## 22.7 Estimating sample sizes: one mean

For a specified level of confidence, the width of a CI depends on the size of the sample. In general, larger samples produce more precise estimates of the parameter (Sect. 5.2), and hence narrower CIs.

To determine the sample size needed to estimate a sample mean with a given precision for a 95% CI is at least

$\left( \frac{2 \times s}{\text{Margin of error}}\right)^2.$

Always round up the results of a sample size calculation.

Example 22.6 (Sample size estimation) For the NHANES data, What size sample is needed to estimate the direct HDL cholesterol levels to within 0.02 mmol/L, with 95% confidence?

Since we would like to estimate the population mean give-or-take 0.02 mmol/L, the ‘margin of error’ that we would like is 0.02. So, using $$s=0.39926$$, the required sample size is at least

$\left( \frac{2 \times 0.39926}{0.02}\right)^2 = 1594.085;$ at least 1595 Americans are needed. (Remember to always round up in sample size calculations.)