An R Introduction to Statistics

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, zα∕2 is the 100(1 α∕2) percentile of the standard normal distribution.

n = (zα∕2)-p(1−-p)


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.


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

> zstar = qnorm(.975) 
> p = 0.5 
> E = 0.05 
> zstar^2  p  (1p) / E^2 
[1] 384.15


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.