Suppose that the variables X1, , Xn form a random sample from the normal distribution with unknown mean and unknown variance 2, and a t test at a given level of significance 0 is to be carried out to test the following hypotheses H0 0, H1 0 Let (, 2 ) denote the power function of this t test, and assume that (1, 21) and (2, 22) are values of the parameters such that 1 0 1 2 0 2 Show that (1, 21 ) (2, 22 )

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, and a t test at a given level of significance α0 is to be carried out to test the following hypotheses:
H0: μ ≤ μ0,
H1: μ>μ0.
Let π(μ, σ2|δ) denote the power function of this t test, and assume that (μ1, σ21) and (μ2, σ22) are values of the parameters such that
μ1 − μ0/σ1 = μ2 − μ0/σ2.
Show that π(μ1, σ21|δ) = π(μ2, σ22|δ).

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