Hi tutor,

could you help with question b and c? Thanks!

5. (Central Limit Theorem, Hypothesis Test, and Power of Test) Let {(Xi, Yi)}._, be i.i.d. random vectors, which means that (X1, Yi), (X2, Y2),..., (Xn, Yn) are mutually independent random vectors and the random vectors are identically distributed. (This does not mean that Xi and Yi are independent.) The null hypothesis of interest is Ho : EX; = EY; and the alternative hypothesis is H1 : EXi > EYi. Suppose for simplicity that Var(Xi - Yi) is known to be 2. (a) Show that Vn(Xn - Yn - (EX; - EY;)) V2 +d N(0, 1), where Xn = " EL_, Xi and Yn = " Et, Ya. (HINT: Use the Central Limit Theorem.) (b) Using (a), find a test statistic T and critical value c E R such that under the null 3 hypothesis, as n - co, P{T > c} - 0.05. (c) Show that under the alternative hypothesis such that EXi = EYi + 1> EYi, (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, P(T> c} > 1 - E, under the alternative hypothesis in (1).)