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.
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.
College Algebra (MindTap Course List)
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ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.7: Probability
Problem 39E: Assume that the probability that an airplane engine will fail during a torture test is 12and that...
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![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,Y₁) = 1/2.
Define D₁ = X – Yį for all i € {1,...,n}. Finally, let X, Y and Ď be the averages of the X₁, Yį
and Di, respectively.
a) Show that Cov(Xį, Yi) = 0²/2.
b) Determine E(D₁) and Var(Di).
c) Consider the test for Ho: x = μy which uses the test statistic
-D
O
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2F449c13db-a7ce-43e3-befd-e5e7cb0b8a70%2Fc5gq03_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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,Y₁) = 1/2.
Define D₁ = X – Yį for all i € {1,...,n}. Finally, let X, Y and Ď be the averages of the X₁, Yį
and Di, respectively.
a) Show that Cov(Xį, Yi) = 0²/2.
b) Determine E(D₁) and Var(Di).
c) Consider the test for Ho: x = μy which uses the test statistic
-D
O
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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