2. Crazy Cauchy: The Cauchy distribution (aka a t distribution with one degree of freedom) is outlier prone (extremely heavy tailed). In R, Cauchy random variables can be simulated using rcauchy(). For example, the R code z <- rcauchy(500,0,1)

will generate 500 Cauchy random variables with a median of 0 and a spread of 1

(meaning 50% of the density function lies between –1 and + 1).

(a) Simulate 50 Cauchy random variables with median 0 and spread 1 and compute a confidence interval for the mean.

(b) Simulate 500 Cauchy random variables with median 0 and spread 1 and compute a confidence interval for the mean.

(c) Simulate 5000 Cauchy random variables with median 0 and spread 1 and compute a confidence interval for the mean.

(d) Simulate 50,000 Cauchy random variables with median 0 and spread 1 and compute a confidence interval for the mean.

(e) What might you conclude about the central limit theorem and the Cauchy distribution based on the results in (a)–(d).