Shrinking the margin from 2pp to half a point

Question

You currently sample 2{,}500 orders and estimate a fill rate with a worst-case margin of about \pm 2 percentage points at 95% confidence. Your boss now wants the margin down to \pm 0.5 percentage points. How many orders must you sample now?

Accepted Answer

The margin obeys E \propto 1/\sqrt{n}, so n \propto 1/E^2. Shrinking the margin by a factor of \frac{2}{0.5} = 4 requires multiplying n by 4^2 = 16: n{	ext{new}} = 16 	imes 2{,}500 = 40{,}000. Check the endpoints directly with the worst-case formula E = 1.96 	imes 0.5/\sqrt{n}. At n = 2{,}500: 0.98/50 = 0.0196 \approx 2pp. At n = 40{,}000: 0.98/200 = 0.0049 \approx 0.5pp. Consistent. So going from a \pm 2pp to a \pm 0.5pp margin costs a sixteenfold increase in data, from 2{,}500 to 40{,}000 orders. Takeaway. Precision is bought at quadratically rising prices: cutting the margin by a factor…

Shrinking the margin from 2pp to half a point

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