• Find the required sample size for a desired margin of error and population’s confidence interval.

Sample Size Needed for Proportions

Researchers can use the margin of error formula, [latex]ME = z^{*} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex], to determine the minimum sample size needed to produce a given margin of error by solving for [latex]n[/latex].

The rearranged formula to find the sample size needed for proportion:

[latex]n = \hat{p}(1-\hat{p})(\frac{z^{*}}{ME})^{2}[/latex]

Notice that this formula requires the researcher to know the value of [latex]\hat{p}[/latex], which is unknown. However, researchers often have preliminary data or prior research that can be used to estimate [latex]\hat{p}[/latex].

If there is no way to estimate [latex]\hat{p}[/latex], researchers will find the largest possible n by setting [latex]\hat{p}[/latex] to [latex]0.5[/latex]. (Try it! The largest value you can get for [latex]\hat{p}(1-\hat{p})[/latex]) is [latex]0.25[/latex] when you set [latex]\hat{p}[/latex] to [latex]0.5[/latex].)

We can also use our statistical tool to help us calculate the sample size.

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Using the conservative [latex]\hat{p} = 0.5[/latex] approach always yields a larger than necessary sample size.