This is a simulation exercise designed to illustrate the robustness of the t-test to the assumption of normality.

(a) Generate 100,000 samples of size n  5 from a standard normal distribution ( ). For each sample find the 95% two-sided t CI on the mean. For each sample determine if the CI includes the true population mean of   0. Let X be the number of intervals for which the true mean is captured in the interval. Compute the ratio (X100,000) and multiply

  0, 2  1 this ratio by100. This is the coverage of the t CI based on your simulation. The coverage should be close to 95%.

(b) Repeat part

(a) but generate the 100,000 samples from a chi-square distribution with one degree of freedom. This distribution is very skewed with a long tail to the right and does not look much like a normal distribution. (Hint: To generate random variables remember that the square of a standard normal random variable is a random variable.) In computing the coverage remember that the mean of a chi-square random variable with one degree of freedom is unity. What is the coverage of these CIs?

Is it close to 95%?