Sample size for a plus-or-minus 3-point poll

Question

You are designing a poll and want a 95\% margin of error of at most \pm 3 percentage points. You do not know the true support, so plan for the worst case \hat{p} = 0.5. Use z = 1.96. How many people must you survey?

Accepted Answer

The margin of error for a proportion is 	ext{ME} = z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. You do not yet know \hat{p}, so use the value that makes the margin largest: \hat{p} = 0.5, giving \hat{p}(1-\hat{p}) = 0.25. This guarantees the poll is accurate enough no matter how the vote actually splits. Set the margin to 0.03 and solve for n. Rearranging, n = \frac{z^2 \,\hat{p}(1-\hat{p})}{	ext{ME}^2} = \left(\frac{1.96}{2 	imes 0.03}ight)^2 = \left(\frac{1.96}{0.06}ight)^2 = (32.67)^2 \approx 1067.1. Round up to be safe: n = 1068. This is why national polls so often use around 1000 …

Sample size for a plus-or-minus 3-point poll

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