Let (X1, Yi), , (Xn, Yn) be a random sample that follows the regression model
Yi = ̠Xi + ei, i = 1, , n,
where the distribution of Xi is unspecified, but ei are independent of Xi and are normally distributed with mean zero and known variance σ^2. Within the class of tests such that the conditional probability of the type I error given (X1, , Xn) is no greater than α, can you find the uniformly most powerful test for testing the null hypothesis β = 0 against the alternative hypothesis β > 0? If so, give an explicit critical region for the test.