Recall that the Hotelling 7"-statistic based on X1, ...; Xn (which are assumed iid Np(, Z) with _ invertible) is defined as T' (X, p) = n(X -U) S (X) (X -H), where S( X ) is the (unbiased) sample covariance matrix based on X1, . .., X,. (i) What is 7"(X, p) in the case p = 1 and what is its distribution? (ii) Write the formula for 7"(X, ) in terms of the maximum likelihood estimate ) of E instead of S and specify its distribution. (iii) Suppose the data are transformed to Y = AXk + a, where A is a fixed invertible matrix and a is a fixed vector. (a) What is the mean py and covariance matrix Ey of Y,? (b) Determine the sample covariance matrix S(Y ) in terms of S(X ) and write T"(Y, My) in terms of T* ( X, #)