2 = p 3 H 1 : Not all p j are equal 2. Contingency Table: OpinionUnderGradFacultyTotals Favor Oppose Totals

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Example (continued) OpinionUnderGradFacultyTotals Favor Oppose Totals Compute Expected Frequencies (100)(120)/200=60 (50)(80)/200=20 All expected frequencies are large.

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Example (continued) 4. Compute Test Statistic: f 0 f e (f 0 - f e )(f 0 - f e ) 2 (f 0 - f e ) 2 / f e Test Statistic  2 =

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions: Example Solution H 0 : p 1 = p 2 = p 3 H 1 : Not all p j are equal Decision: Conclusion: df = (c – 1)(r - 1) = = 2 Reject  =.01  Do Not Reject H 0. Since  2 =12.792, there is sufficient evidence of a difference in attitude among the groups.

© 2004 Prentice-Hall, Inc. Chap  2 Test for c Proportions in PHStat PHStat | c-Sample Tests | Chi-Square Test … Example Solution in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap Marascuilo Procedure Used when the Test for c Proportions is Rejected Compares All Pairs of Groups The Marascuilo Multiple Comparison Procedure: Compute among all pairs of groups The critical range for a pair is A pair is considered significantly different if critical range

© 2004 Prentice-Hall, Inc. Chap Marascuilo Procedure : Example The University is thinking of switching to a trimester academic calendar. A random sample of 100 undergraduates, 50 graduate students and 50 faculty members were surveyed. OpinionUnderGradFaculty Favor Oppose Totals Using a 1% level of significance, which groups have a different attitude?

© 2004 Prentice-Hall, Inc. Chap Marascuilo Procedure : Solution At 1% level of significance, there is evidence of a difference in attitude between graduate students and faculty. Excel Output:

© 2004 Prentice-Hall, Inc. Chap  2 Test of Independence Shows If a Relationship Exists between 2 Factors of Interest One sample drawn Each factor has 2 or more levels of responses Does not show nature of relationship Does not show causality Similar to Testing p 1 = p 2 = … = p c Used Widely in Marketing Uses Contingency Table

© 2004 Prentice-Hall, Inc. Chap  2 Test of Independence: Example A survey was conducted to determine whether there is a relationship between architectural style (Split-Level or Ranch) and geographical location (Urban or Rural). Given the survey data, test at the  =.01 level to determine whether there is a relationship between location and architectural style.

© 2004 Prentice-Hall, Inc. Chap House Location House StyleUrbanRuralTotal Split-Level Ranch Total  2 Test of Independence: Example 1. Set Hypotheses: H 0 : The 2 categorical variables (Architectural Style and Location) are independent H 1 : The 2 categorical variables are related 2. Contingency Table: Levels of Variable 2 Levels of Variable 1 (continued)

© 2004 Prentice-Hall, Inc. Chap  2 Test of Independence: Example (continued) 3. Computing Expected Frequencies Statistical independence : P(A and B) = P(A)·P(B) Compute marginal (row & column) probabilities & multiply for joint probability Expected frequency is sample size times joint probability House Location UrbanRural House Style Obs.Exp.Obs.Exp. Total Split-Level Ranch Total · ·

© 2004 Prentice-Hall, Inc. Chap f 0 f e (f 0 - f e )(f 0 - f e ) 2 (f 0 - f e ) 2 / f e  2 Test of Independence: Example (continued) 4. Calculate Test Statistic:  2 Test Statistic = All expected frequencies are large, i.e. > 1.

© 2004 Prentice-Hall, Inc. Chap  2 Test of Independence: Example Solution H 0 : The 2 categorical variables (Architectural Style and Location) are independent H 1 : The 2 categorical variables are related Decision: Conclusion: df = (r - 1)(c - 1) = 1 Reject  = Reject H 0 at  =.01. Since  2 =8.404, there is evidence that the choice of architectural design and location are related.

© 2004 Prentice-Hall, Inc. Chap  2 Test of Independence in PHStat PHStat | c-Sample Tests | Chi-Square Test … Example Solution in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap Chapter Summary Performed Z Test for Differences in Two Proportions (Independent Samples) Discussed   Test for Differences in Two Proportions (Independent Samples) Addressed  2 Test for Differences in More Than Two Proportions (Independent Samples) Illustrated Marascuilo Procedure Described  2 Test of Independence