Contingency TableContingency TableTesting for Independence Between Two Categorical VariablesTesting for Equality of Proportions Between Two SamplesUsing TI CalculatorTest of Independence ExamplesTraffic Cameras ExampleSmoking Ban ExampleCommuting and Stress ExampleTest of Homogeneity ExamplesBody Weight Classification ExampleConsumer Recall Classification ExampleContingency Table Test of Independence Examples Test of Homogeneity ExamplesChapter 9Testing Equality of FrequenciesJ.C. WangJC Wang (WMU) Stat2160 S2160, Chapter 9 1 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesGoal and ObjectivesGoal: To learn about contingency table testsObjectives: To learn about testing for independence of twocategorical variables and about testing for homogeneouspopulations on proportions. Also to learn about χ2distribution.JC Wang (WMU) Stat2160 S2160, Chapter 9 2 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesGoal and ObjectivesGoal: To learn about contingency table testsObjectives: To learn about testing for independence of twocategorical variables and about testing for homogeneouspopulations on proportions. Also to learn about χ2distribution.JC Wang (WMU) Stat2160 S2160, Chapter 9 2 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesOutline1Contingency TableContingency TableTesting for Independence Between Two Categorical VariablesTesting for Equality of Proportions Between Two SamplesUsing TI Calculator2Test of Independence ExamplesTraffic Cameras ExampleSmoking Ban ExampleCommuting and Stress Example3Test of Homogeneity ExamplesBody Weight Classification ExampleConsumer Recall Classification ExampleJC Wang (WMU) Stat2160 S2160, Chapter 9 3 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesOutline1Contingency TableContingency TableTesting for Independence Between Two Categorical VariablesTesting for Equality of Proportions Between Two SamplesUsing TI Calculator2Test of Independence ExamplesTraffic Cameras ExampleSmoking Ban ExampleCommuting and Stress Example3Test of Homogeneity ExamplesBody Weight Classification ExampleConsumer Recall Classification ExampleJC Wang (WMU) Stat2160 S2160, Chapter 9 4 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesContingency TabledefinitionCategorical variables are organized into categories and the data arethe frequency counts of the various categories. When frequency countdata are classified according to two or more variables or populations,they are called cross-tabulated data and are displayed in acontingency table.JC Wang (WMU) Stat2160 S2160, Chapter 9 5 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesContingency Tabledata summaryPopulationRow cat. (or Column cat. variable)variable 1 2 · · · c TotalA A1 A2 · · · Ac row-1 totalB B1 B2 · · · Bc row-2 total..................r r1 r2 · · · rc row-r totalTotal col-1 col-2 · · · col-c Grandtotal total total TotalJC Wang (WMU) Stat2160 S2160, Chapter 9 6 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesContingency Tabletest statistic and its computationHypotheses tests on such data are performed using a χ2teststatistic which is approximatedly chi-square distributed.It has degree of freedom, df = (r − 1)(c − 1), wherer = number of rows and c = number of columns.test statistic = χ2=X(O − E)2E, whereO = Observed Frequency, andE = Expected =(row total) × (column total)Grand TotalJC Wang (WMU) Stat2160 S2160, Chapter 9 7 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesContingency Tabletest statistic and its computationHypotheses tests on such data are performed using a χ2teststatistic which is approximatedly chi-square distributed.It has degree of freedom, df = (r − 1)(c − 1), wherer = number of rows and c = number of columns.test statistic = χ2=X(O − E)2E, whereO = Observed Frequency, andE = Expected =(row total) × (column total)Grand TotalJC Wang (WMU) Stat2160 S2160, Chapter 9 7 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesContingency Tabletest statistic and its computationHypotheses tests on such data are performed using a χ2teststatistic which is approximatedly chi-square distributed.It has degree of freedom, df = (r − 1)(c − 1), wherer = number of rows and c = number of columns.test statistic = χ2=X(O − E)2E, whereO = Observed Frequency, andE = Expected =(row total) × (column total)Grand TotalJC Wang (WMU) Stat2160 S2160, Chapter 9 7 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesThe Chisquare Distributionxχdf=12 distribution0 5 10 15xχdf=22 distribution0 5 10 15xχdf=32 distribution0 5 10 15xχdf=42 distribution0 5 10 15xχdf=62 distribution0 5 10 15Various Chisquare Distributions0 5 10 15χ12χ22χ32χ42χ62JC Wang (WMU) Stat2160 S2160, Chapter 9 8 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesThe Chisquare Distributionxχdf=12 distribution0 5 10 15xχdf=22 distribution0 5 10 15xχdf=32 distribution0 5 10 15xχdf=42 distribution0 5 10 15xχdf=62 distribution0 5 10 15Various Chisquare Distributions0 5 10 15χ12χ22χ32χ42χ62JC Wang (WMU) Stat2160 S2160, Chapter 9 8 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesThe Chisquare Distributionxχdf=12 distribution0 5 10 15xχdf=22 distribution0 5 10 15xχdf=32 distribution0 5 10 15xχdf=42 distribution0 5 10 15xχdf=62 distribution0 5 10 15Various Chisquare Distributions0 5 10 15χ12χ22χ32χ42χ62JC Wang (WMU) Stat2160 S2160, Chapter 9 8 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesThe Chisquare Distributionxχdf=12 distribution0 5 10 15xχdf=22 distribution0 5 10 15xχdf=32 distribution0 5 10 15xχdf=42 distribution0 5 10 15xχdf=62 distribution0 5 10 15Various Chisquare Distributions0 5 10 15χ12χ22χ32χ42χ62JC Wang (WMU) Stat2160 S2160, Chapter 9 8 / 65Contingency Table Test of Independence Examples Test of Homogeneity ExamplesThe Chisquare Distributionxχdf=12 distribution0 5 10 15xχdf=22 distribution0 5 10 15xχdf=32 distribution0 5 10 15xχdf=42 distribution0 5 10 15xχdf=62 distribution0 5 10 15Various Chisquare Distributions0 5 10 15χ12χ22χ32χ42χ62JC Wang (WMU) Stat2160 S2160, Chapter 9 8 / 65Contingency Table Test of Independence Examples Test of Homogeneity