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Developing a
Sampling Distribution

Assume there is a population …

Population size N=4

Random variable, X,
is age of individuals

Values of X: 18, 20,
22, 24 (years)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
A
B
C
D
Chap 7-1
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, 11e © 2009 Prentice-Hall, Inc..
Chap 7-2
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)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-3
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
μX
X

i 18  19  19    24


 21
σX 

N
16
2
(
X

μ
)
i

X
N
(18 - 21)2  (19 - 21)2    (24 - 21)2
 1.58
16
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-4
Comparing the Population Distribution
to the Sample Means Distribution
Population
N=4
μ  21
σ  2.236
Sample Means Distribution
n=2
μX  21
σ X  1.58
_
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
18
20
22
24
A
B
C
D
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
X
0
18 19
20 21 22 23
24
_
X
Chap 7-5
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)
σ
σX 
n

Note that the standard error of the mean decreases as
the sample size increases
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-6
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
and
σ
σX 
n
Chap 7-7
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, 11e © 2009 Prentice-Hall, Inc..
Chap 7-8
Sampling Distribution Properties
μx  μ

(i.e.
x is unbiased )
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Normal Population
Distribution
μ
x
μx
x
Normal Sampling
Distribution
(has the same mean)
Chap 7-9
Sampling Distribution Properties
(continued)
As n increases,
Larger
sample size
σ x decreases
Smaller
sample size
μ
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
x
Chap 7-10
Sample Mean Sampling Distribution:
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
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
and
σ
σx 
n
Chap 7-11
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, 11e © 2009 Prentice-Hall, Inc..
Chap 7-12
Sample Mean Sampling Distribution:
If the Population is not Normal
(continued)
Population Distribution
Sampling distribution
properties:
Central Tendency
μx  μ
σ
σx 
n
Variation
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
μ
x
Sampling Distribution
(becomes normal as n increases)
Larger
sample
size
Smaller
sample size
μx
x
Chap 7-13
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, 11e © 2009 Prentice-Hall, Inc..
Chap 7-14
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, 11e © 2009 Prentice-Hall, Inc..
Chap 7-15
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
x
is

… with mean μx = 8

σ
3
…and standard deviation σ x  n  36  0.5
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Chap 7-16
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
Standard Normal
Distribution
Sample
.1554
+.1554
Standardize
?
X
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7.8
μX  8
8.2
x
-0.4
μz  0
0.4
Z
Chap 7-17