Calculating the Sample Size n

Learning Outcomes

  • Calculate the sample size needed for a given error bound for a confidence interval for a population mean based on a normal distribution

Calculating the Sample Size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a population mean when the population standard deviation is known is EBM = [latex]{\left ( z_\frac{a}{2} \right )}{\left ( \dfrac{\sigma}{\sqrt n} \right )}.[/latex]

The formula for sample size is n = [latex]\dfrac{z^2\sigma^2}{EBM^2}[/latex], found by solving the error bound formula for n.

In this formula, z is [latex]z_\frac{a}{2}[/latex], corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

Example 7

The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?

try it 7

The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?