Question Four Part I Consider a random sample X = (X1, . .., Xn) of size n = 10 from a normal distribution NV(0, 1). Note that the density of a N(/, 02) is given by f (z|4, 0) = 1 V2702 - 00 5 X 5 00 ; - 00 0. a) [2 marks] Find a complete and minimal sufficient statistic for this family. O T(X) = O 10 T(X ) = > O 10 T(X ) = [X? i=1 O T(X) = X(10) O 20343/slides/1449.. T(X) = X(1)h) [2 marks] This family has a monotone likelihood ratio (MLRJ in its complete and sufcient statistic T{X]. Select a valid reason that applies. The Neyman-Fisher factorization criterion since ea} = J2? exp ( :32) exp (e9) exp ( $32) 2 go, a) Since 3' belongs to the one-para meter exponential family T(X} has the MLR property. Let 3'\" .'> 3' then the ratio gigll) = EXP G (2W 913+ \"3(ng _ 3H2)T(X}))' is monotonically increasing in the statistic T(X) since 6'" 3" > 0. Let 3'\" > B" then the ratio i = exp (; ((a" ogre) + 10w\" 9'90} is monotonically increasing in the statistic T(X) since 3" 3' 2:: 0. () [1 mark] Determine the form of the UMP size a level test of Ho : 0 1. 4 ( X ) = 1 if P(T(X) k) > 0.5 O 1 if T(X) > k P( X ) = Y if T(X ) = k if T(X ) k O 4 ( X ) = if T(X) > k 0 if T(X ) kd) [2 marks] Compute the threshold constant k in the i from part (c). Note that zo is the upper a x 100 point of the standard normal distribution with mean zero and variance 1 i.e. P(Z > ZQ) = a. Hint: Consider the distribution of X ~ N(0, 1). O k = V10 + 5za O k = 10 + v10za O k = V10 + v5zo O k = 10 + v5za O k = V5 + 10za