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Question 2 Consider the basic one-way model: Yij = H+ a; + E;ij, i = 1, . .. , n; j= 1, ..., a, where Yij is the response, u is the overall (grand) mean, a; = u; µ is the effect of treatment j, iid is the error such that ɛij N N(0, o2). Let Y, be the sample mean of the jth group and Y be the sample grand mean.

Question 2 Consider the basic one-way model: Yij = H+ a; + E;ij, i = 1, . .. , n; j= 1, ..., a, where Yij is the response, u is the overall (grand) mean, a; = u; µ is the effect of treatment j, iid is the error such that ɛij N N(0, o2). Let Y, be the sample mean of the jth group and Y be the sample grand mean.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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Question 2 Consider the basic one-way model: Yij = +a; + Eij, i = 1,... , n; j= 1,..., a, where Yij is the response, µ is the overall (grand) mean, a; = µj – µ is the effect of treatment j, and e %3D is the error such that iid Eij N(0, o²). Let Y, be the sample mean of the jth group and Y be the sample grand mean. (a) Derive the distribution of u; = Y;. Show all the steps. (b) Derive the distribution of u = Y. Show all the steps. (c) how that â¡ = Y; – Y is unbiased estimator of a;. Show all the steps. (d) Show that Cov (Y,, Y) = . Show all the steps. па (e) Derive the variance of â;. Show all the steps. Hint: You may use the following facts without proof: • Var(aX + bY) = a²Var(X) + b²Var(Y)+ 2abCov(X, Y). %3D . Cov (Σ Q, Χ., Σ"b, Y) - ΣΗ Σ"0,0, Cov (X., ). i=D1 • If X and Y are independent random variables, then Cov(X, Y) = 0. • Ej=10j = 0. • If X ~ N(u, o²), then aX + b ~ N(au + b, a²o²). • A linear combination of independent normal random variables is normal.
Transcribed Image Text:Question 2 Consider the basic one-way model: Yij = +a; + Eij, i = 1,... , n; j= 1,..., a, where Yij is the response, µ is the overall (grand) mean, a; = µj – µ is the effect of treatment j, and e %3D is the error such that iid Eij N(0, o²). Let Y, be the sample mean of the jth group and Y be the sample grand mean. (a) Derive the distribution of u; = Y;. Show all the steps. (b) Derive the distribution of u = Y. Show all the steps. (c) how that â¡ = Y; – Y is unbiased estimator of a;. Show all the steps. (d) Show that Cov (Y,, Y) = . Show all the steps. па (e) Derive the variance of â;. Show all the steps. Hint: You may use the following facts without proof: • Var(aX + bY) = a²Var(X) + b²Var(Y)+ 2abCov(X, Y). %3D . Cov (Σ Q, Χ., Σ"b, Y) - ΣΗ Σ"0,0, Cov (X., ). i=D1 • If X and Y are independent random variables, then Cov(X, Y) = 0. • Ej=10j = 0. • If X ~ N(u, o²), then aX + b ~ N(au + b, a²o²). • A linear combination of independent normal random variables is normal.
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