1. Introduction to the Practice of Statistics Fifth Edition Chapter 5: Sampling Distributions Copyright © 2005 by W. H. Freeman and Company David S. Moore George P. McCabe

2. Sampling Distributions 5.1 Sampling Distributions for Counts and Proportions 5.2 The Sampling Distribution of a Sample Mean

3. Basic Terminology The population distribution of a variable gives for a randomly chosen individual from the population, how likely the value of the variable for the individual is in certain ranges. Example: If the variable X (height of American women) is normal with mean, inches, and standard deviation, inches, then how likely is it that a randomly selected American women is over 65 inches tall?

4. More Terminology If we consider all SRSs of size n from the population of American women, what will the distribution of the sample means from each SRS?? This distribution is called the sampling distribution of a sample mean (from samples of size n). In this case, will denote the mean of the sampling distribution by and the standard deviation of the sampling distribution by. Via its shape, center, and spread, the sampling distribution of a statistic generally tells us how likely the statistic is to have certain values, if the statistic is unbiased (centered at the parameter it is meant to estimate), and how much variability the statistic has about its mean.

5. Section 5.2: The Sampling Distribution of a Sample Mean Population (Individuals) Sampling Distribution Of Means (Averages) for n=80 Averages are less variable than individual observations. Averages are more normal than individual observations. The Big Ideas:

6. Properties of the Sampling Distribution of a Sample Mean Remember: IF the population is normal then the sampling distribution of the sample mean (for fixed n is normal) otherwise, via the CLT the sampling distribution of the sample mean becomes approximately normal as n increases! How fast does the CLT work? Let’s check it out via Sampling Sim!

7. The CLT In Action! Individual Measurements (Population Distribution) with Population Mean of 1 Averages (n=2) (Sampling Distribution of Sample Mean when n=2) Averages (n=10) (Sampling Distribution of Sample Mean when n=10) Averages (n=25) (Sampling Distribution of Sample Mean when n=25)

8. Properties of the Sampling Distribution of a Sample Mean

9. Let’s Work Some Problems! Problem 5.34 (page 370) Problem 5.40 (page 371)