Let X1,... ,Xn be a random sample from the uniformed(, + 1) distribution. To test H0: = 0 versus H1: > 0, use

Let X1,... ,Xn be a random sample from the uniformed(θ,θ + 1) distribution. To test H0: θ = 0 versus H1: θ > 0, use the test
reject H0 if Yn ≥ 1 or Y1 > k,
where k is a constant, Y1 = min{X1,..., Xn}, Yn = max{X1,..., Xn}.
(a) Determine k so that the test will have size α.
(b) Find an expression for the power function of the test in part (a).
(c) Prove that the test is UMP size a.
(d) Find values of n and k so that the UMP .10 level test will have power at least .8 if θ > 1.