Section Sample Size
Subsection Determining the Sample Size
How large a sample is necessary to make an accurate estimate? The answer is not simple, since it depends on three things:
The confidence level
The standard deviation
The margin of error
The more confident we want to be, the wider we want our interval, but there is a tradeoff. If an interval is too wide, it provides little information. On the other hand, a narrow interval requires a larger sample size and sampling can be a costly procedure. Hence, it is handy to be able to be able to calculate ahead of time the sample size required to achieve a specified margin of error. With a bit of algebra on the appropriate margin of error from the confidence interval formula, we can determine the sample size.
The formula when working with a mean is \(n=\frac{(z_{\alpha/2})^2\sigma^2}{(ME_{\overline{x}})^2}\) where \(ME_{\overline{x}}\) is the desired margin of error.
Similarly, the formula when working with a proportion is \(n=\frac{(z_{\alpha/2})^2\overline{p}(1-\overline{p})}{(ME_{\overline{p}})^2}\) where \(ME_{\overline{p}}\) is the desired margin of error.
If necessary, round the answer up to obtain a whole number. If you round down, then you won’t achieve the desired margin of error.
Example 7.8.
(Donnelly 8.44)
Determine the sample size needed to construct a \(99\%\) confidence interval to estimate the average GPA for the student population at a college with a margin of error equal to 0.5. Assume the standard deviation of the GPA for the student population is 2.5.
Example 7.9.
(Donnelly 8.47)
A certain region would like to estimate the proportion of voters who intend to participate in upcoming elections. A pilot sample of 50 voters found that 39 of them intended to vote in the election. Determine the additional number of voters that need to be sampled to construct a \(96\%\) interval with a margin of error equal to 0.08 to estimate the proportion.
