AdventHealth University Department of Health and Biomedical Sciences STAT505: Applied Statistics for the Graduate Students Fall 2022 Project #4:  Central Limit Theorem Experiment Instructions: You will need a standard six-sided die and at least six sets of data to complete this project

AdventHealth University

Department of Health and Biomedical Sciences

STAT505: Applied Statistics for the Graduate Students

Fall 2022

Project #4:  Central Limit Theorem Experiment

Instructions: You will need a standard six-sided 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.

1. 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: ______________________

2. Calculate the mean of the population. (Hint: What is the mean outcome for rolling a single die?)

μ = __________________________

3. Calculate the standard deviation of the population. (Hint: What is the standard deviation of all possible outcomes from rolling a single die? Use the formula for finding the standard deviation for discrete probability distribution or Minitab)

σ = ___________________________

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 two other students in your class and tally the frequency of each roll of the die from the combined results. Record your results in a table similar to the following.

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?  Explain.

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 at least five (5) of your classmates. 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.

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? (Hint: Use the sample means you collected in Step 6.)

              μ_? = ________________

How does μ_? compare to μ 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 Minitab statistical software.)

              σ_?= __________________________

              How does σ_? compare to σ from question 3.

              Since our samples were groups of 10 rolls, n = 10.  Using σ from question 3, calculate .

               = __________________

              How does σ_? compare to ?

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.

Repeat Steps 5 – 11using the new sample size of n = 30.

Step 12:           Do your results seem to verify the Central Limit Theorem.