Foradiscretedistributionandateststatistic T with observedvalue tobs and one-sided Ha suchthatlarge T contradicts H0, mid P-value = 1 2

P(T = tobs) + P(T > tobs).

(a) Supposethat P(T = tj) = πj , j = 1, . ..c. Showthat E(mid P-value) = 0.50. (Hint: Show that Σj πj[(πj~2) + πj+1 + ⋯+ πc] = ‰Σj πjŽ2~2. Themid P-valuehasnulldistribution closer touniform(whichisthedistributionofthe P-valueforateststatistichavinga continuousdistribution)thantheordinary P-value.

(b) Fortheresultsin Section 5.1.1 on testingforbiasinselectingmanagers,findthemid P-value.Findthe P-valueforthesedatawiththe z-test of Section 5.2.1, eventhough n is small.(Fordiscretedata, P-valuesforlarge-sampletestsareusuallymoresimilarto mid P-valuesthantoordinary P-valuesusingexactdistributions.)