Let

Let X1, , X" be i.i.d. Bernoulli random variables with unknown parameter p E (0, 1) . Suppose we want to test 3.3 :pE [043,051] vs 111 2\"; [043,051] We want to construct an asymptotic test 1;! for these hypotheses using 317,. . For this problem, we specifically consider the family of tests we\": wherewe reject the null hypothesis if either 3?... c; z 0.5] for some cl and c; that may depend on n, i.e. Wow: = 1 ((17,, cg) where (:1 (2 ) to both be 0. 025 . PELO . 48 , 0.511 ( If applicable , enter q ( alpha) for ya , the I - a-quantile of a standard normal distribution , e .g . enter 9(0. 01 ) for 90. 01 . ) CZ -We will now show that the values we just derived for ci and c2 are in fact too conservative. Recall the expression from part (b) for P, (X/ 0.48 (note the strict inequality), find lim Pp (X, 0.48 (Xn C2). Use the expression you found in part (b) for Pp (X, > C2). n-+co n-00 lim Pp C2) =