Download Chapter 5 5.1 * 5.2: The Standard Normal Distribution

CHS Statistics
5.5: The Central Limit Theorem
Objective: To apply the Central Limit Theorem to the
Normal Model
Link to the Past Section
 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
Sampling Distributions
 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.
The Central Limit Theorem (CLT)
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:
1. The distribution of sample means 𝑥 will, as the sample size increases,
approach a Normal distribution.
2. The mean of the sample means will approach the population mean µ.
3.
The standard deviation of the sample means will approach
𝜎
𝑛
The Central Limit Theorem (cont.)
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).
Example 1
 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 2
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.
Assignment
 P. 263 # 2-20 Even