Intervals For The Population Proportion

in Section 8.5, we saw that based on a random sample of« observations, a 100( I — a)% confidence interval for the population proportion p is given by

¡pj±eji^ ^ - . M*1 -fa p, - Zat2 ---< P < p., + Z„.

n \ n where />, is the observed sample proportion. This interval is centered on the sample proportion and extends a distance llUl - px)

on each side of the sample proportion. Now, this result cannot be used directly to determine the sample size necessary to obtain a confidence interval of some specified width, since it involves the sample proportion, which will not be known at the outset. However, whatever the outcome, px{ 1 — px) cannot be bigger than .25, its value when the sample proportion is .5. Thus, the largest possible value for L is Lgiven by

Suppose, then, that an investigator wants to choose a sufficiently large sample size to guarantee that the confidence interval extends no more than L* on each side of the sample proportion. From Eq. (8.9.2), we have

and squaring yields

.25zap_

This provides the required sample size.

Sample Size for Confidence Intervals for the Population Proportion

Suppose that we take a random sample from a population. Then a 100(1 - a)% confidence interval for the population proportion, extending a distance of at most L- on each side of the sample proportion, can be guaranteed if the number of sample observations is