# The Central Limit Theorem (Stat 5101, Geyer)

## General Instructions

To do each example, just click the "Submit" button. You do not have to type in any R instructions (that's already done for you). You do not have to select a dataset (that's already done for you).

## Normal Population Distribution

If we have a random sample of size n from a normally distributed population, we know the sampling distribution of the sample mean is exactly normal with

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a normal population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed normal density curve that is the theoretical sampling distribution of the sample mean.

• If you increase the simulation size nsim (the number of xbar values used to make the histogram), the histogram gets closer and closer to the black theoretical curve (the exact sampling distribution of the sample mean).

## Gamma Population Distribution

If we have a random sample of size n from a non-normally distributed population, we know the sampling distribution of the sample mean is not exactly normal, only approximately normal for large sample sizes, but we do know the mean and sd are exactly

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a gamma population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed non-normal density curve that is the theoretical sampling distribution of the sample mean and the normal density curve (red) that is the approximately sampling distribution for large sample sizes. For comparison, the population density is shown in another plot.

• If you increase the sample size n, the two theoretical curves get closer and closer together.
• If you increase the simulation size nsim (the number of xbar values used to make the histogram), the histogram gets closer and closer to the black theoretical curve (the exact sampling distribution of the sample mean).
• If you change the shape parameter `alpha` of the gamma distribution we are taking as the population distribution in this example, you change the skewness of the population. The default `alpha` = 1 is an exponential distribution.
• If you change the scale parameter `lambda` of the gamma distribution, none of the shapes in either of the plots changes. Why?

## Bimodal Skewed Population Distribution

If we have a random sample of size n from a non-normally distributed population, we know the sampling distribution of the sample mean is not exactly normal, only approximately normal for large sample sizes, but we do know the mean and sd are exactly

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a bimodal skewed population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed non-normal density curve that is the theoretical sampling distribution of the sample mean and the normal density curve (red) that is the approximately sampling distribution for large sample sizes.

• If you increase the sample size n, the two theoretical curves get closer and closer together.
• If you increase the simulation size nsim (the number of xbar values used to make the histogram), the histogram gets closer and closer to the black theoretical curve (the exact sampling distribution of the sample mean).
• If you change the value of `mu` to any number between zero and one, you make the population distribution more or less skewed. `mu <- 1 / 2` makes a symmetric bimodal population.
• If you change the value of `sigma.normal` to any positive number, you make the width of the peaks of the population wider or narrower.