1. Suppose X1,..., Xn is a random sample from a normal distribution with a known variance of σ2 and an unknown mean of μ. Find the most powerful α-level test of H0 : μ = μ0 versus Ha : μ = μa if (a) μ0 > μa, and (b) μa > μ0.
2. Show that the most powerful test obtained in Example 7.2.1 is uniformly most powerful for testing H0 : μ ≤μ0 versus Ha : μ> μa, but not uniformly most powerful for testing H0 : μ = μ0 versus Ha : μ /= μ0.
3. Suppose X1,..., Xn is a random sample from a U(0, θ) distribution. Find the most powerful α-level test for testing H0 : θ = θ0 versus Ha : θ = θ1, where θ0 < > θ1.