Chi-Square TestsChapter 11Chi-Square TestsChapter 11ObjectivesIn this chapter, you learn:nHow and when to use the chi-square test for contingency tablesObjectivesIn this chapter, youContingency TablesContingency TablesnUseful in situations comparing multiple population proportionsnUsed to classify sample observations according to two or more characteristicsnAlso called a cross-classification table.DCOVAContingency TablesContingency Left-Handed vs. Gender Dominant Hand: Left vs. Right Gender: Male vs. Femalen2 categories for each variable, so this is called a 2 x 2 tablenSuppose we examine a sample of 300 childrenContingency Table ExampleDCOVALeft-Handed vs. GenderContingContingency Table ExampleSample results organized in a contingency table:(continued)GenderHand PreferenceLeftRightFemale12108120Male2415618036264300120 Females, 12 were left handed180 Males, 24 were left handedsample size = n = 300:DCOVAContingency Table ExampleSampl 2 Test for the Difference Between Two ProportionsnIf H0 is true, then the proportion of left-handed females should be the same as the proportion of left-handed malesnThe two proportions above should be the same as the proportion of left-handed people overallH0: 1 = 2 (Proportion of females who are left handed is equal to the proportion of males who are left handed) H1: 1 2 (The two proportions are not the same)DCOVA2 Test for the Difference BetThe Chi-Square Test Statisticnwhere:fo = observed frequency in a particular cellfe = expected frequency in a particular cell if H0 is true (Assumed: each cell in the contingency table has expectedfrequency of at least 5)The Chi-square test statistic is:DCOVAThe Chi-Square Test StatisticwDecision Rule 2 22Decision Rule:If , reject H0, otherwise, do not reject H0The test statistic approximately follows a chi-squared distribution with one degree of freedom0 Reject H0Do not reject H0DCOVADecision Rule22Decision RulComputing the Overall ProportionHere: 120 Females, 12 were left handed180 Males, 24 were left handedi.e., based on all 300 children the proportion of left handers is 0.12, that is, 12%The overall proportion is: DCOVAComputing the Overall ProportFinding Expected FrequenciesnTo obtain the expected frequency for left handed females, multiply the average proportion left handed (p) by the total number of femalesnTo obtain the expected frequency for left handed males, multiply the average proportion left handed (p) by the total number of malesIf the two proportions are equal, then P(Left Handed | Female) = P(Left Handed | Male) = .12i.e., we would expect (.12)(120) = 14.4 females to be left handed(.12)(180) = 21.6 males to be left handedDCOVAFinding Expected FrequenciesToObserved vs. Expected FrequenciesGenderHand PreferenceLeftRightFemaleObserved = 12Expected = 14.4Observed = 108Expected = 105.6120MaleObserved = 24Expected = 21.6Observed = 156Expected = 158.418036264300DCOVAObserved vs. Expected FrequencGenderHand PreferenceLeftRightFemaleObserved = 12Expected = 14.4Observed = 108Expected = 105.6120MaleObserved = 24Expected = 21.6Observed = 156Expected = 158.418036264300The Chi-Square Test StatisticThe test statistic is:DCOVAHand PreferenceLeftRightFemaleDecision RuleDecision Rule:If 3.841, reject H0, otherwise, do not reject H0Here, = 0.7576 12.592, reject H0, otherwise, do not reject H0Here, = 0.709 = 12.592, so do not reject H0 Conclusion: there is not sufficient evidence that meal plan and class standing are related at = 0.05 2 220.05=12.5920 0.05Reject H0Do not reject H0DCOVAExample: Decision and InterprChapter SummaryIn this chapter we discussed:nHow and when to use the chi-square test for contingency tablesChapter SummaryIn this chapter