Applied Statistics and Probability for Engineers

8-40. Consider the hole diameter data in Exercise 8-35.

Construct a 99% two-sided confidence interval for .

8-41. Consider the sugar content data in Exercise 8-37. Find

a 90% lower confidence bound for .

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

8-39. The percentage of titanium in an alloy used in aero-

space castings is measured in 51 randomly selected parts. The

sample standard deviation is s0.37. Construct a 95% two-

sided confidence interval for .

To construct the confidence interval on p, note that

so

This may be rearranged as

(8-23)

The quantity in Equation 8-23 is called the standard error of the point esti-

matorPˆ.Unfortunately, the upper and lower limits of the confidence interval obtained from

1 p 11 p 2 n

Pq Pˆz    2

B

p 11 p 2

n pP

ˆz

(^2) B
p 11 p 2
n
r 1 
P °z 2 
Pˆp
B
p 11 p 2
n
z 2 ¢ 1 
P 1 z 2 Zz 22  1 
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 nhas been taken from a large (possibly infinite)
population and that X(n) observations in this sample belong to a class of interest. Then
is a point estimator of the proportion of the population pthat belongs to this class.
Note that nand pare the parameters of a binomial distribution. Furthermore, from Chapter 4
we know that the sampling distribution of is approximately normal with mean pand vari-
ance if pis not too close to either 0 or 1 and if nis relatively large. Typically, to
apply this approximation we require that npand n(1p) be greater than or equal to 5. We
will make use of the normal approximation in this section.
p 11 p 2 n,
Pˆ
PˆX n
If nis large, the distribution of
is approximately standard normal.
Z
Xnp
1 np 11 p 2

Pˆp
B
p 11 p 2
n
Definition
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