In a crossover trial comparing a new drug to a standard, denotes the probability that the

In a crossover trial comparing a new drug to a standard, π denotes the probability that the new one is judged better. It is desired to estimate π and test H₀: π = 0.5 against H_a: π ≠ 0.5. In 20 independent observations, the new drug is better each time.

a. Find and sketch the likelihood function. Give the ML estimate of

b. Conduct a Wald test and construct a 95% Wald confidence interval for π. Are these sensible?

c. Conduct a score test, reporting the P-value. Construct a 95% score confidence interval. Interpret.

d. Conduct a likelihood-ratio test and construct a likelihood-based 95% confidence interval. Interpret.

e. Construct an exact binomial test and 95% confidence interval. Interpret.

f. Suppose that researchers wanted a sufficiently large sample to estimate the probability of preferring the new drug to within 0.05, with confidence 0.95. If the true probability is 0.90, about how large a sample is needed?