# Understanding the Central Limit Theorem.

Visit the following Web site Central Limit Theorem Applet and read what is posted: http://www.stat.sc.edu/~west/javahtml/CLT.html

You will choose from the pull down menu at the bottom of the page both the number of dice and the number of rolls at a time. When you "click" you will be virtually rolling your dice.

Complete the experiment using the following conditions. Note: You may need to click your Web browser's Refresh or Reload button to reset the experiment. Each time you repeat the experiment, keep track of how many clicks (rolls) it takes to produce a normal distribution:

1 die, 10 rolls at a time

1 die, 100 rolls at a time

1 die, 1000 rolls at a time

2 dice, 10 rolls at a time

2 dice, 100 rolls at a time

2 dice, 1000 rolls at a time

5 dice, 10 rolls at a time

5 dice, 100 rolls at a time

5 dice, 1000 rolls at a time

Discuss the following: a report of how many clicks (rolls) it took under each condition to produce a normal distribution, a brief explanation of what conducting these experiments tells you about the Central Limit Theorem, and how this information may apply to research in which you may be involved.

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#### Solution Preview

The response here does not include graphs and tables please see attachment.

There are multiple ways to understand the CLT in terms of dice. First you could imagine that you have a die. When you roll it there is a probability associated with the outcome. Namely,

Roll 1 2 3 4 5 6

Probability 1/6 1/6 1/6 1/6 1/6 1/6

The mean of which is

1/6(1) + 1/6(2) + 1/6(3) + 1/6(4) + 1/6(5) + 1/6(6) = 3.5

Now imagine rolling the die 2 times or rolling 2 dice once (it doesn't matter in this case) and computed the mean of the dice. Possible outcomes are as follows:

(1,1) the mean of which is 1.

(1,3) the mean of which is 2.

(4,6) the mean of which is 5.

The CLT states that if we were to continue this experiment over and over and to compute the mean each time and to plot a histogram of our results (the means) they would have a normal distribution with a mean themselves approaching 3.5. So, we say that the mean of the sample means approaches the true population mean. ...

#### Solution Summary

I demonstrate the properties of the central limit theorem (CLT) by conducting several dice rolling experiments with a web based java applet. Graphs of the process are included in the explanation.

Central Limit Theorem: understanding the Z Score.

Assume that men's weights are normally distributed with a mean of 172 lbs. and a standard deviation of 29 lbs.

I understand that if one man is selected the probability his weight is less than 182 lbs is .6331

However I don't understand if 25 men are randomly selected what is the probability that they have a mean weight less than 182 lbs.

I know the answer is .9573 but I do not understand how to get it.

Please explain.

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