Exercise 13. Let X₁,..., Xn~ N(a,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 Di Xi - Yi for all i E {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 = √nº. 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
Exercise 13. Let X₁,..., Xn~ N(a,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 Di Xi - Yi for all i E {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 = √nº. 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
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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
Transcribed Image Text:Exercise 13. Let X₁,..., Xn~ N(x, 0²) i.i.d. and Y₁,..., Yn ~ N(µy, 02) i.i.d. with known o².
We assume that for each i the random variables X; and Y; are correlated, with Corr(Xį, Y₁) = 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(D₁).
c) Consider the test for Ho: x = µy which uses the test statistic
Z = √√nD
σ
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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