A review of univariate normal inference theory: consider X1, ,Xn a random sample from an assumed normal distribution with unknown mean y. We want to test the hypothesis H0: p = {.10 and Ha: ,u at #0 (a 2-sided alternative hypothesis). The most powerful test statistic is the \"sample 2 score\": X #n s/x t: -> t2 = 310? p9)(sz)'1()? p0) (student's t distribution) We reject Ha if |t| is large and do not reject ifuo lies in the 100(1 - a)% condence interval ii 131-1 (E) in. In a multivariate setting, this confidence interval becomes a confidence region which extrapolates from the above form to: T2 = 31(3 nu)'(S)'1(X ,uu) (Hotelling's T2) Where we are assuming there are n observations of 39 random variables. The sampling distribution (found by Harold Hotelling) is: Where Fpm-p is an F distribution with p and n p degrees of freedom. This means that the confidence region at 100 (1 a)% confidence of a p dimensional random vector being normally distributed is the ellipsoid formed by all mean vectors that satisfy: _ , _ _ (n - 1339 no: n) (5) la: n) s Wenpros) Questions 5 2 1. Consider the simple data matrix X = 9 1 evaluate '1"2 and give its distribution 4 4 2. Complete Problem 14.4 and Problem 14.5 and give interpretations of your results. What is a reasonable interpretation of this statistical test? Construct confidence regions for each problem to aide your interpretations. How might this process compare to testing each univariate variable independently and what problems might that incur