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8-5 A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION 267

z 2 2

n a b p11 p2 (8-26)
˛
E

p ˆ
An estimate of p is required to use Equation 8-26. If an estimate from a previous sam-
ple is available, it can be substituted for p in Equation 8-26, or perhaps a subjective estimate
can be made. If these alternatives are unsatisfactory, a preliminary sample can be taken, p ˆ
computed, and then Equation 8-26 used to determine how many additional observations are
required to estimate p with the desired accuracy. Another approach to choosing n uses the fact
that the sample size from Equation 8-26 will always be a maximum for p 0.5 [that is,
p(1 p) 0.25 with equality for p 0.5], and this can be used to obtain an upper bound on
n. In other words, we are at least 100(1 )% confident that the error in estimating p by p ˆ
is less than E if the sample size is

2
z 2
n a b 10.252 (8-27)
E

EXAMPLE 8-7 Consider the situation in Example 8-6. How large a sample is required if we want to be 95%
p ˆ
confident that the error in using to estimate p is less than 0.05? Using 0.12 as an initial
p ˆ
estimate of p, we find from Equation 8-26 that the required sample size is
z 0.025 2 1.96 2
n a b p ˆ 11 p ˆ2 a b 0.1210.882 163
E 0.05

If we wanted to be at least 95% confident that our estimate p ˆ of the true proportion p was
within 0.05 regardless of the value of p, we would use Equation 8-27 to find the sample size

z 0.025 2 1.96 2
n a b 10.252 a b 10.252 385
E 0.05

Notice that if we have information concerning the value of p, either from a preliminary sam-
ple or from past experience, we could use a smaller sample while maintaining both the desired
precision of estimation and the level of confidence.
One-Sided Confidence Bounds
We may find approximate one-sided confidence bounds on p by a simple modification of
Equation 8-25.

The approximate 100(1 )% lower and upper confidence bounds are
p ˆ 11 p ˆ2 p ˆ 11 p ˆ2
p ˆ z B n p and p p ˆ z B n (8-28)

respectively.