5. {Central Limit Theorem, Hypothesis Test, and Power of Test) Let {(X,,Y,-) \"3:1 be i.i.d. random vectors, which means that (X1, Yl), (X2, 172),..., (Xn, Y\") are mutually independent random vectors and the random vectors are identically distributed. (This does not mean that X,- and Y,- are independent.) The null hypothesis of interest is H0 2 EX 1' = EY; and the alternative hypothesis is H1 : EX, > EY,-. Suppose for simplicity that Var(X,- Y,) is known to be 2. (a) Show that ms, a (EX,- Ev, where X\" = % 121 X, and Y\(b) Using (a), find a test statistic 7 and critical value c E R such that under the null 3 hypothesis, as n - oo, P{T> c'} - 0.05. (c) Show that under the alternative hypothesis such that EXi = EYi + 1 > EY, (1) the power of the test (7, c) converges to 1 as n - co. (HINT. For this, it is sufficient to show that for any small & > 0, there exists a sufficiently large no such that for all n 2 no, PT> c} > 1-E, under the alternative hypothesis in (1).)