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random variables X, and Y, are correlated, with Corr(X₁, Y₁) € {1,...,n}. Finally, let X, Y and D be the averages of the = 0²/2. Var(D₂). 0 Hz Hy which uses the test statistic 2= √n². f and only if |Z| > 1.96. Show that P(type I error) = 5%.

random variables X, and Y, are correlated, with Corr(X₁, Y₁) € {1,...,n}. Finally, let X, Y and D be the averages of the = 0²/2. Var(D₂). 0 Hz Hy which uses the test statistic 2= √n². f and only if |Z| > 1.96. Show that P(type I error) = 5%.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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Exercise 13. Let X₁,..., Xn~ N(2,02) i.i.d. and Y₁,..., Yn~ N(Hy, 02) 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 - Y, for all i E {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Yi and D₁, respectively. a) Show that Cov(X₁,Y)= 02/2. b) Determine E(D₁) and Var(D;). c) Consider the test for Ho: ₂ = Hy which uses the test statistic Z = √n- and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%.
Transcribed Image Text:Exercise 13. Let X₁,..., Xn~ N(2,02) i.i.d. and Y₁,..., Yn~ N(Hy, 02) 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 - Y, for all i E {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Yi and D₁, respectively. a) Show that Cov(X₁,Y)= 02/2. b) Determine E(D₁) and Var(D;). c) Consider the test for Ho: ₂ = Hy which uses the test statistic Z = √n- and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%.
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