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b. Consider the simple regression model y; = Bo + B1xi + €i, i = 1,...,n with e; ~ independent. Develop the likelihood ratio test for testing Ho : B1 = 0 against Ha : B1 # 0. N(0, ơ). The e's are

b. Consider the simple regression model y; = Bo + B1xi + €i, i = 1,...,n with e; ~ independent. Develop the likelihood ratio test for testing Ho : B1 = 0 against Ha : B1 # 0. N(0, ơ). The e's are

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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Part B

a. Let X1,..., Xn be i.i.d. random variables from N(u1, o) and Y1,..., Xm be i.i.d. random variables from N(u2, 0) with o unknown. Suppose we want to test the hypothesis Ho : µ1 = µ2. Under Ho the test statistic follows a central t distribution with n+m - 2 degrees of freedom. Why? Explain how the power of the test is computed in this problem. You will need to determine the noncentrality parameter and include it in your answer. Note: Assume Ha : µ1 > µ2. b. Consider the simple regression model yi independent. Develop the likelihood ratio test for testing Ho : B1 Bo + B1xi + €;, i = 1, ... , n with e; ~ N(0, 0). The es are O against Ha : B1 # 0. c. Suppose X = (X1, X2,..., Xn)' follows a multivariate normal distribution with E(X;) = M, var(X;) = o2, and that the covariance matrix of X can be expressed as E = o?V, where o? is known and V is nxn symmetric matrix of known constants. Find a confidence interval for u using 1-a confidence level. d. Let Q ~ beta(;,(n – k – 2)), where n and k are integers. Show that (n – k – 2) follows an F distribution. What are the degrees of freedom?
Transcribed Image Text:a. Let X1,..., Xn be i.i.d. random variables from N(u1, o) and Y1,..., Xm be i.i.d. random variables from N(u2, 0) with o unknown. Suppose we want to test the hypothesis Ho : µ1 = µ2. Under Ho the test statistic follows a central t distribution with n+m - 2 degrees of freedom. Why? Explain how the power of the test is computed in this problem. You will need to determine the noncentrality parameter and include it in your answer. Note: Assume Ha : µ1 > µ2. b. Consider the simple regression model yi independent. Develop the likelihood ratio test for testing Ho : B1 Bo + B1xi + €;, i = 1, ... , n with e; ~ N(0, 0). The es are O against Ha : B1 # 0. c. Suppose X = (X1, X2,..., Xn)' follows a multivariate normal distribution with E(X;) = M, var(X;) = o2, and that the covariance matrix of X can be expressed as E = o?V, where o? is known and V is nxn symmetric matrix of known constants. Find a confidence interval for u using 1-a confidence level. d. Let Q ~ beta(;,(n – k – 2)), where n and k are integers. Show that (n – k – 2) follows an F distribution. What are the degrees of freedom?
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