Project : Chapter 7 Central Limit Theorem Experiment
Directions: 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, 1,2,3,4,5,6. 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?
μ = _____________
Calculate the standard deviation of the population. (Hint: What is the standard deviation of all possible outcomes from rolling a single die?)
s = _____________
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
Step 3: Draw a bar graph of these frequencies.
Step 4: Does the distribution appear to t be a normal distribution? Is this what you expected from question 1?
The Central Limit Theorem is not about individual rolls like we just look at, but is about 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 our 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 combine results in a table like the one that follows.
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 mean? (Hint: Use the sample means you collected in Step 6.)
μ!” = ___________________
How does μ!” compare to μ from question 2?
Sample Mean
Frequency
Step 11: What is the standard deviation of the sample mean? (Again, go back to the sample means you collected in Step 6 and use a calculator or statistical software.)
𝜎!̅ = ____________
How does 𝜎!̅ compare to 𝜎 from question 3?
Since our samples were groups of 10 rolls, n=10. Using 𝜎 σ/√𝑛.
σ/√𝑛 = _____________
How does 𝜎!̅ compare to σ/√𝑛?
from question 3, calculate
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 three properties of the Central Limit Theorem?
