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```Basic Business Statistics
(10th Edition)
Chapter 7
Sampling Distributions
Chap 7-1
Chapter Topics

Sampling Distribution of the Mean

The Central Limit Theorem

Sampling Distribution of the Proportion

Sampling from Finite Population
Chap 7-2
Why Study Sampling
Distributions

Sample Statistics are Used to Estimate
Population Parameters


Problem: Different Samples Provide Different
Estimates



E.g., X  50 estimates the population mean 
Large sample gives better estimate; large sample
costs more
How good is the estimate?
Approach to Solution: Theoretical Basis is
Sampling Distribution
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
Chap 7-4
Developing Sampling
Distributions

Suppose There is a Population …

Population Size N=4


B
C
Random Variable, X,
is Age of Individuals
Measured in Years
Values of X : 18, 20,
22, 24
A
D
Chap 7-5
Developing Sampling
Distributions
(continued)
Summary Measures for the Population Distribution
N
 

P(X)
Xi
i 1
.3
N

18  20  22  24
.2
 21
4
N
 
X
i
i 1
N


.1
0
2
 2 .2 3 6
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Chap 7-6
Developing Sampling
Distributions
All Possible Samples of Size n=2
st
1
O bs
2
18
nd
O b s e rva tio n
20
22
24
1 8 1 8 ,1 8 1 8 ,2 0 1 8 ,2 2 1 8 ,2 4
(continued)
16 Sample Means
2 0 2 0 ,1 8 2 0 ,2 0 2 0 ,2 2 2 0 ,2 4
1st
2 n d O b s e rva tio n
2 2 2 2 ,1 8 2 2 ,2 0 2 2 ,2 2 2 2 ,2 4
O bs
18
20
22
24
18
18
19
20
21
20
19
20
21
22
22
20
21
22
23
24
21
22
23
24
2 4 2 4 ,1 8 2 4 ,2 0 2 4 ,2 2 2 4 ,2 4
Nn = 42 = 16
Samples Taken with
Replacement
Chap 7-7
Developing Sampling
Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st
2 n d O b s e rva tio n
O bs 1 8
18
20
22
24
18
19
20
21
20
19
20
21
22
22
20
21
22
23
24
21
22
23
24
.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 
n
X
i 1
N
N
X 

n
18  19  19 
i

 24
 21
16
n
X
i
 X

2
i 1
N
n
 18  21 
2
  19  21  
2
  24  21 
2
 1.58
16
Chap 7-9
Comparing the Population with
Its Sampling Distribution
Population
N=4
  21
  2.236
.3
PX
Sample Means Distribution
n=2
 X  21  X  1.58

.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
D X
PX

_
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, 
X
decreases
Chap 7-11
Unbiasedness (  X   )
f
X 
Unbiased

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

X
Chap 7-13
Effect of Large Sample
For sam pling w ith replacem ent:
A s n increases, 
f
X 
X
decreases
Larger
sample size
Smaller
sample size

X
Chap 7-14
When the Population is Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 

n
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
  10
X  
  50
Variation
X 

n
Sampling Distributions
n4
n  30
X 5

 X  50
X
 1.8
X
Chap 7-16
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
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
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
.1 9 1 5
7 .8
8 .2
X  8
X
 0 .5
0 .5
Z  0
Z
Chap 7-19
Population Proportion

Categorical Variable



E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a
Characteristic  p 
Sample Proportion Provides an Estimate


 p
pS 
X

num ber of successes
n
sam ple size
If Two Outcomes, X Has a Binomial
Distribution

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:

Sampling Distribution
p  p
0
.2
.4
.6
8
1
ps
S

Standard error:


pS

p 1  p 
n
p = population proportion
Chap 7-21
Standardizing Sampling
Distribution of Proportion
Z 
p S   pS

pS

pS  p
p 1  p 
n
Standardized
Normal Distribution
Sampling Distribution

Z 1
pS
p
S
pS
Z  0
Z
Chap 7-22
Example:
n  200
p  .4


p S   pS

P  p S  .43   P

 p
S




.43  .4 
 P  Z  .87   .8078
.4  1  .4  

200

Standardized
Normal Distribution
Sampling Distribution

P  p S  .43   ?
Z 1
pS
 p .4 3
S
pS
0 .8 7
Z
Chap 7-23
Sampling from Finite Population
(CD ROM Topic)

Modify Standard Error if Sample Size (n) is
Large Relative to Population Size (N )



n  .0 5 N o r n / N  .0 5
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
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
Chap 7-25
```
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