David S. Moore • George P. McCabe
Introduction to the
Practice of Statistics
Fifth Edition
Chapter 5:
Sampling Distributions
Copyright © 2005 by W. H. Freeman and Company
Sampling Distributions
5.1 Sampling Distributions for Counts and
Proportions
5.2 The Sampling Distribution of a Sample
Mean
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,  X  63 inches, and standard deviation,  X  3 inches, then how
likely is it that a randomly selected American women is over 65 inches
tall?
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  X and the
standard deviation of the sampling distribution by  X .
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.
Section 5.2: The Sampling Distribution of a Sample Mean
Population (Individuals)
Sampling
Distribution Of
Means (Averages)
for n=80
The Big Ideas:
•Averages are less variable than individual observations.
•Averages are more normal than individual observations.
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!
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)
The CLT In Action!
Averages (n=25)
(Sampling Distribution of
Sample Mean when n=25)
Properties of the Sampling Distribution of a Sample Mean
Let’s Work Some Problems!
• Problem 5.34 (page 370)
• Problem 5.40 (page 371)