The2018General SocialSurveyasked1136subjectswhethertheybelieveinheavenandwhether they believeinhell.Ofthem,804said yes to both,209said no to both,113said yes to heaven and no to hell,and10said no to heavenand yes to hell.

(a) Showthedataina2×2 contingencytable.Denotethepopulationproportionssaying yes by π1 for heavenand π2 for hell.Find ˆπ1 and ˆπ2. Explainwhythesamplesare dependent rather thanindependent,sothetestof H0: π1 = π2 in Section5.4.1 is notvalid.

(b) Explainwhy H0: π1 = π2 is equivalentto H0: π12 = π21, where πij denotes thepopulation proportionforthecellinrow i and column j of thecontingencytable.Showhowtouse a one-sampletestforaproportiontotest H0 for thesedata,usingthedataintwoofthe four cellsofthecontingencytable.41