Chapter 11Chapter 11: Chi–Square : Chi–Square and F Distributionsand F DistributionsSection 1Section 1: Chi-Square: Tests of : Chi-Square: Tests of

IndependenceIndependence

Chi–Square DistributionChi–Square Distribution

- - Greek letter “Chi”Greek letter “Chi” Chi-Square denoted by Chi-Square denoted by graph is not symmetricalgraph is not symmetrical depends on degrees of freedomdepends on degrees of freedom as df increase, graph becomes more as df increase, graph becomes more

symmetricalsymmetrical

2

Chi-Square Test of Independence – used to determine whether two characteristics are related

Formulas Formulas

E =E =

df = (r -1)(c – 1) df = (r -1)(c – 1) E

EO 22 )(

SizeSample

totalColumntotalRow ))((

ExampleExample Use a chi-square test to determine Use a chi-square test to determine

whether there is a relationship between whether there is a relationship between students’ political views and their opinion students’ political views and their opinion about a controversial issue. Use a 5% about a controversial issue. Use a 5% level of significancelevel of significance

Ho: Political view and opinion are independent

H1: Political view and opinion are related

Contingency tableContingency table

DemocratDemocrat RepublicanRepublican IndependentIndependent

approveapprove 1010 1515 2020

disapprovedisapprove 99 22 1616

undecidedundecided 88 22 1818

SolutionSolution

OO EE (O – E)(O – E) (O – E)(O – E)22 (O – E)(O – E)22/E/E

1010 12.1512.15 -2.15-2.15 4.62254.6225 0.38050.3805

1515 8.558.55 6.456.45 41.602541.6025 4.86584.8658

2020 24.324.3 -4.3-4.3 18.4918.49 0.76090.7609

99 7.297.29 1.711.71 2.92412.9241 0.40110.4011

22 5.135.13 -3.13-3.13 9.79699.7969 1.90971.9097

1616 14.5814.58 1.421.42 2.01642.0164 0.13830.1383

88 7.567.56 0.440.44 0.19360.1936 0.02560.0256

22 5.325.32 -3.32-3.32 11.022411.0224 2.07192.0719

1818 15.1215.12 2.882.88 8.29448.2944 0.54860.548611.102