Confidence Intervals and Sample Size

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Sample Size for
Proportions
Objective 5. Determine the
minimum sample size for
finding a confidence interval
for a proportion.
Example 7–11
Solution
From the Snapshot, pˆ � 0.12 (i.e., 12%), and n � 200,000. Since za/2 � 1.96, substituting
in the formula
�0.12��0.88�
yields 0.12 � 1.96� 200,000 � p � 0.12 � 1.96� �0.12��0.88�
200,000
0.119 � p � 0.121
Hence, one can say with 95% confidence that the true percentage of boats named Serenity
is between 11.9% and 12.1%.
To find the sample size needed to determine a confidence interval about a proportion,
use this formula:
Formula for Minimum Sample Size Needed for Interval Estimate of a Population
Proportion
n � pˆqˆ � z ��2
E � 2
If necessary, round up to obtain a whole number.
This formula can be found by solving the maximum error of estimate value for n:
E � z ��2� pˆqˆ
n
Section 7–4 Confidence Intervals and Sample Size for Proportions 349
pˆ � z��2� pˆqˆ
n � p � pˆ � z��2� pˆqˆ
n
There are two situations to consider. First, if some approximation of pˆ is known
(e.g., from a previous study), that value can be used in the formula.
Second, if no approximation of pˆ is known, one should use pˆ � 0.5. This value will
give a sample size sufficiently large to guarantee an accurate prediction, given the confidence
interval and the error of estimate. The reason is that when pˆ and qˆ are each 0.5,
the product pˆ qˆ is at maximum, as shown here.
pˆ
qˆ
pˆ qˆ
0.1 0.9 0.09
0.2 0.8 0.16
0.3 0.7 0.21
0.4 0.6 0.24
0.5 0.5 0.25
0.6 0.4 0.24
0.7 0.3 0.21
0.8 0.2 0.16
0.9 0.1 0.09
A researcher wishes to estimate, with 95% confidence, the proportion of people who
own a home computer. A previous study shows that 40% of those interviewed had a
7–25