# 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 Section

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 Section

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 $$n=451$$.