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

1.7 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 = 0.50 against H_α: π ≠ 0.50. In 20 independent observations, the new drug is better each time.

a. Find and sketch the likelihood function. Is it close to the quadratic shape that large-sample normal approximations utilize?

b. Give the ML estimate of π. 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. Interpret.