# hw-8

Homework 8 in STAT400: Computational Statistics @ CSU

## Assignment

Be sure to set.seed(400) at the beginning of your homework.

1. 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?

2. 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.

1. 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.
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.
3. Repeat problems a. and b. but set $$\pi=0.1$$ when you simulate the data.
4. Repeat problems a. and b. but set $$\pi=0.5$$ when you simulate the data.
5. 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.