# Math Help - Likelihood ratio test and hypothesis testing

1. ## Likelihood ratio test and hypothesis testing

Hello,

Please help me solve this problem

1. Let X1, · · · ,Xn and Y1, · · · , Yn be independent random samples from
N(μ1, sigma1^2) and N(μ2, sigma2^2), respectively.
(i) Assume sigma1 = sigma2 = 1. Find a uniformly most powerful test for testing
H0 : μ1 = μ2 against Ha : μ1 > μ2 with significance level = 0.1;
(ii) Assume sigma1^2 = sigma2^2=sigma^2, but sigma is unknown. Find the likelihood ratio test for testing H0 : μ1 = μ2 = 0 against Ha : all possible alternatives with exact significance level = 0.1;
(iii) Assume μ1, μ2, sigma1, sigma2 are unknown. Find the likelihood ratio test for testing H0 : μ1 = μ2, sigma1 = sigma2 against Ha : all possible alternatives with approximate significance level = 0.1;
(iv) Assume sigma1^2 = sigma2^2 = 5/2. In order to test H0 : μ1−μ2 = 0 against Ha :μ1 − μ2 = 1 we suppose the loss function is such that L(1, 1) = L(0, 0) = 0 and L(1, 0) = L(0, 1) = 2. Find the minimax test and evaluate the power function of this test at μ1 − μ2 = 0 and μ1 − μ2 = 1.

Thanks for any help!