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IE 440
PROCESS IMPROVEMENT
THROUGH PLANNED EXPERIMENTATION
Sampling Distributions
Dr. Xueping Li
University of Tennessee
© 2003 Prentice-Hall, Inc.
Chap 7-1
Chapter Topics

Sampling Distribution of the Mean

The Central Limit Theorem

Sampling Distribution of the Proportion

Sampling from Finite Population
© 2003 Prentice-Hall, Inc.
Chap 7-2
Why Study Sampling
Distributions


Sample Statistics are Used to Estimate
Population Parameters
 E.g., X  50 estimates the population mean 
Problem: Different Samples Provide Different
Estimates



Large sample gives better estimate; large sample
costs more
How good is the estimate?
Approach to Solution: Theoretical Basis is
Sampling Distribution
© 2003 Prentice-Hall, Inc.
Chap 7-3
Sampling Distribution


Theoretical Probability Distribution of a
Sample Statistic
Sample Statistic is a Random Variable


Sample mean, sample proportion
Results from Taking All Possible Samples of
the Same Size
© 2003 Prentice-Hall, Inc.
Chap 7-4
Developing Sampling
Distributions

Suppose There is a Population …

Population Size N=4


B
C
Random Variable, X,
is Age of Individuals
Values of X: 18, 20,
22, 24 Measured in
Years
A
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D
Chap 7-5
Developing Sampling
Distributions
(continued)
Summary Measures for the Population Distribution
N

X
i 1
P(X)
i
.3
N
18  20  22  24

 21
4
N
 
 X
i 1
© 2003 Prentice-Hall, Inc.
i

N
.2
.1
0
2
 2.236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Chap 7-6
Developing Sampling
Distributions
All Possible Samples of Size n=2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
(continued)
16 Sample Means
22 22,18 22,20 22,22 22,24
1st 2nd Observation
Obs 18 20 22 24
24 24,18 24,20 24,22 24,24
18 18 19 20 21
16 Samples Taken
with Replacement
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
© 2003 Prentice-Hall, Inc.
Chap 7-7
Developing Sampling
Distributions
(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
© 2003 Prentice-Hall, Inc.
.3
P X 
.2
.1
0
_
18 19
20 21 22 23
24
X
Chap 7-8
Developing Sampling
Distributions
(continued)
Summary Measures of Sampling Distribution
N
X 
X
i 1
N
i
18  19  19 

16
N
X 
 X
i 1
i
 X 
© 2003 Prentice-Hall, Inc.
 21
2
N
18  21  19  21
2

 24
16
2

  24  21
2
 1.58
Chap 7-9
Comparing the Population with
Its Sampling Distribution
Population
N=4
  21
P X 
  2.236
Sample Means Distribution
n=2
 X  21
.3
.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
© 2003 Prentice-Hall, Inc.
D X
P X 
 X  1.58
_
18 19
20 21 22 23
24
X
(24)
Chap 7-10
Properties of Summary Measures

X  



I.e., X is unbiased
Standard Error (Standard Deviation) of the
Sampling Distribution  X is Less Than the
Standard Error of Other Unbiased Estimators
For Sampling with Replacement or without
Replacement from Large or Infinite Populations:
X 


n
As n increases,
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X
decreases
Chap 7-11
Unbiasedness (  X   )
f X 
Unbiased

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Biased
X
X
Chap 7-12
Less Variability
Standard Error (Standard Deviation) of the
Sampling Distribution  X is Less Than the
Standard Error of Other Unbiased Estimators
f  X  Sampling
Distribution
of Median
Sampling
Distribution of
Mean

© 2003 Prentice-Hall, Inc.
X
Chap 7-13
Effect of Large Sample
For sampling with replacement:
As n increases,  X decreases
f X 
Larger
sample size
Smaller
sample size
© 2003 Prentice-Hall, Inc.

X
Chap 7-14
When the Population is Normal
Population Distribution
Central Tendency
X  
Variation
X 
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
n
  10
  50
Sampling Distributions
n4
n  16
X 5
 X  2.5
 X  50
X
Chap 7-15
When the Population is
Not Normal
Population Distribution
Central Tendency
X  
Variation
X 
© 2003 Prentice-Hall, Inc.

n
  10
  50
Sampling Distributions
n4
n  30
X 5
 X  1.8
 X  50
X
Chap 7-16
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
© 2003 Prentice-Hall, Inc.
Chap 7-17
How Large is Large Enough?

For Most Distributions, n>30

For Fairly Symmetric Distributions, n>15

For Normal Distribution, the Sampling
Distribution of the Mean is Always Normally
Distributed Regardless of the Sample Size

This is a property of sampling from a normal
population distribution and is NOT a result of the
central limit theorem
© 2003 Prentice-Hall, Inc.
Chap 7-18
Example:   8
 =2
n  25
P  7.8  X  8.2   ?
 7.8  8 X   X 8.2  8 
P  7.8  X  8.2   P 



X
2 / 25 
 2 / 25
 P  .5  Z  .5  .3830
Standardized
Normal Distribution
Sampling Distribution
2
X 
 .4
25
Z 1
.1915
7.8
© 2003 Prentice-Hall, Inc.
8.2
X  8
X
0.5
Z  0
0.5
Z
Chap 7-19
Population Proportions

Categorical Variable



E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a
Characteristic  p 
Sample Proportion Provides an Estimate


 p
X number of successes
pS  
n
sample size
If Two Outcomes, X Has a Binomial
Distribution

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Possess or do not possess characteristic
Chap 7-20
Sampling Distribution of
Sample Proportion

Approximated by
Normal Distribution


np  5
n 1  p   5
f(ps)
.3
.2
.1
0
Mean:
p  p

Sampling Distribution
0
.2
.4
.6
8
1
ps
S

Standard error:

p 
S
© 2003 Prentice-Hall, Inc.
p 1  p 
n
p = population proportion
Chap 7-21
Standardizing Sampling
Distribution of Proportion
Z
pS   pS
p
S
p 1  p 
n
Standardized
Normal Distribution
Sampling Distribution
p

pS  p
Z 1
S
p
© 2003 Prentice-Hall, Inc.
S
pS
Z  0
Z
Chap 7-22
Example:
n  200
p  .4
P  pS  .43  ?

 p 
.43  .4
S
pS

P  pS  .43  P

  pS
.4 1  .4 

200

Standardized
Normal Distribution
Sampling Distribution
p
Z 1
S
© 2003 Prentice-Hall, Inc.


  P  Z  .87   .8078



 p .43
S
pS
0 .87
Z
Chap 7-23
Sampling from Finite Sample


Modify Standard Error if Sample Size (n) is
Large Relative to Population Size (N )

n  .05N or n / N  .05

Use Finite Population Correction Factor (fpc)
Standard Error with FPC

 N n
X 
n N 1

© 2003 Prentice-Hall, Inc.
P 
S
p 1  p  N  n
n
N 1
Chap 7-24
Chapter Summary




Discussed Sampling Distribution of the Sample
Mean
Described the Central Limit Theorem
Discussed Sampling Distribution of the Sample
Proportion
Described Sampling from Finite Populations
© 2003 Prentice-Hall, Inc.
Chap 7-25