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When 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 \(n=\left ( \dfrac \right )^2 \tilde(1-\tilde
)\)
\(M\) 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 \(\tilde
=0.50\) because it is most conservative and it will give use the largest sample size calculation.
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 \(\tilde
=0.50\) for a conservative estimate.
For a 95% confidence interval, \(z^*=1.960\)
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 \(n=601\).
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:
Again, we should round up to 451. In order to construct a 95% confidence interval with a margin of error of 4%, given \(\tilde
=.25\), we should obtain a sample of at least \(n=451\).
Note that when we changed \(\tilde
\) in the formula from .50 to .25, the necessary sample size decreased from \(n=601\) to \(n=451\).