This exercise is designed to compare the performance of

Chapter 5, Problem 9E

(choose chapter or problem)

This exercise is designed to compare the performance  of the Agresti-Coull confidence interval for a pro- portion (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.

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.Repeat part (a), using n = 17.Repeat part (a), using n = 40.The performance of the traditional confidence  interval does not improve steadily as the sam- ple 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?For which sample sizes does the Agresti–Coull in- terval have greater coverage probability than does  the traditional one? For which sample size are the coverage probabilities nearly equal?Other things being equal, a narrower confidence interval is better than a wider one. Which method pro- duces confidence intervals with the narrower mean  width?

Equation transcription:

Text transcription:

X_{i}^{*}