Exercise 13. Let X₁,..., Xn~ N(x, 02) i.i.d. and Y₁,..., Yn~ N(y, o2) i.i.d. with known o². We assume that for each i the random variables X, and Y; are correlated, with Corr(X₁, Yi) = 1/2. Define D₁ = X₁ - Yį 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(X₁, Yi) = 02/2. b) Determine E(D₁) and Var(D₂). c) Consider the test for Ho: x=y which uses the test statistic Z=√n-. O 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 hand.

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