Sample size determination for the estimation of proportion

Just as we did with the mean, we can also determine the sample size for estimating the population proportion p.

We know that confidence interval for p is given by

where - is the sample proportion.

This interval is centered on the sample proportion and extends a distance L:

This result can not be used directly to determine the sample size n necessary to obtain a confidence interval of some specific width, since it involves , which is not known. But whatever the outcome, can not be bigger than 0.25, its value when the sample proportion is 0.5. Thus, the largest possible value for L is given by

Using basic algebra, we obtain

and squaring yields

Definition:

Let a random sample be selected from a normal population.

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 sample size is

Example:

A public health survey is to be designed to estimate the proportion p of a population having defective vision. How many persons should be examined if the public health doctor wishes to be 98 % certain that error of estimation is below 0.05?

Solution:

The public health doctor wants the 98 % confidence interval to be

Therefore . The value of for a 98 % confidence level is 2.33.

After substituting we obtain that the required sample size is

.

Thus, if the doctor takes a sample of 543 persons, the estimate of p will be within 0.05 of the population proportion.

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