8.1.1.3  Computing Necessary Sample Size
8.1.1.3  Computing Necessary Sample SizeWhen we begin a study to estimate a population parameter we typically have an idea as how confident we want to be in our results and within what degree of accuracy. This means we get started with a set level of confidence and margin of error. We can use these pieces to determine a minimum sample size needed to produce these results by using algebra to solve for \(n\):
 Finding Sample Size for Estimating a Population Proportion
 \(n=\left ( \frac{z^*}{M} \right )^2 \tilde{p}(1\tilde{p})\)

\(M\) is the margin of error
\(\tilde p\) is an estimated value of the proportion
If we have no preconceived idea of the value of the population proportion, then we use \(\tilde{p}=0.50\) because it is most conservative and it will give use the largest sample size calculation.
Example: No Estimate
We want to construct a 95% confidence interval for \(p\) with a margin of error equal to 4%.
Because there is no estimate of the proportion given, we use \(\tilde{p}=0.50\) for a conservative estimate.
For a 95% confidence interval, \(z^*=1.960\)
\(n=\left ( \frac{1.960}{0.04} \right )^2 (0.5)(10.5)=600.25\)
This is the minimum sample size, therefore we should round up to 601. In order to construct a 95% confidence interval with a margin of error of 4%, we should obtain a sample of at least \(n=601\).
Example: Estimate Known
We want to construct a 95% confidence interval for \(p\) with a margin of error equal to 4%. What if we knew that the population proportion was around 0.25?
The \(z^*\) multiplier for a 95% confidence interval is 1.960. Now, we have an estimate to include in the formula:
\(n=\left ( \frac{1.960}{0.04} \right )^2 (0.25)(10.25)=450.188\)
Again, we should round up to 451. In order to construct a 95% confidence interval with a margin of error of 4%, given \(\tilde{p}=.25\), we should obtain a sample of at least \(n=451\).
Note that when we changed \(\tilde{p}\) in the formula from .50 to .25, the necessary sample size decreased from \(n=601\) to \(n=451\).