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BA 201
Lecture 13
Sample Size Determination and
Ethical Issues
Topics
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Confidence Interval Estimation for the
Mean (  Unknown)
Confidence Interval Estimation for the
Proportion
Determining Sample Size
Confidence Interval Estimation and
Ethical Issues
Confidence Interval for 
(  Unknown)
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Assumptions
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Population standard deviation is unknown
If population is normal, small sample can be used
If population is not normal, use large sample
Use Student’s t Distribution
Confidence Interval Estimate
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X  t / 2,n 1
S
S
   X  t / 2,n 1
n
n
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5)
0
Z
t
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
0 2.920
t
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
Confidence Interval for 
(  Unknown) in PHStat
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PHStat | Confidence Interval | Estimate
for the Mean, sigma unknown
Example in Excel Spreadsheet
Confidence Interval Estimate
for Proportion
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Assumptions
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Two categorical outcomes
Population follows Binomial distribution
Normal approximation can be used if
np  5 and n 1  p   5
Confidence Interval Estimate
pS  Z / 2
pS 1  pS 
 p  pS  Z / 2
n
pS 1  pS 
n
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 
 p  ps  Z / 
n
ps 1  ps 
n
.08 1  .08 
.08 1  .08 
.08  1.96
 p  .08  1.96
400
400
.053  p  .107
Confidence Interval Estimate
for Proportion in PHStat
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PHStat | Confidence Interval | Estimate
for the Proportion …
Example in Excel Spreadsheet
Elements of Confidence
Interval Estimation
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Level of Confidence 1   100%
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Precision (Sampling Error)
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Confidence in which the interval will
contain the unknown population parameter
Closeness to the unknown parameter
Cost (Sample Size)
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Cost required to obtain a sample of size n
Level of Confidence
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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 the
unknown parameter
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A Specific Interval Will Either Contain or Not
Contain the Parameter
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No probability involved in a specific interval
Factors Affecting Interval
Width (Precision)
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Data Variation
 Measured by 
Sample Size
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X
Intervals Extend from
X  Z / 2 X to X  Z / 2 X
Sampling Error = Z / 2
n
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Level of Confidence
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100 1    %
© 1984-1994 T/Maker Co.
X
Determining Sample Size
(Cost)
Too Big:
Too small:
• Requires
too many
resources
• Won’t do
the job
Determining Sample Size for
Mean
pp.284-286
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
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
2
Round Up
Obtaining Sample Size in
PHStat
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PHStat | Sample Size |
Determination for the mean…
PHStat | Sample Size |
Determination for the proportion…
Here’s an Excel spreadsheet that
contains the result of the previous
examples:
pp.290-291
Ethical Issues
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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
Some Examples of Good and
Bad Reporting
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Here is a link
reporting
Here is a link
Here is a link
practices
Here is a link
to a really terrible
to a really bad practice
to some not so bad
to a better practice
Summary
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Discussed Confidence Interval Estimation
for the Mean (  Unknown)
Addressed Determining Sample Size
Discussed Confidence Interval Estimation
and Ethical Issues