Module 13: Sample Size Determination and Intro to Hypothesis Testing
This module is divided into 2 distinct parts:
I.
Sample Size Determination when estimating:
A.
The Population Mean (
)
B. The Population Proportion (P)
Note: When ever calculating samples sizes, ALWAYS round up to the
next whole
number. You can't, for example, sample .25 of a person.
II. Introduction to Hypothesis testing
The first part is fairly straightforward and just a matter of applying a formula based on desired criteria. Hypothesis testing is
more involved and involves strict adherence to an eight-step process.

Module 13: Sample Size Determination and Intro to Hypothesis Testing
I.
Sample Size Determination
A.
The Population Mean (
)
When estimating
, the sample size is determined using the following formula:
n =
Where:
Based on the level of confidence exactly as in Modules 11 and 12 (see next slide to refresh your memory)
=
The standard deviation.
If unknown,
= 1/4 the range.
E =
The amount of error we're willing to live with. The smaller the error, the larger the sample
size required.
As a statistician, however, the larger the sample size, the more costly the
study. Therefore it's a tradeoff.
We know larger sample sizes increase accuracy and reduce
error, but is the reduction in error worth the
increase in cost? This, of course, is a
subjective question.
•

Module 13: Sample Size Determination and Intro to Hypothesis Testing
I.
Sample Size Determination
Factors for various confidence levels:
•
Confidence
α
α/2
Z
α/2
Z
α/2
0.99
0.01
0.005
Z
.005
2.5758
=NORMSINV(0.005)*-1
0.98
0.02
0.01
Z
.01
2.3263
=NORMSINV(0.01)*-1
0.96
0.04
0.02
Z
.02
2.0537
=NORMSINV(0.02)*-1
0.95
0.05
0.025
Z
.025
1.9600
=NORMSINV(0.025)*-1
0.90
0.10
0.05
Z
.05
1.6449
=NORMSINV(0.05)*-1
0.85
0.15
0.075
Z
.075
1.4395
=NORMSINV(0.075)*-1
0.80
0.20
0.10
Z
.10
1.2816
=NORMSINV(0.1)*-1

Module 13: Sample Size Determination and Intro to Hypothesis Testing
I.
Sample Size Determination
Example 1
: A researcher wants to estimate the average monthly expenditure om bread by a family in Chicago. She wants
to be 90% confident of her results and wants to be within $1.00 of the actual figure. The standard deviation of monthly
bread purchases in Chicago is known to be $4.00.
How large a sample size should she take?
From the table on slide 3, we know that
= 1.6449since she desires a 90% level of confidence.
From the text of the question we know
= 4
From the text of the question, we also know E = 1.
E = 1 since she wants to be within $1 of the actual number.
That is, in
effect, the amount of error she's willing to live with.
Plugging this information into our formula yields:
n == = = 43.2911.
Therefore n =
44*
* Again, with sample size, we always round up to the next whole number.
•

Module 13: Sample Size Determination and Intro to Hypothesis Testing
I.
Sample Size Determination
Example 2:
Suppose you want to estimate the average age of all Boeing 727s currently in active domestic service in the
U.S.
You want to be 95% confident of your results and want the estimate to be within 2 years of the actual age. The first
Boeing 727 was placed in active service 40 years ago, but none of the planes in the U.S. domestic fleet are older than 25
years. How large a sample size should you take?

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- Spring '14
- DebraACasto
- Null hypothesis, Statistical hypothesis testing