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Suppose X is a single observation from a Beta(θ, 1) distribution, and consider the hypotheses H0...

Suppose X is a single observation from a Beta(θ, 1) distribution, and consider the hypotheses H0 : θ ≥ 2 vs H1 : θ < 2.

(a) Consider the test with rejection region R = {X < c}. Derive the power function of this test (as a function of θ and c).

(b) Find the value of c so that the test has size α = 0.01.

(c) Find the probability of a Type II error when θ = 1.

(d) If the observed data is x = 0.2, what is the corresponding p-value?

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