1. 10.1: Two-Way Tables and the Chi-Square Test Statistics Chap 10:Inference for Tables and Means

2. Two-Way Tables We use two-way tables to examine the relationship between two categorical variables. A categorical variable places an individual into one of several groups or categories.

3. Two-Way Tables Row Variable Column Variable Purdue University Faculty 1998-99 FemaleMaleTotal Assistant Professors126213339 Associate Professors149411560 Full Professors60662722 Total33512861621

4. Two-Way Tables Frequency counts Purdue University Faculty 1998-99 FemaleMaleTotal Assistant Professors126213339 Associate Professors149411560 Full Professors60662722 Total33512861621 Joint frequencies: Entries in the body of the table Marginal frequencies: Entries in the “Total” row or column

5. Two-Way Tables Relative frequencies Purdue University Faculty 1998-99 FemaleMaleTotal Assistant Professors0.0780.1310.209 Associate Professors0.0920.2540.345 Full Professors0.0370.4080.445 Total0.2070.7931.000 Conditional frequencies:The relative frequencies in the body of the table

6. Two-Way Tables Relative frequencies for rows Purdue University Faculty 1998-99 FemaleMaleTotal Assistant Professors0.3720.6281.000 Associate Professors0.2660.7341.000 Full Professors0.0830.9171.000 Total0.2070.7931.000

7. Two-Way Tables Relative frequencies for columns Purdue University Faculty 1998-99 FemaleMaleTotal Assistant Professors0.3760.1660.209 Associate Professors0.4450.3200.345 Full Professors0.1790.5150.445 Total1.000

8. Simpson’s Paradox An association or comparison that holds for all of several groups can disappear or even reverse direction when the data are combined to form a single group.

9. Simpson’s Paradox University Admissions MaleFemaleTotal Admit352055 Deny454085 Total8060 Frequency counts University Admissions MaleFemaleTotal Admit0.440.330.39 Deny0.560.670.61 Total1.00 Relative frequencies

10. Simpson’s Paradox Engineering Admissions MaleFemaleTotal Admit301040 Deny301040 Total602080 Frequency counts English Admissions MaleFemaleTotal Admit51015 Deny153045 Total204060

11. Simpson’s Paradox Engineering Admissions MaleFemaleTotal Admit0.50 Deny0.50 Total1.00 Relative frequencies English Admissions MaleFemaleTotal Admit0.25 Deny0.75 Total1.00

12. Activity 10.1: Sports Preference Freq Counts BasketballFootballSoccerOtherTotal Girls Boys Total

13. Activity 10.1: Sports Preference Rel Freq BasketballFootballSoccerOtherTotal Girls Boys Total

14. Exercises p 557-559: 10.1, 10.3, 10.5

15. Example 10.4: Treating Cocaine Addiction Frequency counts GroupTreatmentSuccessFailureTotal 1Desipramine141024 2Lithium61824 3Placebo42024 Total244872

16. Expected Counts

17. Exercises p 562-559: 10.6, 10.8, 10.10

18. Exercise 10.10 p 562-559: 10.6, 10.8, 10.10

19. Chi-square Statistic

20. Example 10.5: The Cocaine Study ObservedExpected TreatmentSuccessFailureSuccessFailure Desipramine1410816 Lithium618816 Placebo420816

21. Chi-square Statistic

22. The Chi-square Distributions

23. Figure 10.2 The density curves for three members of the chi-square family of distributions. The sampling distributions of chi-square statistics belong to this family.

24. The Chi-square Distributions

25. Example 10.6: The Cocaine Study, Conclusion ObservedExpected TreatmentSuccessFailureSuccessFailure Desipramine1410816 Lithium618816 Placebo420816

26. Exercises p 566: 10.11, 10.13, 10.15

27. Using the chi-square test You can safely use the chi-square test when no more than 20% of the expected counts are less than 5 and all individual expected counts are 1 or greater.

28. Example 10.7: Do angry people have more heart disease? Anger Score LowModerateHigh Sample size31104731633 CHD count5311027 CHD percent1.7%2.3%4.3%

29. Example 10.7: Do angry people have more heart disease? Anger Score LowModerateHighTotal CHD5311027190 No CHD305746216068284 Total311047316338474

30. Example 10.7: Do angry people have more heart disease? ObservedExpected LowModerateHighLowModerateHigh CHD531102769.73106.0814.19 No CHD 305746216063040.274624.92618.81

31. Exercises p 571: 10.16, 10.18, 10.20

32. Application 10.1: Matching Products with Markets: A Study in Economics p 571

33. Return of the M&M’s Color Observed (O) Expected (E) Blue(.24)( )= Orange(.20) Green(.16) Yellow(.14) Red(.13) Brown(.13) Total

34. Exercises p 571: 10.16, 10.18, 10.20

35. Exercises p 572-574: 10.22, 10.26

36. Exercise: Preventing domestic violence A study conducted in Charlotte, NC, tested the effectiveness of three police responses to spousal abuse: (1) advise and possibly separate the couple, (2) issue a citation to the offender, and (3) arrest the offender. Police officers were trained to recognize eligible cases. When presented with an eligible case, a police officer called the dispatcher, who would randomly select one of the three available treatments to be administered. There were 650 cases in the study. Each case was classified according to whether the abuser was subsequently arrested within 6 months of the original incident.

37. Exercise: Preventing domestic violence No subsequent arrest Subsequent arrest Advise and separate18725 Citation18143 Arrest17539 Is there sufficient evidence from this study to conclude that the treatment imposed on an abuser is related to the likelihood of the abuser’s subsequent arrest? Give appropriate evidence to support our position.