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

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.

Two-Way Tables Row Variable Column Variable Purdue University Faculty FemaleMaleTotal Assistant Professors Associate Professors Full Professors Total

Two-Way Tables Frequency counts Purdue University Faculty FemaleMaleTotal Assistant Professors Associate Professors Full Professors Total Joint frequencies: Entries in the body of the table Marginal frequencies: Entries in the “Total” row or column

Two-Way Tables Relative frequencies Purdue University Faculty FemaleMaleTotal Assistant Professors Associate Professors Full Professors Total Conditional frequencies:The relative frequencies in the body of the table

Two-Way Tables Relative frequencies for rows Purdue University Faculty FemaleMaleTotal Assistant Professors Associate Professors Full Professors Total

Two-Way Tables Relative frequencies for columns Purdue University Faculty FemaleMaleTotal Assistant Professors Associate Professors Full Professors Total1.000

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.

Simpson’s Paradox University Admissions MaleFemaleTotal Admit Deny Total8060 Frequency counts University Admissions MaleFemaleTotal Admit Deny Total1.00 Relative frequencies

Simpson’s Paradox Engineering Admissions MaleFemaleTotal Admit Deny Total Frequency counts English Admissions MaleFemaleTotal Admit51015 Deny Total204060

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

Activity 10.1: Sports Preference Freq Counts BasketballFootballSoccerOtherTotal Girls Boys Total

Activity 10.1: Sports Preference Rel Freq BasketballFootballSoccerOtherTotal Girls Boys Total

Exercises p : 10.1, 10.3, 10.5

Example 10.4: Treating Cocaine Addiction Frequency counts GroupTreatmentSuccessFailureTotal 1Desipramine Lithium Placebo42024 Total244872

Expected Counts

Exercises p : 10.6, 10.8, 10.10

Exercise p : 10.6, 10.8, 10.10

Chi-square Statistic

Example 10.5: The Cocaine Study ObservedExpected TreatmentSuccessFailureSuccessFailure Desipramine Lithium Placebo420816

Chi-square Statistic

The Chi-square Distributions

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.

The Chi-square Distributions

Example 10.6: The Cocaine Study, Conclusion ObservedExpected TreatmentSuccessFailureSuccessFailure Desipramine Lithium Placebo420816

Exercises p 566: 10.11, 10.13, 10.15

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.

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

Example 10.7: Do angry people have more heart disease? Anger Score LowModerateHighTotal CHD No CHD Total

Example 10.7: Do angry people have more heart disease? ObservedExpected LowModerateHighLowModerateHigh CHD No CHD

Exercises p 571: 10.16, 10.18, 10.20

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

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

Exercises p 571: 10.16, 10.18, 10.20

Exercises p : 10.22, 10.26

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.

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.