Equation 8-23 contain the unknown parameter p. However, as suggested at the end of Section
8-2.5, a satisfactory solution is to replace pby in the standard error, which results in
(8-24)
This leads to the approximate 100(1)% confidence interval on p.
PqPˆz 2 ̨
B
Pˆ 11 Pˆ 2
n pP
ˆz
(^2) B
Pˆ 11 Pˆ 2
n
r 1
Pˆ
266 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE
This procedure depends on the adequacy of the normal approximation to the binomial. To
be reasonably conservative, this requires that npand n(1p) be greater than or equal to 5. In
situations where this approximation is inappropriate, particularly in cases where nis small,
other methods must be used. Tables of the binomial distribution could be used to obtain a con-
fidence interval for p. However, we could also use numerical methods based on the binomial
probability mass function that are implemented in computer programs.
EXAMPLE 8-6 In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish that
is rougher than the specifications allow. Therefore, a point estimate of the proportion of bear-
ings in the population that exceeds the roughness specification is
A 95% two-sided confidence interval for pis computed from Equation 8-25 as
or
which simplifies to
Choice of Sample Size
Since is the point estimator of p, we can define the error in estimating pby as
Note that we are approximately 100(1)% confident that this error is less
than For instance, in Example 8-6, we are 95% confident that the sample
proportion differs from the true proportion pby an amount not exceeding 0.07.
In situations where the sample size can be selected, we may choose nto be 100 (1)%
confident that the error is less than some specified value E. If we set
and solve for n, the appropriate sample size is
Ez 2 ̨ 1 p 11 p 2 n
pˆ0.12
z 2 ̨ 1 p 11 p 2 n.
E 0 pPˆ 0.
Pˆ Pˆ
0.05p0.19
0.121.96
B
0.12 1 0.88 2
85
p0.121.96
B
0.12 1 0.88 2
85
pˆz0.025
B
pˆ 11 pˆ 2
n ppˆz0.025^ ̨B
pˆ 11 pˆ 2
n
pˆx n 10 85 0.12.
If is the proportion of observations in a random sample of size nthat belongs to a
class of interest, an approximate 100(1)% confidence interval on the proportion
pof the population that belongs to this class is
(8-25)
where z 2 is the upper 2 percentage point of the standard normal distribution.
pˆz 2
B
pˆ 11 pˆ 2
n ppˆz^2 ̨B
pˆ 11 pˆ 2
n
pˆ
Definition
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