Business Statistics:
A First Course
(3rd Edition)
Chapter 6
Sampling Distributions
and
Confidence Interval Estimation
© 2003 Prentice-Hall, Inc.
Chap 6-1
Chapter Topics

Sampling Distribution of the Mean

Estimation Process




Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean
(  Known)
Determining Sample Size
© 2003 Prentice-Hall, Inc.
Chap 6-2
Chapter Topics



(continued)
Confidence Interval Estimation for the Mean
(  Unknown)
Confidence Interval Estimation for the
Proportion
Confidence Interval Estimation and Ethical
Issues
© 2003 Prentice-Hall, Inc.
Chap 6-3
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 6-4
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 6-5
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
© 2003 Prentice-Hall, Inc.
D
Chap 6-6
Developing Sampling
Distributions
(continued)
Summary Measures for the Population Distribution
N

X
i 1
P(X)
i
N
18  20  22  24

 21
4
N
 
 X
i 1
© 2003 Prentice-Hall, Inc.
i

N
.3
.2
.1
0
2
 2.236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Chap 6-7
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 6-8
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 6-9
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 6-10
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
D
(18)
(20)
(22)
(24)
© 2003 Prentice-Hall, Inc.
X
P X 
18 19
 X  1.58
20 21 22 23
24
X
Chap 6-11
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 6-12
Unbiasedness
f X 
Unbiased

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Biased
X
X
Chap 6-13
Less Variability
f X 
Sampling
Distribution
of Median
Sampling
Distribution of
Mean

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X
Chap 6-14
Effect of Large Sample
f X 
Larger
sample size
Smaller
sample size

© 2003 Prentice-Hall, Inc.
X
Chap 6-15
When the Population is Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 

n
Sampling Distributions
n4
n  16
X 5
© 2003 Prentice-Hall, Inc.
 X  2.5
 X  50
X
Chap 6-16
When the Population is Not
Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 

n
Sampling Distributions
n4
n  30
X 5
© 2003 Prentice-Hall, Inc.
 X  1.8
 X  50
X
Chap 6-17
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 6-18
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
© 2003 Prentice-Hall, Inc.
Chap 6-19
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 6-20
Estimation Process
Population
Mean, , is
unknown
Random Sample
Mean
X = 50
I am 95%
confident that 
is between 40 &
60.
Sample
© 2003 Prentice-Hall, Inc.
Chap 6-21
Point Estimates
Estimate Population
Parameters …
Mean
Proportion
Variance
Difference
© 2003 Prentice-Hall, Inc.

p
with Sample
Statistics
X
PS

1   2
2
S
2
X1  X 2
Chap 6-22
Interval Estimates

Provides Range of Values




Take into consideration variation in sample
statistics from sample to sample
Based on observation from 1 sample
Give Information about Closeness to Unknown
Population Parameters
Stated in terms of level of confidence

© 2003 Prentice-Hall, Inc.
Never 100% sure
Chap 6-23
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
© 2003 Prentice-Hall, Inc.
Proportion
 Unknown
Chap 6-24
Confidence Interval for 
(  Known)

Assumptions




Population standard deviation is known
Population is normally distributed
If population is not normal, use large sample
Confidence Interval Estimate

© 2003 Prentice-Hall, Inc.
X  Z / 2

n
   X  Z / 2

n
Chap 6-25
Elements of Confidence Interval
Estimation

Level of Confidence


Precision (Range)


Confidence that the interval will contain the
unknown population parameter
Closeness to the unknown parameter
Cost

Cost required to obtain a sample of size n
© 2003 Prentice-Hall, Inc.
Chap 6-26
Level of Confidence


Denoted by 100 1   %
A Relative Frequency Interpretation
 In the long run, 100 1    % of all the confidence
intervals that can be constructed will contain
(bracket) the unknown parameter

A Specific Interval Will Either Contain or Not
Contain the Parameter

No probability involved in a specific interval
© 2003 Prentice-Hall, Inc.
Chap 6-27
Interval and Level of Confidence
Sampling Distribution of the
_ Mean
  Z / 2 X
Intervals
extend from
 /2
X
1
X  
X  Z X
X
1   100%
of intervals
constructed
contain  ;
to
X  Z X
© 2003 Prentice-Hall, Inc.
  Z / 2 X
 /2
100 % do
Confidence Intervals
not.
Chap 6-28
Factors Affecting Interval Width
(Precision)

Data Variation


Measured by 
Sample Size


X 
Intervals Extend from
X  Z X to X  Z X

n
Level of Confidence
 100 1   %
 
© 1984-1994 T/Maker Co.
© 2003 Prentice-Hall, Inc.
Chap 6-29
Determining Sample Size (Cost)
Too Big:
Too small:
• Requires
more resources
• Won’t do
the job
© 2003 Prentice-Hall, Inc.
Chap 6-30
Determining Sample Size for
Mean
What sample size is needed to be 90% confident of
being correct within ± 5? A pilot study suggested that
the standard deviation is 45.
1.645  45
Z
n

2
2
Error
5
2
2
2
2
  219.2  220
Round Up
© 2003 Prentice-Hall, Inc.
Chap 6-31
Determining Sample Size for
Mean in PHStat


PHStat | Sample Size | Determination for the
Mean …
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-32
Confidence Interval for 
(  Unknown)

Assumptions





Population standard deviation is unknown
Population is normally distributed
If population is not normal, use large sample
Use Student’s t Distribution
Confidence Interval Estimate

© 2003 Prentice-Hall, Inc.
X  t / 2,n 1
S
S
   X  t / 2,n 1
n
n
Chap 6-33
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5)
0
© 2003 Prentice-Hall, Inc.
Z
t
Chap 6-34
Degrees of Freedom (df )


Number of Observations that are Free to
Vary after Sample Mean has been
Calculated
degrees of freedom
= n -1
Example

Mean of 3 numbers is 2
X1  1 (or any number)
= 3 -1
=2
X 2  2 (or any number)
X 3  3 (cannot vary)
© 2003 Prentice-Hall, Inc.
Chap 6-35
Student’s t Table
Upper Tail Area
df
.25
.10
.05
Let: n = 3
df = n - 1 = 2
 = .10
/2 =.05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
 / 2 = .05
3 0.765 1.638 2.353
t Values
© 2003 Prentice-Hall, Inc.
0 2.920
t
Chap 6-36
Example
A random sample of n  25 has X  50 and S  8.
Set up a 95% confidence interval estimate for 
S
S
X  t / 2,n1
   X  t / 2,n1
n
n
8
8
50  2.0639
   50  2.0639
25
25
46.69    53.30
© 2003 Prentice-Hall, Inc.
Chap 6-37
Confidence Interval for 
(  Unknown) in PHStat


PHStat | Confidence Interval | Estimate for
the Mean, sigma unknown
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-38
Confidence Interval Estimate for
Proportion

Assumptions




Two categorical outcomes
Population follows Binomial distribution
Normal approximation can be used if
np  5 and n 1  p   5
Confidence Interval Estimate

© 2003 Prentice-Hall, Inc.
pS  Z / 2
pS 1  pS 
pS 1  pS 
 p  pS  Z / 2
n
n
Chap 6-39
Example
A random sample of 400 Voters showed 32
preferred Candidate A. Set up a 95% confidence
interval estimate for p.
ps  Z / 
ps 1  ps 
ps 1  ps 
 p  ps  Z / 
n
n
.08 1  .08 
.08 1  .08 
.08  1.96
 p  .08  1.96
400
400
.053  p  .107
© 2003 Prentice-Hall, Inc.
Chap 6-40
Confidence Interval Estimate for
Proportion in PHStat


PHStat | Confidence Interval | Estimate for
the Proportion …
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-41
Determining Sample Size for
Proportion
Out of a population of 1,000, we randomly
selected 100 of which 30 were defective. What
sample size is needed to be within ± 5% with
90% confidence?
Z p 1  p  1.645  0.3 0.7 
n

2
2
Error
0.05
 227.3  228
2
© 2003 Prentice-Hall, Inc.
2
Round Up
Chap 6-42
Determining Sample Size for
Proportion in PHStat


PHStat | Sample Size | Determination for the
Proportion …
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-43
Ethical Issues




Confidence Interval (Reflects Sampling Error)
Should Always be Reported Along with the
Point Estimate
The Level of Confidence Should Always be
Reported
The Sample Size Should be Reported
An Interpretation of the Confidence Interval
Estimate Should Also be Provided
© 2003 Prentice-Hall, Inc.
Chap 6-44
Chapter Summary






Discussed Sampling Distribution of the Sample
Mean
Illustrated Estimation Process
Discussed Point Estimates
Addressed Interval Estimates
Discussed Confidence Interval Estimation for
the Mean (  Known)
Addressed Determining Sample Size
© 2003 Prentice-Hall, Inc.
Chap 6-45
Chapter Summary



(continued)
Discussed Confidence Interval Estimation for
the Mean (  Unknown)
Discussed Confidence Interval Estimation for
the Proportion
Addressed Confidence Interval Estimation and
Ethical Issues
© 2003 Prentice-Hall, Inc.
Chap 6-46