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

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Chapter1: Combinatorial Analysis
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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, Yi
and 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 statistic
nD.
σ
Z = √n
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.
Transcribed Image Text: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, Yi and 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 statistic nD. σ Z = √n 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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