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Chapter 8
Sampling Distributions of a
Sample Mean
Section 2
1
BIG Idea
When the objective of a statistical
investigation is to make an inference
about the population mean µ, it is natural
to consider the sample mean as an
estimate of µ.
To understand how inferential procedures
based on the sample mean work, we
must first study how sampling variability
causes the sample mean to vary in value
from one sample to the next. The
behavior of the sample mean is described
by its sampling distribution.
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Sampling Distribution
The distribution of a statistic is called its
sampling distribution.
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Example
Consider a population that consists of the
numbers 1, 2, 3, 4 and 5 generated in a
manner that the probability of each of those
values is 0.2 no matter what the previous
selections were. This population could be
described as the outcome associated with a
spinner such as given below. The distribution is
next to it.
x
1
2
3
4
5
4
p(x)
0.2
0.2
0.2
0.2
0.2
Example
If the sampling distribution for the means of
samples of size two is analyzed, it looks like
Sample
1, 1
1, 2
1, 3
1, 4
1, 5
2, 1
2, 2
2, 3
2, 4
2, 5
3, 1
3, 2
3, 3
5
1
1.5
2
2.5
3
1.5
2
2.5
3
3.5
2
2.5
3
Sample
3, 4
3, 5
4, 1
4, 2
4, 3
4, 4
4, 5
5, 1
5, 2
5, 3
5, 4
5, 5
3.5
4
2.5
3
3.5
4
4.5
3
3.5
4
4.5
5
1
1.5
2
2.5
3
3.5
4
4.5
5
frequency
1
2
3
4
5
4
3
2
1
25
p(x)
0.04
0.08
0.12
0.16
0.20
0.16
0.12
0.08
0.04
Example
The original distribution and the sampling
distribution of means of samples with n=2
are given below.
1
2
3
4
Original distribution
5
1
2
3
5
Sampling distribution
n=2
6
4
Example
Sampling distributions for n=3 and n=4 were
calculated and are illustrated below.
1
2
3
4
5
Sampling distribution n = 3
7
1
2
3
4
5
Sampling distribution n = 4
Simulations
To illustrate the general
behavior of samples of
fixed size n, 10000
samples each of size 30,
60 and 120 were
generated from this
uniform distribution and
the means calculated.
Probability histograms
were created for each of
these (simulated)
sampling distributions.
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2
3
4
Means (n=30)
2
Notice all three of these
look to be essentially
normally distributed.
Further, note that the
variability decreases as
the sample size increases. 2
3
4
3
4
Means (n=60)
Means (n=120)
© 2008 Brooks/Cole, a division of Thomson Learning, Inc.
Simulations
To further illustrate the general behavior of
samples of fixed size n, 10000 samples each of
size 4, 16 and 32 were generated from the
positively skewed distribution pictured below. It
represents the length of overtime games played
in the NHL from 1970 – 1993.
Skewed distribution
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Notice that these sampling distributions all all skewed,
but as n increased the sampling distributions became
more symmetric and eventually appeared to be almost
normally distributed.
Terminology
Let x denote the mean of the observations
in a random sample of size n from a
population having mean µ and standard
deviation . Denote the mean value of the
distribution by  x and the standard deviation
of the distribution by  x (called the standard
error of the mean), then the rules on the
next two slides hold.
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Properties of the Sampling
Distribution of the Sample Mean.
Rule 1:  x  
Rule 2: 
x

n
This rule is approximately correct as
long as no more than 5% of the
population is included in the sample.
Rule 3: When the population distribution is
normal, the sampling distribution of x
is also normal for any sample size n.
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Central Limit Theorem.
Rule 4: When n is sufficiently large, the
sampling distribution of x is
approximately normally
distributed, even when the
population distribution is not
itself normal.
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Illustrations of Sampling
Distributions
Population
n =4
n=9
n = 16
Symmetric normal like population
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© 2008 Brooks/Cole, a division of Thomson Learning, Inc.
Illustrations of Sampling
Distributions
Population
n=4
n=10
n=30
Skewed population
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More about the Central Limit
Theorem.
The Central Limit Theorem can safely
be applied when n exceeds 30.
If n is large or the population distribution
is normal, the standardized variable
x  X x  
z

X
 n
has (approximately) a standard normal
(z) distribution.
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Example
Hot Dogs!
A hot dog manufacturer asserts that one of its brands has
an average fat content of µ=18 g per hot dog. Consumers would
not be unhappy if the content was less than 18 but would be if
it exceeded 18 grams. Suppose the standard deviation of the x
distribution is 1.
A testing organization is asked to analyze 36 hot dogs.
What is the mean of the sample?
What is the standard deviation of the sample mean?
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Example
Suppose that we randomly select a
sample of 64 measurements from a
population having a mean equal to 20
and a standard deviation equal to 4.
a.) Describe the shape of the sampling
distribution of the sample mean x
b.) Find the mean and the standard
deviation of the sampling distribution of
the sample mean.
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Hospital Example
The average length of a hospital stay in
the US is µ=9 days with a standard
deviation of 3 days. Assume a random
sample of 100 patients is obtained and
the mean stay for the 100 patients is
obtained. What is the probability that the
average length of stay for this group of
patients will be less than 9.6 days?
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Hospital Example Continued
Step 1: Find the mean and standard
deviation for the sample.
Step 2: Find the z score for the value of
9.6.
Step 3: Use the standard normal
distribution table to find the answer.
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ACT Example
The scores of students on the ACT
college entrance exam in a recent year
had a Normal distribution with µ=18.6 and
a standard deviation of 5.9.
a.) What is the probability that a single
student randomly chosen from all those
taking the test scores 21 or higher?
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ACT Example Continued
b.) Now take a simple random sample of
50 students who took the test. What is
the probability that the mean score of
these students is 21 or higher?
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Example
A food company sells “18 ounce” boxes
of cereal. Let x denote the actual amount
of cereal in a box of cereal. Suppose that
x is normally distributed with µ = 18.03
ounces and  = 0.05.
a) What proportion of the boxes will
contain less than 18 ounces?
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Example - continued
b) A case consists of 24 boxes of cereal.
What is the probability that the mean
amount of cereal (per box in a case)
is less than 18 ounces?
The central limit theorem states that the
distribution of x is normally distributed so
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