Question

Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,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-Ynare i.i.d. with normal distribution, show that (exactly) (X-Y)-(ux - y) t(n-1), (14) Sin where s2 = 2 (X; -Y)-(X-Y))2 denotes the sample variance. (ii) Our critical region for rejecting the null hypothesis Ho: Mx = My will be of the form (15) '" " " 'l s/ n for some rejection threshold >0. Show that type I error = P(reject Ho: Mx =my; Ho) (16) p/1X- IS. =PCS 20; Ho) =P (Itin - 1)/20). (17) (iii) Conclude that approximately for large n) type I error sa 02 2a/2 02 ta/z(n-1). (18) (iv) Show that the critical region of rejecting Houx = My in favor of H:My My with significance level a is given by X- (19) c = {(x2 -- Xmı V1 -- »Ym) | 1* - 20}, SIM 2 la/2n-1).

Answer 1

let X, Y be Rvg. denoting that e[x] = Mx and Ely] = py full hypothesis Hoi Hx = Hy Hi Ex & My X-Y a normal distribution U mung (x - Y;) v B. M ( 13 , M , Sa+, g• 8) + = | X; - Y; N. NO. = fx-My, n + 6y²- 28 Oyby Thus, Di = x;-Yin NC8, 9²), where S 8 = Mx - My = 6 + 6y+-2866 Then , on NCS, Tot Faro (8-8) ~ N(0, 1) Also, assuming bad and by unknown, 60% is abo unkno con e independent of F where so sad Co. - . : Ź ( xs-4; -(8-5)) Top 02 IM VM, We known , itOne N (0,3) vin : ind s Mane w trained

Then I s 96,00 m (5-2) n tay is expected that t :) citical region for rejecting the null hypothesis Ho: My = My will be of the form * C={(2,-- AM, 4---4m2 1 5 0 20% i Note that Triton a completely known as Symmetric about one Ho : Kaye - My w Ho :8=0; it should be close to b If obserred Ti so, we deject Ho : Szo in favour of His ²0 - Hi Mx & My # T-kroor = P40 C Ho is rected] Poo[171>o] 11. Pero [ 1 vnco. -51 >o] - P[ 1 von > ] Ho: 8=0, Yoo w towe Pismo {ltne) >o] | 7 concluding that the approximately for large in type 1 error s & 6 o 7212 6 os tal2 Co-1) If Pe-ol termel>o] = o = tnu in 412

If P8-0-{ ito-11 >0] = a's a => >, o'stint – 0 for large Samples, E (ton) =0 - 2 as no ii) SJ tom N(0,4) - T-I error peso [lto11> o] Śd for large Sample, now tor N HCO,1) (2 oso = 7 ( In the same way as ) Ciren that 18 - >, tal2 (0-1) of observed ITIDE we reject Hoi Max My 1. Ho 8=0 in favour of the Mx & My 6 H: 8to at level a, where Pte Litl >] .. o = thing, where I = rn (x-4) Pino [ -) : -] = d,