Let

Hg 1:16 [043,051] vs H1 :pE [(143,051] We want to construct an asymptotic test 1;! for these hypotheses using 317,. . For this problem, we specifically consider the family of tests niche: wherewe reject the null hypothesis if either X... (:2 2 0.51 for some (:1 and c; that may depend on :1, Le. Wrap; = 1((fn C2) O a = max pE[0.48,0.51] max (Pp (Xn c2)) ) O a = max pE[0.48,0.51] Pp (Xn C2) O a = max pE[0.48,0.51] (Pp (Xn C2 )Use the central limit theorem and the approximation Up (1 p) Pd % forp E [0.48, 0.51] to approximate PJJ (if, (:2) for large n. Express your answers as a formula in terms of (:1 , (:3, n and p. (Write Phi for the cdf of a Normal distribution, c_1 for q. and c_z for (2.} PP (in (:2) maximized? PP (f... > c2) ismaxat p: Next, we combine the results from parts (a} and {b}. Apply the inequality max (f (x) + g (x)) s max f (x) + max 3 (x) to the expression for the {asymptotic} level (I obtained in part (a) and use I I x the results from part (b) to give an upper bound on 0:. Express your answer as a formula in terms of cl , 02, and n. (Write Phi for the cdf of a Normal distribution, c_1 for CI. and I:_2 for 62.} as (Food for thought: Is this upper bound tight? A bound is tight if equality may be achieved.}