Exercise 13. Let X₁,..., Xn~ N(Hz, 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 X₁ and Y; are correlated, with Corr(X₁, Y₁) = 1/2.Define D₂ = X₁ - Yį for all i € {1,...,n}. Finally, let X, Y and D be the averages of the Xi, Yiand Di, respectively.a) Show that Cov(X₁, Yi) = 0²/2.b) Determine E(D₂) and Var(D₂).c) Consider the test for Ho: a = μy which uses the test statisticnD.σZ = √nand 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 onehand, and the non-paired test for comparing the means of two populations on the otherhand.

Answered: y. ov(X₁, Y₁) = 0²/2. (D;) and Var(D;).…

A First Course in Probability (10th Edition)

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y. ov(X₁, Y₁) = 0²/2. (D;) and Var(D;). test for Ho: x=y which uses the test statistic Z = √nD. jects Ho, if and only if |Z| > 1.96. Show that P(type I error) ee sentences, discuss the differences between the test from part e non-paired test for comparing the means of two populations