Let W = YU+ (1-Y)V, where Y ~ Bern(0), U ~ N(u1, of), V ~ N(u2, 62) and Y, U, V are independent, i.e., W is a mixture of normal distributions and we know E(W) = Qui + (1 -0)M2. Suppose we have W1, W2,..., Wn being independent copies of W. We are interested in testing the hypotheses Ho : Of1 + (1 - 0) u2 = 0 and Ha : Oui + (1-0) u2 * 0 using a two-sided t-test.(c) (4 points) Let a : 0.05. Calculate the number of replications needed to estimate Type I error probability of the test with a margin of error of at most 0.01 at the conservative approximate 99% condence level. (d) (4 points) Test your function by estimating the type I error probability of the test H0 : 9m + (1 9) 1,12 : 0 and Ha : 9p] +(1 (9)112 7e 0 whenn = 200,9 = 0.5,,u1= -1,#2 =1,0'12= 0.52,o=1,a = 0.05 using reps calculated above. Create a 99% Wald CI for the Type I error probability. (6) (4 points) Test your function by producing a plot of a simulated estimate of the power curve of the test H0 : 9,111+(l9),u2 : 0andHa : 9,111+(10),u2 7 0by using 112 200,0 : 0.5,,111 : l,,0'12: 052, 0'22 : 1,,LL2 E {1, 1.05, 1.1, . . . ,2} using reps calculated above