Page 315 - Applied statistics and probability for engineers

Section 8-4/Large-Sample Conidence Interval for a Population Proportion 293

Choice of Sample Size
Because ˆ is the point estimator of p, we can dei ne the error in estimating p by ˆ as
P
P
ˆ
E = | p P . Note that we are approximately 100(1 – α)% coni dent that this error is less
−
|
p − )
1
(
than z /α 2 p / n. For instance, in Example 8-8, we are 95% coni dent that the sample
proportion ˆ p = .0 12 differs from the true proportion p by an amount not exceeding 0.07.
In situations when the sample size can be selected, we may choose n to be 100(1 – α)%
conident that the error is less than some specii ed value E. If we set E = z /α 2 p − p / n and

)
(
1
solve for n, the appropriate sample size is
Sample Size for a

Speciied Error on a 2
Binomial Proportion n = ⎛ ⎜ ⎝ z / ⎞ ⎟ p − p) (8-24)
α 2
(1
E ⎠
An estimate of p is required to use Equation 8-24. If an estimate ˆ p from a previous sam-
ple is available, it can be substituted for p in Equation 8-24, or perhaps a subjective estimate
can be made. If these alternatives are unsatisfactory, a preliminary sample can be taken, ˆ p
computed, and then Equation 8-24 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 8-24 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
.
ˆ
n. In other words, we are at least 100(1 – α)% conident that the error in estimating p by P is

less than E if the sample size is
⎛ z / ⎞ 2

n = ⎜ ⎝ α 2 ⎟ (0 25. ) (8-25)
E ⎠

Example 8-9 Crankshaft Bearings Consider the situation in Example 8-8. How large a sample is required if we

want to be 95% conident that the error in using ˆ p to estimate p is less than 0.05? Using ˆ p = 0.12 as
an initial estimate of p, we ind from Equation 8-24 that the required sample size is

⎛ ⎞ 2 ⎛ . 1 96 ⎞ 2
n = ⎜ ⎝ z .0 025 ⎟ p(1 − ˆ p) = ⎜ ⎝ . 0 05 ⎟ ⎠ 0 12 . (0 88 ) ≅ 163
ˆ
.
E ⎠

If we wanted to be at least 95% conident that our estimate ˆ p of the true proportion p was within 0.05 regardless of the
value of p, we would use Equation 8-25 to ind the sample size

⎛ ⎞ 2 ⎛ . 1 96 ⎞ 2
n = ⎜ ⎝ z .0 025 ⎟ (0 . ) =25 ⎜ ⎝ . 0 05 ⎟ (0 . ) ≅25 385
E ⎠
⎠
Practical Interpretation: Notice that if we have information concerning the value of p, either from a preliminary
sample or from past experience, we could use a smaller sample while maintaining both the desired precision of estima-
tion and the level of coni dence.

One-Sided Confi dence Bounds
We may i nd approximate one-sided coni dence bounds on p by using a simple modii cation
of Equation 8-23.