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

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 Definition: Sampling Distributions 7.1 Central Limit Theorem

HAWKES LEARNING SYSTEMS math courseware specialists Properties of the Central Limit Theorem: Sampling Distributions 7.1 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: 1.The sampling distribution will approximate a normal distribution, regardless of the shape of the original distribution. Larger sample sizes will produce a better approximation. 2.The mean of a sampling distribution,, equals the mean of the population. 3.The standard deviation of a sampling distribution,, equals the standard deviation of the population divided by the square root of the sample size.

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

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? Calculate the standard deviation of the sampling distribution: HAWKES LEARNING SYSTEMS math courseware specialists 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.” Solution: Sampling Distributions 7.1 Central Limit Theorem

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? Calculate the mean of the sampling distribution: HAWKES LEARNING SYSTEMS math courseware specialists Solution: Sampling Distributions 7.1 Central Limit Theorem Property 2 states: “The mean of the sampling distribution equals the mean of the population.”  217

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? Calculate the standard deviation: HAWKES LEARNING SYSTEMS math courseware specialists 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.” Solution: Sampling Distributions 7.1 Central Limit Theorem