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

2. © 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

3. © 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

4. © 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

5. © 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

6. © 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?

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

8. © 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  = 0.01. There is evidence of a difference in proportions. Z  290. Z 0 2.58-2.58.005 Reject H 0 0.005

9. © 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

10. © 2004 Prentice-Hall, Inc. Chap 12-10 Confidence Interval for Differences in Two Proportions The Confidence Interval for Differences in Two Proportions

11. © 2004 Prentice-Hall, Inc. Chap 12-11 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.

12. © 2004 Prentice-Hall, Inc. Chap 12-12 Confidence Interval for Differences in Two Proportions: Solution We are 99% confident that the difference between two proportions is somewhere between 0.0294 and 0.3909.

13. © 2004 Prentice-Hall, Inc. Chap 12-13  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

14. © 2004 Prentice-Hall, Inc. Chap 12-14  2 Test for Two Proportions: Contingency Table Evaluation Method Perception12Total Fair6349112 Unfair153348 Total 7882160 Contingency Table (Observed Frequencies) for Comparing Fairness of Performance Evaluation Methods 2 Populations Levels of Variable

15. © 2004 Prentice-Hall, Inc. Chap 12-15  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 Fair6349112 Unfair153348 Total 7882160

16. © 2004 Prentice-Hall, Inc. Chap 12-16 The  2 Test Statistic

17. © 2004 Prentice-Hall, Inc. Chap 12-17 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 6354.6 8.4 70.56 1.293 49 57.4 -8.4 70.56 1.229 1523.4 -8.4 70.56 3.015 3324.6 8.4 70.56 2.868 Sum = 8.405 Observed Frequencies Expected Frequencies

18. © 2004 Prentice-Hall, Inc. Chap 12-18  2 Test for Two Proportions: Finding the Critical Value  06.635 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 ….05 1... … 0.0043.841 20.0100.1035.991.025.01 5.024 7.378 6.635 9.210 … … …

19. © 2004 Prentice-Hall, Inc. Chap 12-19  2 Test for Two Proportions: Solution H 0 : p 1 - p 2 = 0 H 1 : p 1 - p 2  0 Test Statistic = 8.405 Decision: Conclusion: 6.635  0 Reject  =.01 Reject at  = 0.01. 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.

20. © 2004 Prentice-Hall, Inc. Chap 12-20  2 Test for Two Proportions in PHStat PHStat | Two-Sample Tests | Chi-Square Test for Differences in Two Proportions … Example Solution in Excel Spreadsheet

21. © 2004 Prentice-Hall, Inc. Chap 12-21  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

22. © 2004 Prentice-Hall, Inc. Chap 12-22  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

23. © 2004 Prentice-Hall, Inc. Chap 12-23  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 63 20 37 Oppose 37 30 13 Totals 100 50 50 Test at the.01 level of significance to determine if there is evidence of a difference in attitude between the groups.

24. © 2004 Prentice-Hall, Inc. Chap 12-24  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 63 20 37 120 Oppose 37 30 13 80 Totals 100 50 50 200

25. © 2004 Prentice-Hall, Inc. Chap 12-25  2 Test for c Proportions: Example (continued) OpinionUnderGradFacultyTotals Favor 60 30 30 120 Oppose 40 20 20 80 Totals 100 50 50 200 3. Compute Expected Frequencies (100)(120)/200=60 (50)(80)/200=20 All expected frequencies are large.

26. © 2004 Prentice-Hall, Inc. Chap 12-26  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 6360 3 9.15 2030 -10 100 3.3333 3730 7 49 1.6333 3740 -3 9.225 3020 10 100 5 1320 -7 49 2.45 Test Statistic  2 = 12.792

27. © 2004 Prentice-Hall, Inc. Chap 12-27  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) = 3 - 1 = 2 Reject  =.01  2 09.210 Do Not Reject H 0. Since  2 =12.792, there is sufficient evidence of a difference in attitude among the groups.

28. © 2004 Prentice-Hall, Inc. Chap 12-28  2 Test for c Proportions in PHStat PHStat | c-Sample Tests | Chi-Square Test … Example Solution in Excel Spreadsheet

29. © 2004 Prentice-Hall, Inc. Chap 12-29 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

30. © 2004 Prentice-Hall, Inc. Chap 12-30 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 63 20 37 Oppose 37 30 13 Totals 100 50 50 Using a 1% level of significance, which groups have a different attitude?

31. © 2004 Prentice-Hall, Inc. Chap 12-31 Marascuilo Procedure : Solution At 1% level of significance, there is evidence of a difference in attitude between graduate students and faculty. Excel Output:

32. © 2004 Prentice-Hall, Inc. Chap 12-32  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

33. © 2004 Prentice-Hall, Inc. Chap 12-33  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.

34. © 2004 Prentice-Hall, Inc. Chap 12-34 House Location House StyleUrbanRuralTotal Split-Level6349112 Ranch153348 Total 7882160  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)

35. © 2004 Prentice-Hall, Inc. Chap 12-35  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-Level6354.64957.4112 Ranch1523.43324.648 Total 78 82 160 78·112 160 82·112 160

36. © 2004 Prentice-Hall, Inc. Chap 12-36 f 0 f e (f 0 - f e )(f 0 - f e ) 2 (f 0 - f e ) 2 / f e 6354.6 8.4 70.56 1.292 4957.4 -8.4 70.56 1.229 1523.4 -8.4 70.56 3.015 3324.6 8.4 70.56 2.868 8.404  2 Test of Independence: Example (continued) 4. Calculate Test Statistic:  2 Test Statistic = 8.404 All expected frequencies are large, i.e. > 1.

37. © 2004 Prentice-Hall, Inc. Chap 12-37  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  =.01 06.635 Reject H 0 at  =.01. Since  2 =8.404, there is evidence that the choice of architectural design and location are related.

38. © 2004 Prentice-Hall, Inc. Chap 12-38  2 Test of Independence in PHStat PHStat | c-Sample Tests | Chi-Square Test … Example Solution in Excel Spreadsheet

39. © 2004 Prentice-Hall, Inc. Chap 12-39 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