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# Central Limit Theorem: Teacher's Guide

Summary: This module is the complementary teacher's guide for the Central Limit Theorem chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

The Central Limit Theorem (CLT) is considered to be one of the most powerful theorems in all of statistics and probability. It states that if you draw samples of size nn and average (or sum) them, you will get a distribution of averages (or sums) that follow a normal distribution.

## Example 1

Suppose μμ and σσ are the original mean and standard deviation of the population from which each sample of size nn is drawn. Let X-X-= the random variable for the average of nn samples. Let xx = the random variable for the number of nn samples

X~N(μ, σn )X~N(μ, σn)

Σx~N (nμ,n σ)Σx~N(nμ,nσ)

## The Dice Experiment

At the beginning of the chapter, there is a dice experiment. Together with the students, do the experiment. The example consists of rolling 10 times each, 1 die, 2 dice, 5 dice, and 10 dice and averaging the faces. Draw graphs (histograms are OK). This experiment, most of the time, shows that, as the number of dice increase, the graph looks more and more bell-shaped. Because the samples taken are usually small, you will not necessarily get a perfect bell-shaped curve. However, the students should get the idea.

## Example 2

It can be shown that the average amount of money one person spends on one trip to a particular supermarket is $51. The averages follow an exponential distribution. ### Problem 1 Find the probability that the average of 40 samples is more than$60.

#### Solution

Let X-X-= the average amount of money that 40 people spend. Have the students draw the appropriate picture, labeling the x-axis with X-X-. The mean μ=51μ=51 and the standard deviation μ=51μ=51. If you are using the TI-83/84 series, use the function normalcdf(60, 10^99, 51, 51/40).

The 75th percentile for the average amount spent by 40 people at the supermarket is $56.44. This means that 75% of the people spend no more than$56.44 and 25% spend no less than that amount.

This can be calculated by using the TI-83/84 function InvNorm(.75, 51, 51/ 40).

## Calculate Sums

You can also do examples for sums. We, the authors, do not do sums because of time (we are on a quarter system). Help the students to find the probability that the total (sum) amount of money spent by 10 people at the supermarket is less than \$500. Also, help them do a percentile problem.

## Z-score Formulas

If you want to teach the z-score formulas for averages and sums, they are:

• z = value-μ (-σ n) z = value-μ (-σ n)
• z=value -nμ nσz=value -nμ nσ

## Assign Practice

Assign the Practice in class to be done in groups.

## Assign Homework

Assign Homework. Suggested homework: (averages) 1a - f, 3, 5, 9, 10, 11a - d, f, k, 13a-c,g-j, 16, 17, 19 - 23

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