# Sampling Size of Population Proportion

The quality of a sample survey can be improved by increasing the sample size. The
formula below provide the sample size needed under the requirement of population
proportion interval estimate at (1 − α) confidence level, margin of error E, and
planned proportion estimate p. Here, z_{α∕2} is the 100(1 − α∕2) percentile of the
standard normal distribution.

#### Problem

Using a 50% planned proportion estimate, find the sample size needed to achieve 5% margin of error for the female student survey at 95% confidence level.

#### Solution

Since there are two tails of the normal distribution, the 95% confidence level would
imply the 97.5^{th} percentile of the normal distribution at the upper tail. Therefore,
z_{α∕2} is given by qnorm(.975).

#### Answer

With a planned proportion estimate of 50% at 95% confidence level, it needs a sample size of 385 to achieve a 5% margin of error for the survey of female student proportion.