This exercise is designed to compare the performance of the Agresti-Coull confidence interval for a proportion (expression 5.5 on page 339) with that of the traditional confidence interval (expression 5.8 on page 341). We will use sample sizes of n = 10, n = 17, and n = 40, with p = 0.5.
a. Generate 10,000 observations X∗i , each from a binomial distribution with n = 10 and p = 0.5. For each observation, compute the upper and lower limits for both the Agresti-Coull 95% confidence interval and the traditional one. For each confidence interval, compute its width (upper limit − lower limit). Use the simulated data to estimate the coverage probability and mean width for both the Agresti-Coull and the traditional confidence interval.
b. Repeat part (a), using n = 17.
c. Repeat part (a), using n = 40.
d. The performance of the traditional confidence interval does not improve steadily as the sample size increases; instead it oscillates, so that the coverage probability can be better for a smaller sample than for a larger one. Compare the coverage probabilities for the traditional method for sample sizes of 17 and of 40. Do your results confirm this fact?
e. For which sample sizes does the Agresti-Coull interval have greater coverage probability than does the traditional one? For which sample size are the coverage probabilities nearly equal?