# Question

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

(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.

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