Section 7.1 Central Limit Theorem Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Section 7.1 Central Limit Theorem

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Definition: Sampling distribution for sample means – describes the means of all possible samples of a particular sample size from a specified population.

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Properties of the Central Limit Theorem: For any given population with mean, , and standard deviation, , a sampling distribution of the sample mean, with sample sizes of at least 30, will have the following three characteristics: The sampling distribution will approximate a normal distribution, regardless of the shape of the original distribution. Larger sample sizes will produce a better approximation. The mean of a sampling distribution, , equals the mean of the population. The standard deviation of a sampling distribution, , equals the standard deviation of the population divided by the square root of the sample size. It is also known as Standard Error:

The sampling distribution will approximate a normal distribution : HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem The sampling distribution will approximate a normal distribution : Property 1 states: The sampling distribution will approximate a normal distribution, regardless of the shape of the original distribution. Larger sample sizes will produce a better approximation.

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Estimate the mean of the population: If the mean of a given sampling distribution is = 85, what is an estimate for the mean of the population? Solution: Property 2 states: “The mean of the sampling distribution equals the mean of the population.”  = 85

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Calculate the standard deviation of the sampling distribution: If the standard deviation of a given population distribution is  = 9, and a sampling distribution is created from the population distribution with sample sizes of n = 100, what is the standard deviation of the sampling distribution? Solution: Property 3 states: “The standard deviation of a sampling distribution equals the standard deviation of the population divided by the square root of the sample size.”

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Calculate the mean of the sampling distribution: An internet source shows that the average one-way fare for business travel is $217, the lowest in five years. If 215 samples of size 45 are collected from across the U.S., what would you expect the average of the sampling distribution to be? Solution: Property 2 states: “The mean of the sampling distribution equals the mean of the population.” = 217

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Calculate the standard deviation: A study of elementary school students reports that children begin reading at age 5.7 years on average, with a standard deviation of 1.1 years. If a sampling distribution is created using samples of size 55, what would be the standard deviation of the sampling distribution? Solution: Property 3 states: “The standard deviation of a sampling distribution equals the standard deviation of the population divided by the square root of the sample size.”

HAWKES LEARNING SYSTEMS math courseware specialists Sampling Distributions 7.1 Central Limit Theorem Calculate the standard deviation of the sampling distribution: “The standard deviation of a sampling distribution equals the standard deviation of the population divided by the square root of the sample size.”