For the test described in Exercise 6.3.5, obtain the distribution of the test statistic under general alternatives. If computational facilities are available, sketch this power curve for the case when θ₀ = 1, n = 10, μ = 0, and α = 0.05.

Exercise 6.3.5

Let X₁,X₂, . . . , X_n be a random sample from a N(μ₀, σ² = θ) distribution, where 0 < θ < ∞and μ₀ is known. Show that the likelihood ratio test of H₀ : θ = θ₀ versus H₁ : θ ≠ θ₀ can be based upon the statistic W = Σⁿ_i=1(X_i − μ0)²/θ₀. Determine the null distribution of W and give, explicitly, the rejection rule for a level α test.