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```STATISTICS FOR
MANAGERS
University of Management and Technology
1925 North Lynn Street
Arlington, VA 22209
Voice: (703) 516-0035 Fax: (703) 516-0985
Website: www.umtweb.edu
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1 of 26
Chapter 7, STAT125
CHAPTER 7
Sampling Distributions
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2 of 26
Chapter 7, STAT125
Chapter Topics
Sampling Distribution of the Mean
The Central Limit Theorem
Sampling Distribution of the Proportion
Sampling from Finite Population
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3 of 26
Chapter 7, STAT125
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
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4 of 26
Chapter 7, STAT125
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
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5 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
Suppose There is a Population …
Population Size N=4
Random Variable, X,
is Age of Individuals
Values of X: 18, 20,
22, 24 Measured in
Years
B
C
D
A
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6 of 26
Chapter 7, STAT125
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
i

N
.2
.1
0
2
 2.236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
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7 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
(continued)
All Possible Samples of Size n=2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
16 Sample Means
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
24 21 22
23
24
16 Samples Taken
with Replacement
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8 of 26
Chapter 7, STAT125
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
.3
20 19 20 21 22
.2
22 20 21 22 23
.1
24 21 22 23 24
0
P X 
_
18 19
20 21 22 23
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24
X
9 of 26
Chapter 7, STAT125
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 
 21
2
N
18  21  19  21
2

 24
2

  24  21
16
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2
 1.58
10 of 26
Chapter 7, STAT125
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
A
B
C
D
(18)
(20)
(22)
(24)
X
0
P X 
 X  1.58
_
18 19
20 21 22 23
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24
X
11 of 26
Chapter 7, STAT125
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,
 X decreases
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12 of 26
Chapter 7, STAT125
Unbiasedness (
X   )
f X 
Unbiased

Biased
X
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X
13 of 26
Chapter 7, STAT125
Less Variability
Standard Error (Standard Deviation) of the Sampling
Distribution  is Less Than the Standard Error of Other
X
Unbiased Estimators
f  X  Sampling
Distribution
of Median
Sampling
Distribution of
Mean

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X
14 of 26
Chapter 7, STAT125
Effect of Large Sample
For sampling with replacement:
As n increases,  X decreases
f X 
Larger
sample size
Smaller
sample size

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X
15 of 26
Chapter 7, STAT125
When the Population is Normal
Population Distribution
  10
Central Tendency
X  
  50
Variation
X 

n
Sampling Distributions
n4
X 5
n  16
 X  2.5
Visit UMT online at www.umtweb.edu  X  50
X
16 of 26
Chapter 7, STAT125
When the Population is
Not Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 

n
Sampling Distributions
n4
X 5
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n  30
 X  1.8
 X  50
X
17 of 26
Chapter 7, STAT125
Central Limit Theorem
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
As Sample
Size Gets
Large
Enough
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X
18 of 26
Chapter 7, STAT125
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
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19 of 26
Chapter 7, STAT125
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
Sampling Distribution
2
X 
 .4
25
Standardized
Normal Distribution
Z 1
.1915
0.5
0.5
8.2
X
 X  8 Visit UMT online at www.umtweb.edu Z  0
7.8
Z
20 of 26
Chapter 7, STAT125
Population Proportions
 p
Categorical Variable
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a Characteristic
Sample Proportion Provides an Estimate
 p
X number of successes
pS  
n
sample size
If Two Outcomes, X Has a Binomial Distribution
Possess or do not possess characteristic
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21 of 26
Chapter 7, STAT125
Sampling Distribution of
Sample Proportion
Approximated by
Normal Distribution
np  5
n 1  p   5
Mean:
p  p
S
Standard error:
p 
S
p 1  p 
n
Sampling Distribution
f(ps)
.3
.2
.1
0
0
.2
.4
.6
8
1
ps
p = population proportion
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22 of 26
Chapter 7, STAT125
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
S
pS
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Z  0
Z
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Chapter 7, STAT125
n  200
Example:
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


  P  Z  .87   .8078



Z 1
S
 p .43
S
pS
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0 .87
Z
24 of 26
Chapter 7, STAT125
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
X 
P 
S

n
N n
N 1
p 1  p  N  n
n
N 1
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25 of 26
Chapter 7, STAT125
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