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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 5.1: Sampling Distributions for Counts and Proportions
Goal: Estimate the proportion, p, of a population that belongs to a particular
category (i.e. find the proportion of Americans that “approve” of the job GW is
doing as President).
Take a random sample of Americans, of size n and count the number of
Americans in the sample that “approve.” Suppose we poll 110 Americans and
45 “approve.”
What are the values of X, n, and p for this example?
Notation:
X: The number (or count) of items in sample that are in the category.
n: The sample size
p : The sample proportion (i.e. the proportion of the sample in the category)
Binomial Distributions for Sample Counts - X
(page 335)
Some Common Binomial Settings: Coins, Dice, etc.
Consider Example 5.2….
Does the example in the previous slide constitute a binomial setting?
If so, identify, what is a “success,” n, and p for the example….
See technical note on the middle of page 337!
Finding Probabilities for Binomially Distributed Counts
Let’s work a few examples (coin, dice, etc.)!
Calculator Commands:
•binompdf(n, p, value) gives probability (likelyhood) that X=value.
•binomcdf(n, p ,value) gives probability that X is less than or equal
to the value, i.e. that X=0 or X=1 or …X=value.
Finding Probabilities for Sample Counts
For the scenario in Problem 5.6, how likely is it that your sample
will contain:
(a) Exactly 4 “successes.” Hint: binompdf(n,p,value)
(b) At most 4 “successes.” Hint: binomcdf(n,p,value)
(c) Between 4 and 8, inclusive, “successes.”
(d) On average, how many “successes” will your sample have? (see
next slide!)
The Binomial Mean and Standard Deviation
Answer to Part (d) on Previous Slide?
Finding Probabilities for Sample Proportions
Example 5.8 – Converting a probability about a sample proportion
to a probability about a sample count. Note that the sample
proportion is not binomial since it is not a count! How is the sample
proportion distributed? What are the values defined above for this
example?
Notice the shape,
center, and variability
of the sampling
distribution of
for
pn=2500.
Let’s Use Two Cool Applets to Visualize Each of These:
•Sampling Distribution of X: Applet on CD (CLT Binomial)
•Sampling Distribution of p : Sampling Sim
The Sampling Distribution of p
Let’s rework Example 5.8 using the Normal Approximation!
(Example 5.10 page 345, see illustration, next slide…)
Sampling Distribution of p
Normal Approximation of
Sampling Distribution of p
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
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)
Let’s Work Some Problems!
• Problem 5.34 (page 370)
• Problem 5.40 (page 371)