Suppose that a single observation X is taken from a uniform density on [0, θ], and consider testing H0 : θ = 1 versus H1 : θ = 2.
a. Find a test that has significance level α = 0. What is its power?
b. For 0 < α < 1, consider the test that rejects when X ∈ [0, α]. What is its significance level and power?
c. What is the significance level and power of the test that rejects when X ∈ [1 − α, 1]?
d. Find another test that has the same significance level and power as the previous one.
e. Does the likelihood ratio test determine a unique rejection region?
f. What happens if the null and alternative hypotheses are interchanged—H0 : θ = 2 versus H1 : θ = 1?
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