Sample size for a proportion using a prior estimate

Question

You want to estimate a proportion to within a margin of error of $E = 0.02$ , with 95% confidence. A pilot study suggests the true rate is near $p = 0.2$ .

Using the pilot estimate, what is the minimum sample size?

Accepted Answer

Setup. Use n = (z^{})^{2}\,p(1-p)/E^{2} with z^{} = 1.96 and the prior p = 0.2, so p(1-p) = 0.2 	imes 0.8 = 0.16. Compute. n = \frac{(1.96)^{2}(0.16)}{(0.02)^{2}} = \frac{3.8416 	imes 0.16}{0.0004} = \frac{0.614656}{0.0004} = 1536.6. Round up: n = 1537. Interpret. Because p = 0.2 gives p(1-p) = 0.16, smaller than the worst case 0.25, using the pilot estimate lets you get away with fewer respondents than the conservative p = 0.5 would require. Takeaway. A credible prior estimate away from 0.5 lowers p(1-p) and therefore the required sample. Use it when you trust it; fall back to 0.5 when you…

Sample size for a proportion using a prior estimate

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