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Basic Business Statistics
(8th Edition)
Chapter 7
Sampling Distributions
© 2002 Prentice-Hall, Inc.
Chap 7-1
Chapter Topics

Sampling distribution of the mean

Sampling distribution of the proportion

Sampling from finite population
© 2002 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 
Problems: Different samples provide different
estimates



Large samples give better estimates; large sample
costs more
How good is the estimate?
Approach to solution: Theoretical basis is
sampling distribution
© 2002 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
© 2002 Prentice-Hall, Inc.
Chap 7-4
Developing Sampling
Distributions

Assume 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
© 2002 Prentice-Hall, Inc.
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
© 2002 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
© 2002 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
© 2002 Prentice-Hall, Inc.
P(X)
.3
.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 
© 2002 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
Sample Means Distribution
n=2
Population
N=4
  21
  2.236
 X  21
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
© 2002 Prentice-Hall, Inc.
D X
 X  1.58
_
18 19
20 21 22 23
24
X
(24)
Chap 7-10
Properties of Summary Measures

X  



e.g.: 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:

© 2002 Prentice-Hall, Inc.
As n increases,
X
decreases
X 

n
Chap 7-11
Unbiasedness
P(X)
Unbiased

© 2002 Prentice-Hall, Inc.
Biased
X
X
Chap 7-12
Less Variability
P(X)
Sampling
Distribution
of Median
Sampling
Distribution of
Mean

© 2002 Prentice-Hall, Inc.
X
Chap 7-13
Effect of Large Sample
Larger
sample size
P(X)
Smaller
sample size

© 2002 Prentice-Hall, Inc.
X
Chap 7-14
When the Population is Normal
Population Distribution
Central Tendency
X  
Variation
X 

n
Sampling with
Replacement
© 2002 Prentice-Hall, Inc.
  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 

n
Sampling with
Replacement
© 2002 Prentice-Hall, Inc.
  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
© 2002 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
© 2002 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
© 2002 Prentice-Hall, Inc.
8.2
X  8
X
0.5
Z  0
0.5
Z
Chap 7-19
Population Proportions

Categorical variable



 p
e.g.: Gender, voted for bush, college degree
Proportion of population that has a
characteristic p
 
Sample proportion provides an estimate
X number of successes
pS  
n
sample size

If two outcomes, X has a binomial distribution

Possess or do not possess characteristic
© 2002 Prentice-Hall, Inc.
Chap 7-20
Sampling Distribution
of Sample Proportion

Approximated by
normal distribution


np  5
n 1  p   5
P(ps)
.3
.2
.1
0
Mean:
p  p

Sampling Distribution
0
.2
.4
.6
8
1
ps
S

Standard error:

p 
S
© 2002 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
© 2002 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
© 2002 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


P 
S
© 2002 Prentice-Hall, Inc.
p 1  p  N  n
n
N 1
Chap 7-24
Chapter Summary



Discussed sampling distribution of the
sample mean
Described sampling distribution of the
sample proportion
Discussed sampling from finite
populations
© 2002 Prentice-Hall, Inc.
Chap 7-25