Showthatwith y successes in n binary trials,the95%scoreconfidenceinterval(4.7)for : (a) Hasmidpoint(4.9)thatisapproximately = (y+2)~(n+4), thatis,thesampleproportion after weaddtwosuccessesandtwofailurestothedataset. (b)

Showthatwith y successes in n binary trials,the95%scoreconfidenceinterval(4.7)for π:

(a) Hasmidpoint(4.9)thatisapproximately ˜π = (y+2)~(n+4), thatis,thesampleproportion after weaddtwosuccessesandtwofailurestothedataset.

(b) Hassquareofthecoefficientof zα~2 in thetermthatisaddedtoandsubtractedfromthis midpointtoformtheconfidenceintervalapproximatelyequalto

This isaweighted average,withweight n~(n + 4) for thevarianceofasampleproportion when π = ˆπ and weight 4~(n + 4) for thevarianceofasampleproportionwhen π = 1~2, using n + 4 in placeoftheusualsamplesize n in thedenominatorofeachvariance.(The Waldconfidenceinterval ˆπ ± zα~2 »
ˆπ(1 − ˆπ)~n with ˆπ replaced by (y + 2)~(n + 4) and n replaced by (n+4) is calledthe Agresti–Coullconfidenceinterval. Itperformssimilarly to thescoreintervalandmuchbetterthantheWaldconfidenceinterval(4.8).)23

Transcribed Image Text:

1 (1 A) n+4 n n+4 n+4