5.5: The Central Limit TheoremObjective: To apply the Central Limit Theorem to the Normal Model

CHS Statistics

Before, we took the individual values and compared them to others using the Normal model

Today, we will work with data in which each value represents the mean of some other sample

Link to the Past Section

The sampling distribution of a mean is the probability distribution of sample means, with all samples having the same sample size n.

In general, the sampling distribution of any particular statistic is the probability distribution of that statistic.

Sampling Distributions

Given:1. The random variable x has a distribution (which may or may

not be Normal) with a mean µ and standard deviation σ.2. Samples of all of the same size n are randomly selected from

the population of x values. (The samples are selected so that all possible samples of size n have the same chance of being selected.)

Conclusions:3. The distribution of sample means will, as the sample size

increases, approach a Normal distribution.4. The mean of the sample means will approach the population

mean µ.5. The standard deviation of the sample means will approach

The Central Limit Theorem (CLT)

Practical Rules Commonly Used:1. For sample sizes larger than 30 (within context), the

distribution of the sample means can be approximated reasonably well by a Normal distribution. The approximation gets better as the sample size n becomes larger.

2. If the original probability is itself Normally distributed, then the sample means will be Normally distributed for any sample size n (not just the values of n larger than 30).

The Central Limit Theorem (cont.)

In a study of work patterns in one population, the mean family times devoted to child care is 23.08 hours with a standard deviation of 15.58 hours. If 55 subjects are randomly selected from this population, find the probability that the mean of this sample group is between 23.08 hours and 25 hours.

Example 1

We used women's weights, which are Normally distributed with a mean of 143 lbs. and a standard deviation of 29 lbs. Find the probability that: If 1 woman is randomly selected, her weight is greater than 150 lbs.

 

If 36 different women are randomly selected, their mean weight is greater than 150 lbs.

Example 2

P. 263 # 2-20 Even

Assignment