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Exercise 2 (Paired test, known variance of the difference). Let X, Y be RVs. Denote E[X] = / and E[Y] = wy. Suppose we want to test the null hypothesis Ho : #x - wy = 0 against the alternative hypothesis H1 : #x - My = -3. Suppose we have i.i.d. pairs (X1, Y1). .... (Xn, Y,) from the joint distribution of (X, Y). Further assume that we know the value of oz = Var(X - Y). (i) Noting that X1 - Y1, ..., Xn -Y, are i.i.d. with finite variance o', show using CIT that, as n - co, Z:= ( x - Y) - (HX - HY) olyn N(0, 1). (2) (ii) Our critical region for rejecting the null hypothesis Ho : #x -My = 0 in favor of H : Hx-My = -3 will be of the form C= ((x1,".., *m. ), ".. . ym)lx-p0). (3) for some rejection threshold 0 > 0. Show that the power function of this test is given by K(up) = P(reject Ho : Hx = HY ; Hx -HY = HD) (4) =P(X - Y S0; HX - HY = HD) (5) 5 0 - HD =P ZS on HX - HY = HD =P N(0, 1) s 0 - HD olvn) (6) (iii) Conclude that (approximately for large n) Type I error s a olyn S-Zar (7) Type II error s p 0-3 olyn (8) (iv) Suppose we want both the type I and II errors to be at most 0.05. Find conditions for 0 and n that would guarantee this