# Stat 3011 (Geyer) In-Class Examples (Central Limit Theorem)

## 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).

## Exponential 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 an exponential 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).

## 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.