Question

1. Let X1,...,xn vid N (4,1). Suppose, we are interested in testing the following hypotheses Ho : M = Mo VS H, :μ# μο The goal of this exercise is for you to understand that, in this setting, we can construct a good test, which is not however UMP. (a) Suppose you were testing Ho :μ = μo Vs H, :μ< μο. Show that the uniformly most powerful a-level test would reject H, if 21-a Χ4 μο vn Call this test Tj. (b) Similarly, argue that if you were testing Ho :u= uo vs H1 :p > Mo, the uniformly most powerful a-level test would reject Ho if X> μο- Ζα Vn Call this test T2.

Answer 1

and we are Solution Güven that, x,, X2, interested Хи in testing w NCH, 1), the following hypothesis Hi : M & Mo Hoi Ma Mo VS The pdf of x is I {(x1²_)? L (2, 4) i Xi ERR Lohnehe By Neymann Pearson hemma, an Unibornly most powerful test at level a is given by Llt, Me) Mo > K 212, =) (van) >k ( 3 / 4 h é £ {(xi-M1) Ge- E(3c-4.) 2 e-{elfali-4,*-(361-4053) =) > K Taking log on both sides -) to - 1 E( 47 - M - 2xi (Mi-fo)) > link =) -2 2 (M²M) + (Me - Mo) {xi > link (le - Mo) {xi > Ank & m (Mitolo?) = k'

и. i (i) <c =) (Mi-Mo) Exisk' where, k'= lnk +1 n (MM) a) For testing, Ho: M=lo vs Hiile lo The uniformly most powerful test is given by Elle-Mo) Exik' ( from (11) since, Mi <mo c' where, ' is determined such that Pesc) or Pro (X-lo c-lo Yon Von =) Pu- (Z < Colle) = 2 'n =) e-lo = - Zira You or ' Ć - Mo 2er Hence, we redect Ho, it go 8 Mo - Zuz or 2 < .

b): For For testing testing, Ho M = lo us Hi: M7 Mo The uniformly most powerful test is (Mi-lo) Exi (from Il > k' Exi s or 1x 1 م > where, e' is determined such that Pue la sc')=a or or Pue (e-Mo > colo こく You You Pelo (23 C-le 'ra -) C-Ma - - Zbax n ve he) = or da Mo Zol In Hence, we reject to is Y > Mo - 2x on