Let X1, X2, , Xn be a random sample of size n from the normal distribution N(, 02), where 02 is known but is unknown (a) Find the likelihood ratio test for H0 0 against H1 0 Show that this critical region for a test with significance level is given by 0 z 20 n (b) Test H0 59 against H1 59 when 2 22...

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Let X1, X2, . . . , Xn be a random sample of size n from the

Question:

Let X1, X2, . . . , Xn be a random sample of size n from the normal N(μ, σ02), where σ02 is known but μ is unknown.
(a) Find the likelihood ratio test for H0: μ = μ0 against H1: μ ≠ μ0. Show that this critical region for a test with significance level α is given by |− μ0| > zα/2σ0/√n.
(b) Test H0: μ = 59 against H1: μ ≠ 59 when σ2 = 225 and a sample of size n = 100 yielded = 56.13. Let α = 0.05.
(c) What is the p-value of this test? Note that H1 is a two-sided alternative. Distribution
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