Consider testing H0 : < 0 versus H1 : > 0 using 0-1 loss, where

Consider testing H0 : μ < 0 versus H1 : μ > 0 using 0-1 loss, where X ~ n(μ, 1). Let δc be the test that rejects H0 if X > c. For every test in this problem, there is a δC in the class of tests (δc, -∞ ≤ c ≤ ∞} that has a uniformly smaller (in μ) risk function. Let δ be the test that rejects H0 if 1 < X < 2. Find a test δC that is better than δ. (Either prove that the test is better or graph the risk functions for δ and δC and carefully explain why the proposed test should be better.)