For a flip of a coin, let denote the probability of a head. An experiment tests

For a flip of a coin, let π denote the probability of a head. An experiment tests H₀: π = 0.5 against H_a: π ≠ 0.5, using n = 5 independent flips.

a. Show that the true null probability of rejecting H₁ at the 0.05 significance level is 0.0 for the exact binomial test and using the large-sample score test.

b. Suppose that truly π = 0.5. Explain why the probability that the 95% Clopper—Pearson confidence interval contains π equals 1.0.

(Is there any possible y for which both one-sided tests of H₀: π = 0.5 have P-value ≤ 0.025?)