Homework 8 in STAT400: Computational Statistics @ CSU

Be sure to `set.seed(400)`

at the beginning of your homework.

Sign up for a GitHub account (https://github.com) and request an educational account (https://education.github.com/benefits). What is your GitHub user name?

*Coverage for a two-sided CI for a proportion \(\pi\).*If you have a sample of data that consists of \(0\)’s and \(1\)’s, you may want to estimate the proportion of \(1\)’ based on the sample. In this problem we will compare the properties of two different estimators of the proportion \(\pi\). The goal of the problem is to compare the coverage for confidence intervals computed using the two different estimators.

Let \(p =\) the proportion of \(1\)’s in a sample of \(0\)’s and \(1\)’s. So, \(p\) estimates \(\pi\). Compute the coverage for a \(95%\) CI for \(\pi\) using the approaches below.

95% Confidence Intervals for \(\pi\):

**Method 1**: Standard approach - use \(p \pm z_{0.975}\sqrt{p(1 - p)/n}\), where \(z_{0.975}\) = the \(0.975\) quantile from a \(N(0,1)\).**Method 2**: This method uses a different estimator for \(\pi\). First, add 2 successes and 2 failures to your data and then use the interval from Method 1. Note that you need to adjust both \(p\) and \(n\) from Method 1.For the following, let \(m = 1000\) in your Monte Carlo estimations.

- Simulate \(n=20\) observations from a Bernoulli distribution with \(\pi=0.05\). Compare the empirical coverage for methods 1 and 2. Use the same data to compare methods 1 and 2.
- Simulate \(n=100\) observations from a Bernoulli distribution with \(\pi=0.05\). Compare the empirical coverage for methods 1 and 2. Use the same data to compare methods 1 and 2.
- Repeat problems a. and b. but set \(\pi=0.1\) when you simulate the data.
- Repeat problems a. and b. but set \(\pi=0.5\) when you simulate the data.
- Summarize your findings in a table. Which method do you recommend based on these results?

Note: Method 2 is from the article “Approximate Is Better than”Exact" for Interval Estimation of Binomial Proportions," by Alan Agresti and Brent A. Coull,

*The American Statistician*, Vol. 52, No. 2 (May, 1998), pp. 119-126

Turn in in a pdf of your homework to canvas. Your .Rmd file on rstudio.cloud will also be used in grading, so be sure they are identical and reproducible.