Confidence Intervals and Sample Size

lu49076_ch07.qxd 5/20/2003 3:16 PM Page 348
348 Chapter 7 Confidence Intervals and Sample Size
Confidence Intervals
Example 7–9
Example 7–10
7–24
estimator, pˆ is unbiased, consistent, and relatively efficient. But as with means, one is
not able to decide how good the point estimate of p is. Therefore, statisticians also use
an interval estimate for a proportion, and they can assign a probability that the interval
will contain the population proportion.
To construct a confidence interval about a proportion, one must use the maximum error
of estimate, which is
E � z ��2� pˆqˆ
n
Confidence intervals about proportions must meet the criteria that np � 5 and nq � 5.
Formula for a Specific Confidence Interval for a Proportion
pˆ � z��2� pˆqˆ
n � p � pˆ � z��2� pˆqˆ
n
when np and nq are each greater than or equal to 5.
Rounding Rule for a Confidence Interval for a Proportion Round off to three decimal places.
A sample of 500 nursing applications included 60 from men. Find the 90% confidence
interval of the true proportion of men who applied to the nursing program.
Solution
Since a � 1 � 0.90 � 0.10 and za/2 � 1.65, substituting in the formula
pˆ � z��2� pˆqˆ
n � p � pˆ � z��2� pˆqˆ
n
when pˆ � 60/500 � 0.12 and qˆ � 1 � 0.12 � 0.88, one gets
0.12 � 0.024 � p � 0.12 � 0.024
0.096 � p � 0.144
or 9.6% � p � 14.4%
Hence, one can be 90% confident that the percentage of applicants who are men is
between 9.6% and 14.4%.
When a specific percentage is given, the percentage becomes pˆ when it is changed
to a decimal. For example, if the problem states that 12% of the applicants were men,
then pˆ � 0.12.
A survey of 200,000 boat owners found that 12% of the pleasure boats were named
Serenity. Find the 95% confidence interval of the true proportion of boats named Serenity.
Source: USA TODAY Snapshot.
0.12 � 1.65 � �0.12��0.88�
500
� p � 0.12 � 1.65 � �0.12��0.88�
500