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1. Let X₁j, X2j,.,Xajj represent independent random samples of size a; from a normal distribution with means μj and variances σ², j = 1,2, ..., b. (a) (b) (c) Show that baj b aj ΣΣ (xi) - x)² = (xi) −x.;)² +Σaj(ׂ; −x.)² j=1 [=1 J=1 (=1 =1 - Σ or Q' = Q3+Q4. Here X ==X/a; and X = Xij/a;. =1 aj If μ₁ = μ₂ == μb, show that Q/02 and 03/02 have chi-square distributions. Prove that Q3 and Q4 are independent, and hence Q4/02 also has a chi-square distribution. Note: give all respective degrees of freedom. If the likelihood ratio A is used to test Ho: H₁₂ == Mb =μ, μ unspecified and σ² unknown, against all possible alternatives, show that A is equivalent to the computed F≥ c, where F (2)=1a; - b)Q4 (b - 1)Q3 (d) Determine the distribution of F when Ho is true. (e) If H1, H2, are not equal, what are the distributions of Q3/0², Q4/0² and F?

1. Let X₁j, X2j,.,Xajj represent independent random samples of size a; from a normal distribution with means μj and variances σ², j = 1,2, ..., b. (a) (b) (c) Show that baj b aj ΣΣ (xi) - x)² = (xi) −x.;)² +Σaj(ׂ; −x.)² j=1 [=1 J=1 (=1 =1 - Σ or Q' = Q3+Q4. Here X ==X/a; and X = Xij/a;. =1 aj If μ₁ = μ₂ == μb, show that Q/02 and 03/02 have chi-square distributions. Prove that Q3 and Q4 are independent, and hence Q4/02 also has a chi-square distribution. Note: give all respective degrees of freedom. If the likelihood ratio A is used to test Ho: H₁₂ == Mb =μ, μ unspecified and σ² unknown, against all possible alternatives, show that A is equivalent to the computed F≥ c, where F (2)=1a; - b)Q4 (b - 1)Q3 (d) Determine the distribution of F when Ho is true. (e) If H1, H2, are not equal, what are the distributions of Q3/0², Q4/0² and F?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. Let X₁j, X2j,.,Xajj represent independent random samples of size a; from a normal distribution with means μj and variances σ², j = 1,2, ..., b. (a) (b) (c) Show that baj b aj ΣΣ (xi) - x)² = (xi) −x.;)² +Σaj(ׂ; −x.)² j=1 [=1 J=1 (=1 =1 - Σ or Q' = Q3+Q4. Here X ==X/a; and X = Xij/a;. =1 aj If μ₁ = μ₂ == μb, show that Q/02 and 03/02 have chi-square distributions. Prove that Q3 and Q4 are independent, and hence Q4/02 also has a chi-square distribution. Note: give all respective degrees of freedom. If the likelihood ratio A is used to test Ho: H₁₂ == Mb =μ, μ unspecified and σ² unknown, against all possible alternatives, show that A is equivalent to the computed F≥ c, where F (2)=1a; - b)Q4 (b - 1)Q3 (d) Determine the distribution of F when Ho is true. (e) If H1, H2, are not equal, what are the distributions of Q3/0², Q4/0² and F?
Transcribed Image Text:1. Let X₁j, X2j,.,Xajj represent independent random samples of size a; from a normal distribution with means μj and variances σ², j = 1,2, ..., b. (a) (b) (c) Show that baj b aj ΣΣ (xi) - x)² = (xi) −x.;)² +Σaj(ׂ; −x.)² j=1 [=1 J=1 (=1 =1 - Σ or Q' = Q3+Q4. Here X ==X/a; and X = Xij/a;. =1 aj If μ₁ = μ₂ == μb, show that Q/02 and 03/02 have chi-square distributions. Prove that Q3 and Q4 are independent, and hence Q4/02 also has a chi-square distribution. Note: give all respective degrees of freedom. If the likelihood ratio A is used to test Ho: H₁₂ == Mb =μ, μ unspecified and σ² unknown, against all possible alternatives, show that A is equivalent to the computed F≥ c, where F (2)=1a; - b)Q4 (b - 1)Q3 (d) Determine the distribution of F when Ho is true. (e) If H1, H2, are not equal, what are the distributions of Q3/0², Q4/0² and F?
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