# 8.1.1.3 - Computing Necessary Sample Size

8.1.1.3 - Computing Necessary Sample Size

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
$n=\left ( \frac{z^*}{M} \right )^2 \tilde{p}(1-\tilde{p})$

$M$ 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, $z^*=1.960$

$n=\left ( \frac{1.960}{0.04} \right )^2 (0.5)(1-0.5)=600.25$

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:

$n=\left ( \frac{1.960}{0.04} \right )^2 (0.25)(1-0.25)=450.188$

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 $n=451$.

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