Project A: Central Limit Theorem Experiment
You will need a standard sixsided die and at least six sets of data to complete this project.
Consider the distribution of the possible outcomes from rolling a single die; that is, $1$, $2$, $3$, $4$, $5$, and $6$. Let's use this distribution as our theoretical population distribution. We want to use this population distribution to explore the properties of the Central Limit Theorem. Let's begin by determining the shape, center, and dispersion of the population distribution.

What would you expect the distribution of the outcomes from repeated rolls of a single die to look like; in other words, what is its shape? (Hint: What is the probability of getting each value?)
Shape:

Calculate the mean of the population. (Hint: What is the mean outcome for rolling a single die?)
$\mu =\overline{)\phantom{000000000}}$

Calculate the standard deviation of the population. (Hint: What is the standard deviation of all possible outcomes from rolling a single die?)
$\sigma =\overline{)\phantom{000000000}}$
Let's continue by exploring the distribution of the original population empirically. To do so, follow these steps.
Step 1
Roll your die $60$ times and record each outcome.
Step 2
Combine your results with at least two other students and tally the frequency of each roll of the die from the combined results. Record your results in a table similar to the following.
Outcome  Frequency 

$1$  
$2$  
$3$  
$4$  
$5$  
$6$ 
Step 3
Draw a bar graph of these frequencies.
Step 4
Does the distribution appear to be a normal distribution? Is this what you expected from question 1?
The Central Limit Theorem is not about individual rolls like we just looked at, but is about the averages of sample rolls. Thus we need to create samples in order to explore the properties of the Central Limit Theorem.
Step 5
Return to your original data from Step 1. To create samples from your data you can group the rolls into sets of $10$. For each sequence of $10$ rolls, calculate the mean of that sample. Round your answers to one decimal place. (You should have six sample means.)
Step 6
Combine your sample means with those of as many of your classmates as you can. Record the sample means of each of your classmates' six samples.
Step 7
Tally the frequencies of the sample means from your combined results in a table like the one that follows.
Sample Mean  Frequency 

$1.0$–$1.2$  
$1.3$–$1.5$  
$1.6$–$1.8$  
$1.9$–$2.1$  
$2.2$–$2.4$  
$2.5$–$2.7$  
$2.8$–$3.0$  
$3.1$–$3.3$  
$3.4$–$3.6$  
$3.7$–$3.9$  
$4.0$–$4.2$  
$4.3$–$4.5$  
$4.6$–$4.8$  
$4.9$–$5.1$  
$5.2$–$5.4$  
$5.5$–$5.7$  
$5.8$–$6.0$ 
Step 8
Draw a histogram of the sample means.
Step 9
What is the shape of this distribution?
Step 10
What is the mean of your sample means? (Hint: Use the sample means you collected in Step 6.)
${\mu}_{\stackrel{\_}{x}}=\overline{)\phantom{000000000}}$
How does ${\mu}_{\stackrel{\_}{x}}$ compare to $\mu $ from question 2?
Step 11
What is the standard deviation of the sample means? (Again, go back to the sample means you collected in Step 6 and use a calculator or statistical software.)
${\sigma}_{\stackrel{\_}{x}}=\overline{)\phantom{000000000}}$
How does ${\sigma}_{\stackrel{\_}{x}}$ compare to $\sigma $ from question 3?
Since our samples were groups of $10$ rolls, $n=10$. Using $\sigma $ from question 3, calculate $\frac{\sigma}{\sqrt{n}}$.
$\frac{\sigma}{\sqrt{n}}}=\overline{)\phantom{000000000}$
How does ${\sigma}_{\stackrel{\_}{x}}$ compare to $\sigma $ from question 3, calculate $\frac{\sigma}{\sqrt{n}}$?
The Central Limit Theorem says that the distribution of the sample means should be closer to a normal distribution when the sample size becomes larger. To see this effect, group your original data from Step 1 into two samples of $30$ rolls instead of six sets of $10$. Repeat Steps 5–11 using the new sample size of $n=30$.
Step 12
Do your results seem to verify the Central Limit Theorem?