How many samples to pin down a mean latency?

Question

You want to estimate the mean order-to-fill latency of a venue. Prior data says the latency standard deviation is about $\sigma = 5$ milliseconds. You want a 95% confidence interval with half-width at most $1$ millisecond.

How many independent measurements do you need?

Accepted Answer

For a mean with known standard deviation \sigma, the 95% confidence interval is \bar{x} \pm z\,\sigma/\sqrt{n} with z = 1.96. Requiring half-width E: 1.96\,\frac{\sigma}{\sqrt{n}} \leq E \implies n \geq \left(\frac{1.96\,\sigma}{E}ight)^2. Plug in \sigma = 5, E = 1: n \geq \left(\frac{1.96 	imes 5}{1}ight)^2 = (9.8)^2 = 96.04 \implies n = 97. So about 97 measurements. Note the structure: unlike the proportion case, there is no worst-case p(1-p) to bound, because you already know \sigma. And the same inverse-square scaling applies, tightening the interval to \pm 0.5 ms would need 4	imes …

How many samples to pin down a mean latency?

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