Fora c-category variable,considertesting H0: 1 = 10, ...,c = c0 when counts (y1, ...,yc) haveamultinomialdistribution(2.14)with n = j yj . (a) UsingtheresultthattheMLestimateof j is the

Fora c-category variable,considertesting H0: π1 = π10, ...,πc = πc0 when counts (y1, ...,yc)

haveamultinomialdistribution(2.14)with n = Σj yj .

(a) UsingtheresultthattheMLestimateof πj is the jth sampleproportion yj~n, showthat the likelihood-ratiostatisticfortesting H0 is

with df = c−1 for thelarge-samplechi-squareddistribution. ‰The correspondingPearson statistic is X2 = Σj[(yj − nπj0)2]~(nπj0).Ž

(b) Fortestingthatthesixsidesofadiceareequallylikely,werollthedice100times and obtainscounts(15,20,13,14,17,21)foroutcomes(1,2,3,4,5,6).Conductthe likelihood-ratiotestandinterpretthe P-value.

(c) Simulatetheexactsamplingdistributionofthelikelihood-ratiostatisticin(b),anduse it topreciselyapproximatetheexact P-valueofthetestforthedatashown.

2y, log(y/n),