Suppose that the variables X1, , Xn form a random sample from the normal distribution with unknown mean and unknown variance 2 Let 20 be a given positive number, and suppose that it is desired to test the following hypotheses at a specified level of significance 0 (0 20 Let S2n and suppose that the test procedure to be used specifies that H0 should be rejected if S2n 20 c Also, let (,2 ) denote the power function of this procedure Explain how to choose the constant c so that, regardless of the value of , the following requirements are satisfied (, 2 ) 0 if 2 20, (, 2 ) 0 if 2 20, and (, 2 ) 0 if 2 20

Question: Suppose that the variables X1, . . . , Xn form a random sample from the normal distribution with unknown mean and unknown variance

Suppose that the variables X1, . . . , Xn form a random sample from the normal with unknown mean μ and unknown variance σ2. Let σ20 be a given positive number, and suppose that it is desired to test the following hypotheses at a specified level of significance α0 (0<α0 < 1):
H0: σ2 ≤ σ20,
H1: σ2 > σ20.
Let S2n = and suppose that the test procedure to be used specifies that H0 should be rejected if S2n/σ20 ≥ c. Also, let π(μ,σ2|δ) denote the power function of this procedure. Explain how to choose the constant c so that, regardless of the value of μ, the following requirements are satisfied: π(μ, σ2|δ) < α0 if σ2 < σ20, π(μ, σ2|δ) = α0 if σ2 = σ20, and π(μ, σ2|δ) > α0 if σ2 >σ20.

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