Question Four Part I Consider a random sample X = (X1, . .., Xn ) of size n = 10 from a normal distribution N(0, 10). Note that the density function of a N(u, o?) is given by f (x| M, o ) = - 1 e 20I (I- H) , -00 5 X 5 00; - DO 0. a [2 marks] Find a complete and minimal sufficient statistic for this family. 10 T(X) = > Xi i- 1 O 10 T(X) = >X, O 10 T(X) = [IX? i=1 O T(X) = X(1) O T(X) = Xin)b} [2 marks] This family has a monotone likelihood ratio (MLR) in its complete and sufficient statistic TUE}. Select a valid reason that applies. The Neyman-Fisher factorization criterion since x, a) = 21% exp ( $32) exp (in?) exp ( $92) = h(X]g(t,3] Since f belongs to the one-parameter exponential family T{X) has the MLR property. Let 6'\" 3} I5" then the ratio is monotonically increasing in the statistic TUE] since 6'" 19' 2} I}. Let I?" > 3' then the ratio % = exp (21D ((6'" WM) + mm\" so, is monotonically increasing in the statistic TUE] since 19" 3' > i}. C) [1 mark] Determine the form of the UMP size a level test of Ho : 0 5. P ( X ) = 1 if P(T(X) k) > 0.5 O if T(X) > k 4 ( X) = y if T(X ) = k 0 if T(X) k O 4 (X) = 1 if T(X) > k 0 if T( X ) kd) [2 marks] Compute the threshold constant k in the y from part (c). Note that zo is the upper a x 100 point of the standard normal distribution with mean zero and variance one i.e. P(Z > z.) = a. Hint: Consider the distribution of X ~ N(0, 10). k = 50 + 10za O k = 50 + V10za O k = V50 + 10zxx O k = V10 + 50za O k = V50 + V10zo