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5. we have two independent samples of n observations X1,X2, ,x, and Yi, ½, ,y, We want to test the hypothesis Ho M Hy versus the alternative Hi: Hr y (a) First, assume that the null hypothesis Ho is true and find the MLE for μ Ha-μυ (b) Then plug this estimate into the log likelihood along with the MLEs μ--x and μυ-D to calculate the LRT statistic (c) Is this likelihood ratio test equivalent to the test that rejects the null hypothesis when (x-j)2 > c? Justify your answer
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rl (a) The likelihood function, L(p) = (10n 2(14π)n 2 exp (-10 Σ(r,-)2-14 Σ(у,-μ)2 7n に! dl() 1 12 i=1 i=1 /L rn (b) supl(a) _ (10T)n/2(14m)n/2 exp10 に! に! where μ. x, μ,-y Likelihood ratio=-supnzh L(p) -exp = exp (c) We reject Ho if where c is obtained from size conditioni.e. ~P((ar{x}-ar{y})^2>c|H_0)=alpha

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