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Chapter 7
7-1
Sampling Distributions
Learning Objectives
In this chapter, you learn:
The concept of the sampling distribution
To compute probabilities related to the sample
mean and the sample proportion
The importance of the Central Limit Theorem
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-1
Sampling Distributions
A sampling distribution is a distribution of all of the
possible values of a sample statistic for a given size
sample selected from a population.
For example, suppose you sample 50 students from your
college regarding their mean GPA. If you obtained many
different samples of 50, you will compute a different
mean for each sample. We are interested in the
distribution of all potential mean GPA we might calculate
for any given sample of 50 students.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
Chap 7-2
© 2006 Prentice Hall, Inc.
Chapter 7
7-2
Developing a
Sampling Distribution
Assume there is a population …
A
Population size N=4
B
C
D
Random variable, X,
is age of individuals
Values of X: 18, 20,
22, 24 (years)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-3
Developing a
Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
∑X
µ=
=
P(x)
i
N
.3
18 + 20 + 22 + 24
= 21
4
σ=
∑ (X − µ)
i
N
.2
.1
0
2
= 2.236
18
20
22
24
A
B
C
D
x
Uniform Distribution
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Chap 7-4
© 2006 Prentice Hall, Inc.
Chapter 7
7-3
Developing a
Sampling Distribution
(continued)
Now consider all possible samples of size n=2
1st
Obs
16 Sample
Means
2nd Observation
18
20
22
24
18
18,18
18,20
18,22
18,24
20
20,18
20,20
20,22
20,24
1st 2nd Observation
Obs 18 20 22 24
22
22,18
22,20
22,22
22,24
18 18 19 20 21
24
24,18
24,20
24,22
24,24
20 19 20 21 22
22 20 21 22 23
16 possible samples
(sampling with
replacement)
24 21 22 23 24
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-5
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
_
P(X)
.3
.2
.1
0
18 19
20 21 22 23
24
_
X
(no longer uniform)
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Chap 7-6
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Chapter 7
7-4
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-7
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Chap 7-8
Basic Business Statistics, 10/e
© 2006 Prentice Hall, Inc.
Chapter 7
7-5
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
µX =
σX =
=
∑X
i
=
N
18 + 19 + 21 + L + 24
= 21
16
∑(X − µ
i
X
)2
N
(18 - 21)2 + (19 - 21)2 + L + (24 - 21)2
= 1.58
16
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-9
Comparing the Population with its
Sampling Distribution
Population
N=4
µ = 21
Sample Means Distribution
n=2
µ X = 21
σ = 2.236
_
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
18
20
22
24
A
B
C
D
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
σ X = 1.58
X
0
18 19
20 21 22 23
24
_
X
Chap 7-10
© 2006 Prentice Hall, Inc.
Chapter 7
7-6
Sampling Distribution
of the Mean or Proportion
Sampling
Distributions
Sampling
Distribution of
the Mean
Sampling
Distribution of
the Proportion
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-11
Sample Mean Sampling Distribution:
Standard Error of the Mean
Different samples of the same size from the same
population will yield different sample means
A measure of the variability in the mean from sample to
sample is given by the Standard Error of the Mean:
(This assumes that sampling is with replacement or
sampling is without replacement from an infinite population)
σX =
σ
n
Note that the standard error of the mean decreases as
the sample size increases
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
Chap 7-12
© 2006 Prentice Hall, Inc.
Chapter 7
7-7
Sample Mean Sampling Distribution:
If the Population is Normal
If a population is normal with mean µ and
standard deviation σ, the sampling distribution
of X is also normally distributed with
µX = µ
and
σX =
σ
n
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-13
Z-value for Sampling Distribution
of the Mean
Z-value for the sampling distribution of X :
Z=
where:
(X − µX )
σX
=
( X − µ)
σ
n
X = sample mean
µ = population mean
σ = population standard deviation
n = sample size
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Chap 7-14
© 2006 Prentice Hall, Inc.
Chapter 7
7-8
Sampling Distribution Properties
µx = µ
(i.e.
x is unbiased )
Normal Population
Distribution
µ
x
µx
x
Normal Sampling
Distribution
(has the same mean)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-15
Sampling Distribution Properties
(continued)
As n increases,
Larger
sample size
σ x decreases
Smaller
sample size
µ
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x
Chap 7-16
© 2006 Prentice Hall, Inc.
Chapter 7
7-9
Determining An Interval Including A
Fixed Proportion of the Sample Means
Find a symmetrically distributed interval around µ
that will include 95% of the sample means when µ
= 368, σ = 15, and n = 25.
Since the interval contains 95% of the sample means
5% of the sample means will be outside the interval
Since the interval is symmetric 2.5% will be above
the upper limit and 2.5% will be below the lower limit.
From the standardized normal table, the Z score with
2.5% (0.0250) below it is -1.96 and the Z score with
2.5% (0.0250) above it is 1.96.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-17
Determining An Interval Including A
Fixed Proportion of the Sample Means
(continued)
Calculating the lower limit of the interval
XL = µ+Z
σ
15
= 368 + ( −1.96)
= 362.12
n
25
Calculating the upper limit of the interval
σ
15
= 368 + (1.96)
= 373.88
n
25
95% of all sample means of sample size 25 are
between 362.12 and 373.88
XU = µ + Z
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
Chap 7-18
© 2006 Prentice Hall, Inc.
Chapter 7
7-10
If the Population is not Normal
We can apply the Central Limit Theorem:
Even if the population is not normal,
…sample means from the population will be
approximately normal as long as the sample size is
large enough.
Properties of the sampling distribution:
µx = µ
and
σx =
σ
n
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-19
Central Limit Theorem
As the
sample
size gets
large
enough…
n↑
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
x
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Chap 7-20
© 2006 Prentice Hall, Inc.
Chapter 7
7-11
Central Limit Theorem
Three major elements:
Population distribution
Sample mean (x-bar) is normal distributed
Sample size (n) is large enough
How large of (n) is large enough?
n > 30 for _____________population
n > 15 for ______________population
n is any for _____________population
Practical implications:
E(x-bar) = µ(x-bar) = µ and σ(x-bar) = σ/√n
E(p-bar) = p0 and σ(p-bar) = sqrt[p*(1-p)/sqrt(n)]
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-21
If the Population is not Normal
(continued)
Sampling distribution
properties:
Population Distribution
Central Tendency
µx = µ
Variation
σx =
σ
n
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
µ
x
Sampling Distribution
(becomes normal as n increases)
Larger
sample
size
Smaller
sample size
µx
x
Chap 7-22
© 2006 Prentice Hall, Inc.
Chapter 7
7-12
How Large is Large Enough?
For most distributions, n > 30 will give a
sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15
For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-23
Example
Suppose a population has mean µ = 8 and
standard deviation σ = 3. Suppose a random
sample of size n = 36 is selected.
What is the probability that the sample mean is
between 7.8 and 8.2?
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Chap 7-24
© 2006 Prentice Hall, Inc.
Chapter 7
7-13
Example
(continued)
Solution:
Even if the population is not normally
distributed, the central limit theorem can be
used (n > 30)
… so the sampling distribution of
approximately normal
… with mean µx = 8
…and standard deviation σ x =
x
is
=
3
σ
n
36
= 0.5
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-25
Example
(continued)
Solution (continued):


 7.8 - 8
X -µ
8.2 - 8 
P(7.8 < X < 8.2) = P
<
<

3
σ
3


36
n
36 

= P(-0.4 < Z < 0.4) = 0.3108
Population
Distribution
???
?
??
?
?
?
??
µ=8
Sampling
Distribution
?
Sample
X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Standard Normal
Distribution
.1554
+.1554
Standardize
7.8
µX = 8
8.2
x
-0.4
µz = 0
0.4
Z
Chap 7-26
© 2006 Prentice Hall, Inc.
Chapter 7
7-14
Sampling Distribution
of the Proportion
Sampling
Distributions
Sampling
Distribution of
the Mean
Sampling
Distribution of
the Proportion
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 7-27
Population Proportions
π = the proportion of the population having
some characteristic
Sample proportion ( p ) provides an estimate
of π:
p=
X
number of items in the sample having the characteristic of interest
=
n
sample size
0≤ p≤1
p is approximately distributed as a normal distribution
when n is large
(assuming sampling with replacement from a finite population or
without replacement from an infinite population)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
Chap 7-28
© 2006 Prentice Hall, Inc.
Chapter 7
7-15
Sampling Distribution of p
Approximated by a
normal distribution if:
P(ps)
Sampling Distribution
.3
.2
.1
0
nπ ≥ 5
and
0
.2
.4
.6
8
1
p
n(1 − π ) ≥ 5
where
µp = π
and
σp =
π(1− π )
n
(where π = population proportion)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-29
Z-Value for Proportions
Standardize p to a Z value with the formula:
Z=
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
p −π
=
σp
p −π
π (1− π )
n
Chap 7-30
© 2006 Prentice Hall, Inc.
Chapter 7
7-16
Example
If the true proportion of voters who support
Proposition A is π = 0.4, what is the probability
that a sample of size 200 yields a sample
proportion between 0.40 and 0.45?
i.e.: if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-31
Example
(continued)
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Find σ p : σ p =
π (1− π )
n
=
0.4(1 − 0.4)
= 0.03464
200
0.45 − 0.40 
 0.40 − 0.40
Convert to
P(0.40 ≤ p ≤ 0.45) = P
≤Z≤

standardized
0.03464 
 0.03464
normal:
= P(0 ≤ Z ≤ 1.44)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Basic Business Statistics, 10/e
Chap 7-32
© 2006 Prentice Hall, Inc.
Chapter 7
7-17
Example
Central Limit Thoerem (CLT) online
(continued)
if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Use standardized normal table:
P(0 ≤ Z ≤ 1.44) = 0.4251
Standardized
Normal Distribution
Sampling Distribution
0.4251
Standardize
0.40
0.45
p
0
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
1.44
Z
Chap 7-33
Chapter Summary
Introduced sampling distributions
Described the sampling distribution of the mean
For normal populations
Using the Central Limit Theorem
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Basic Business Statistics, 10/e
Chap 7-34
© 2006 Prentice Hall, Inc.