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8-5 A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION 265

8-39. The percentage of titanium in an alloy used in aero- 8-40. Consider the hole diameter data in Exercise 8-35.
space castings is measured in 51 randomly selected parts. The Construct a 99% two-sided confidence interval for .
sample standard deviation is s 0.37. Construct a 95% two- 8-41. Consider the sugar content data in Exercise 8-37. Find
sided confidence interval for . a 90% lower confidence bound for .

8-5 A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A
POPULATION PROPORTION

It is often necessary to construct confidence intervals on a population proportion. For exam-
ple, suppose that a random sample of size n has been taken from a large (possibly infinite)
population and that X( n) observations in this sample belong to a class of interest. Then
ˆ
P X n is a point estimator of the proportion of the population p that belongs to this class.
Note that n and p are the parameters of a binomial distribution. Furthermore, from Chapter 4
ˆ
we know that the sampling distribution of is approximately normal with mean p and vari-
P
ance p11 p2 n, if p is not too close to either 0 or 1 and if n is relatively large. Typically, to
apply this approximation we require that np and n(1 p) be greater than or equal to 5. We
will make use of the normal approximation in this section.

Definition
If n is large, the distribution of

ˆ
X np P p
Z
1np 11 p2 p 11 p2
B n

is approximately standard normal.

To construct the confidence interval on p, note that

P 1 z 2 Z z 2 2 1

so

ˆ
P p
P ° z 2 z 2 ¢ 1
p11 p2
B n

This may be rearranged as

p11 p2 p11 p2
ˆ
q ˆ
P P z 2 B n p P z 2 B n r 1 (8-23)

The quantity 1p11 p2 n in Equation 8-23 is called the standard error of the point esti-
ˆ
mator P. Unfortunately, the upper and lower limits of the confidence interval obtained from