Calculating the Sample Size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

Recall the margin of error for a population proportion is:

• \(MoE=\left({z}_{\frac{\alpha }{2}}\right)\left(\sqrt{\frac{\hat{p}\hat{q}}{n}}\right)\)
• Solving for n gives you an equation for the sample size.
• \(n=\left(\frac{{z}_{\frac{\alpha }{2}}}{MoE}\right)^{2}\text{p(1-p)}\)

Recall the objective of a CI.  If we are looking to estimate p, then we do not know what is, however it appears in this formula.  So what do we plug in for p?  We have a few options:

• If you have prior information such as a previous sample and can calculate a point estimate \(\left(\hat{p}\right)\) plug it in!
• You can use your best guess at p
• You can use a “conservative” estimate of p, 0.5.*

*Note: Remember that \(\hat{q}=(1-\hat{p})\). But, we do not know \(\hat{p}\) yet. Since we multiply \(\hat{p}\) and \(\hat{q}\) together, we make them both equal to 0.5 because \(\hat{p}\hat{q} = (0.5)(0.5) = 0.25\) results in the largest possible product. (Try other products: (0.6)(0.4) = 0.24; (0.3)(0.7) = 0.21; (0.2)(0.8) = 0.16 and so on). The largest possible product gives us the largest n. This gives us a large enough sample so that we can be CL% confident that we are within three percentage points of the true population proportion.