Let X1, . . . , Xn be a random sample from the exponential distribution with unknown parameter . Let 0 < 0 < 1

Let X1, . . . , Xn be a random sample from the exponential distribution with unknown parameter θ. Let 0 < θ0 < θ1 be two possible values of the parameter. Suppose that we wish to test the following hypotheses:
H0: θ = θ0,
H1: θ = θ1.
For each α0 ∈ (0, 1), show that among all tests δ satisfying α(δ) ≤ α0, the test with the smallest probability of type II error will reject H0 if where c is the α0 quantile of the gamma distribution with parameters n and θ0.