20.8 Estimating sample sizes: one proportion

For a given 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.

Suppose we want our 95% CI for the proportion of smokers (Example 20.7) to be precise to give-or-take \(0.01\) (rather than the \(\pm 0.018\) found from the sample): what size sample is needed? Since we seek a more precise estimate, we’d expect to need a larger sample… but how much larger?

Conservatively, the sample size for a 95% CI needed is at least

\[ \frac{1}{(\text{Margin of error})^2}. \] That is, a sample size of at least \(\displaystyle \frac{1}{0.01^2} = 10\,000\) Americans is needed.

Example 20.8 (Sample size calculations) To estimate the population proportion of Australians that smoke, to within \(0.07\) with 95% confidence, a sample size of at least

\[ \frac{1}{(\text{Margin of error})^2} { = \frac{1}{0.07^2}} \] is needed; at least \(n = 204.0816\) people.

In practice, at least 205 people are needed to achieve this desired level of precision (that is, always round up in sample size calulations).
Always round up the result of the sample size calculation.

The following short video may help explain some of these concepts: