2 Exercise 13. Let X1,..., Xn ~N(x, 02) i.i.d. and Y₁,..., Yn ~N(μy, σ2) i.i.d. with known σ². We assume that for each i the random variables X and Y are correlated, with Corr(Xi, Yi) = 1/2. Define DX - Y₁ for all i = {1,..., n}. Finally, let X, Y and D be the averages of the Xi, Yi and Di, respectively. a) Show that Cov(Xi, Yi) = 02/2. b) Determine E(D) and Var(Di). c) Consider the test for Ho: px = μy which uses the test statistic and which rejects Ho, if and only if | Z|> 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other

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