Question

1. Demonstrate that a sampling distribution of n=4 vs n=20 from a right skewed population distribution approaches normality as the sample size increases. In other words, show the Central Limit Theorem works for a “reasonably large” sample size. Create a population that follows an exponential distribution where the default rate (λ) =1. The population is say N=100,000. Note(s): Use the set.seed function and see how these results might change with different seed number. The suggested number of 100 samples is actually small considering simulations may yield slightly different outcomes, but it is worth exploring the nuance of randomness when using R. Also consider playing around with sample sizes and seeing how the sampling distribution changes. (14 pts)

1. Show the shape of the underlying exponential population distribution.
2. Find the average of 100 random samples of n=4 from the population (Hints for using the sample function and creating a data frame for calculating means is shown in template). Provide: 1) histogram 2) qqplot against theoretical quantiles of normal distribution for the sampling distribution of means. 3) In a sentence or two comment on normality. (6 pts)

Answer 1

As seed increases the shape of distribution becomes more steep or is more skewed to the right which is depicted by histograms of X , seed being 10, 100 and 1000

As sample size increases from 4 to 20, the QQ plot is more normal in the sense that qqplot for sample size 4 is skewed to right and same size 20 has very less skew.