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Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
5-1
Statistics for Business and
Economics
Chapter 5
Sampling Distributions
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
5-2
Content
1. The Concept of a Sampling Distribution
2. Properties of Sampling Distributions:
Unbiasedness and Minimum Variance
3. The Sample Distribution of the Sample
Mean and the Central Limit Theorem
4. The Sampling Distribution of the Sample
Proportion
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Learning Objectives
• Establish that a sample statistic is a random
variable with a probability distribution
• Define a sampling distribution as the probability
distribution of a sample statistic
• Give two important properties of sampling
distributions
• Learn that the sampling distribution of both the
sample mean and sample proportion tends to
be approximately normal
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5.1
The Concept of a Sampling
Distribution
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Parameter & Statistic
A parameter is a numerical descriptive
measure of a population. Because it is
based on all the observations in the
population, its value is almost always
unknown.
A sample statistic is a numerical
descriptive measure of a sample. It is
calculated from the observations in the
sample.
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Common Statistics &
Parameters
Sample Statistic
Population Parameter
Mean
x

Standard
Deviation
s

Variance
s2
2
Binomial
Proportion
^
p
p
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Sampling Distribution
The sampling distribution of a sample
statistic calculated from a sample of n
measurements is the probability distribution
of the statistic.
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Developing
Sampling Distributions
Suppose There’s a Population ...
• Population size, N = 4
• Random variable, x
• Values of x: 1, 2, 3, 4
• Uniform distribution
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Population Characteristics
Summary Measure
N

 xi
i1
N
 2.5
Population Distribution
.3
.2
.1
.0
P(x)
x
1
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2
3
4
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All Possible Samples
of Size n = 2
16 Samples
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1st 2nd Observation
Obs 1
2
3
4
1
1,1 1,2 1,3 1,4
1 1.0 1.5 2.0 2.5
2
2,1 2,2 2,3 2,4
2 1.5 2.0 2.5 3.0
3
3,1 3,2 3,3 3,4
3 2.0 2.5 3.0 3.5
4
4,1 4,2 4,3 4,4
4 2.5 3.0 3.5 4.0
Sample with replacement
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Sampling Distribution
of All Sample Means
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
Sampling Distribution
of the Sample Mean
P(x)
.3
.2
.1
.0
x
1.0 1.5 2.0 2.5 3.0 3.5 4.0
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Summary Measure of
All Sample Means
N
x
1.0  1.5  ...  4.0
X 

 2.5
N
16
i
i1
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Comparison
Population
.3
.2
.1
.0
Sampling Distribution
P(x)
x
1
2
3
4
P(x)
.3
.2
.1
.0
x
1.0 1.5 2.0 2.5 3.0 3.5 4.0
  2.5
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 x  2.5
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5.2
Properties of Sampling
Distributions:
Unbiasedness and
Minimum Variance
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Point Estimator
A point estimator of a population parameter is a
rule or formula that tells us how to use the
sample data to calculate a single number that
can be used as an estimate of the population
parameter.
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Estimates
If the sampling distribution of a sample statistic
has a mean equal to the population parameter
the statistic is intended to estimate, the statistic
is said to be an unbiased estimate of the
parameter.
If the mean of the sampling distribution is not
equal to the parameter, the statistic is said to be
a biased estimate of the parameter.
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Example
Probability Distribution
x
0
2
3
p(x)
1
3
1
3
1
3
 = 1.667
 2 = 1.556
 = 1.247
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Example
Sampling Distribution of x for n = 2
(3 possible samples, each with a sample mean)
x
1
p(x)
1
3
3
2
1
3
5
2
1
3
E(x ) = 1.667 is the same as .
x is an unbiased estimator of .
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5.3
The Sampling Distribution
of a Sample Mean and the
Central Limit Theorem
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Properties of the Sampling
Distribution of x
1. Mean of the sampling distribution equals mean
of sampled population*, that is,
 x  E x   .
2. Standard deviation of the sampling distribution
equals Standard deviation of sampled population
Square root of sample size

.
That is,  x 
n
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Standard Error of the Mean
The standard deviation  x is often referred
to as the standard error of the mean.
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Theorem 5.1
If a random sample of n observations is selected
from a population with a normal distribution, the
sampling distribution of x will be a normal
distribution.
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Sampling from
Normal Populations
• Central Tendency
x  
Population Distribution
 = 10
• Dispersion

x 
n
 = 50
x
Sampling Distribution
– Sampling with
replacement
n=4
 x = 5
n =16
x = 2.5
x- = 50
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x
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Standardizing the
Sampling Distribution of x
z
x  x
x
x


n
Sampling
Distribution
Standardized Normal
Distribution
=1
x
x
x
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 =0
z
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Thinking Challenge
You’re an operations
analyst for AT&T. Longdistance telephone calls
are normally distributed
with  = 8 min. and  = 2
min. If you select random
samples of 25 calls, what
percentage of the sample
means would be between
7.8 & 8.2 minutes?
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Sampling Distribution
Solution*
x   7.8  8
z


2
 .50
25
x   8.2  8
z

 .50

2
Standardized Normal
25
n
Distribution
n
Sampling
Distribution
=1
x = .4
.3830
.1915 .1915
7.8 8 8.2 x
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–.50 0 .50
z
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Sampling from
Non-Normal Populations
• Central Tendency
x  
Population Distribution
 = 10
• Dispersion

x 
n
 = 50
x
Sampling Distribution
– Sampling with
replacement
n=4
 x = 5
n =30
x = 1.8
x- = 50
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x
5-28
Central Limit Theorem
Consider a random sample of n observations
selected from a population (any probability
distribution) with mean μ and standard deviation .
Then, when n is sufficiently large, the sampling
distribution of x will be approximately a normal
distribution with mean  x   and standard
deviation  x   n . The larger the sample size,
the better will be the normal approximation to the
sampling distribution of x .
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Central Limit Theorem
As sample
size gets
large
enough
(n  30) ...
x 

n
sampling
distribution
becomes almost
normal.
x  
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x
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Central Limit Theorem
Example
The amount of soda in cans
of a particular brand has a
mean of 12 oz and a standard
deviation of .2 oz. If you
select random samples of 50
cans, what percentage of the
sample means would be less
than 11.95 oz?
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Central Limit Theorem
Solution*
x   11.95  12
z

 1.77

.2
Sampling
Standardized Normal
n
50
Distribution
Distribution
x = .03
.0384
=1
.4616
11.95 12
x
–1.77 0
z
Shaded area exaggerated
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5.4
The Sampling Distribution
of the Sample Proportion
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Sample Proportion
Just as the sample mean is a good estimator of the
population mean, the sample proportion—denoted
p̂ — is a good estimator of the population
proportion p. How good the estimator p̂ is will
depend on the sampling distribution of the statistic.
This sampling distribution has properties similar to
those of the sampling distribution of x.
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Key Ideas
Properties of Probability Distributions
Sampling distribution of a statistic—the
theoretical probability distribution of the statistic in
repeated sampling
Unbiased estimator—a statistic with a sampling
distribution mean equal to the population
parameter being estimated
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Key Ideas
Properties of Probability Distributions
Central Limit Theorem—the sampling distribution of
the sample mean, x , or the sample proportion, p̂ , is
approximately normal for large n
x is the minimum-variance unbiased estimator
(MVUE) 
p̂ is the MVUE of p
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Key Ideas
Generating the Sampling Distribution of
x
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