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Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = /x and E[Y] = wy. Suppose we want to test the null hypothesis Ho : Ax = MY against the alternative hy- pothesis H1 : Hx # My. Suppose we have i.i.d. pairs (X1, Y1), ...,(Xn, Y,) from the joint distribution of (X, Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1, ..., Xn - Yn are i.i.d. with normal distribution, show that (exactly) _( X - Y) - (ux - HY) _ t(n- 1), S/ Vn (14) where S'=EL En, ((X, - YO) - (X - Y))2 denotes the sample variance. (ii) Our critical region for rejecting the null hypothesis Ho : Hx = My will be of the form C= (X1, ..., Xn, VI, . . . . Ym) Lx - yl. slyn (15) for some rejection threshold 0 > 0. Show that type I error = P(reject Ho : Hx = MY ; Ho) (16) =p X-YI S/ Vn 20; Ho = P(It(n - 1)120). (17) (iii) Conclude that (approximately for large n) type I error sa 02za12 02 ta/2(n-1). (18) (iv) Show that the critical region of rejecting Ho : ux = uy in favor of Hi : My # uy with significance level a is given by IX- YI S/ vn 2 ta/2(n - 1). (19)