Planning Sample Sizes and Power

Sample-size planning is simply deciding, before you collect any data, how many parts you need so the study can actually answer your question. There are two versions. For an estimate, $n = (z\sigma/E)^2$: how noisy the parts are ($\sigma$), how sure you want to be ($z$) and how close you need to get ($E$). For a test, the same shape with $z_\alpha + z_\beta$ (the false-alarm and missed-shift risks you'll accept) and $\delta$ (the smallest shift worth catching) in place of $E$. Both are the standard-error formula solved for $n$, so this is old material put to new use.

Plan first. Every interval and test in this course sharpens as $n$ grows; the question is how sharp it has to be. Too few parts and a real problem slips through; too many and the study burns a week of production.

For an estimate. The margin of error of a mean is $E = z_{\alpha/2}\sigma/\sqrt n$, so $n = (z_{\alpha/2}\sigma/E)^2$, always rounded up. σ comes from past data or a pilot run; if it's a guess, be generous. For a defect rate the same algebra gives $n = z^2p(1-p)/E^2$ (proportion intervals).

For a test: where the power formula comes from. Test $H_0: \mu = \mu_0$ against $H_1: \mu > \mu_0$ with σ known, rejecting when $\bar x > \mu_0 + z_\alpha\sigma/\sqrt n$. If the mean has really moved to $\mu_0 + \delta$, then $\bar x \sim N(\mu_0 + \delta, \sigma^2/n)$ and

For power $1 - \beta$ the argument must equal $z_\beta$, so $\delta\sqrt n/\sigma = z_\alpha + z_\beta$. In pictures: two bell curves for $\bar x$, one centered at $\mu_0$ and one at $\mu_0 + \delta$. The cutoff sits $z_\alpha$ standard errors above the first and $z_\beta$ below the second, so the shift has to span $z_\alpha + z_\beta$ standard errors.

Two-sided tests use $z_{\alpha/2}$ in place of $z_\alpha$; the far tail adds almost nothing to the power.

Trade-offs:

• $n \propto 1/\delta^2$: halving the detectable shift quadruples $n$.
• $n \propto \sigma^2$: a steadier process or a better gauge cuts $n$ directly. Since $\sigma^2_{\text{total}} = \sigma^2_{\text{parts}} + \sigma^2_{\text{gauge}}$ (measurement error), fixing a noisy gauge can beat measuring more parts.
• A smaller α or more power both raise $z_\alpha + z_\beta$: going from 80% to 95% power multiplies $n$ by about 1.75 for a one-sided test at α = 0.05.
• When σ will be estimated by $s$ (a t test), the z formula runs a little low: add two or three parts.
• Comparing two processes with $n$ each, $\operatorname{Var}(\bar x_1 - \bar x_2) = 2\sigma^2/n$, so each needs $n = 2\left((z_{\alpha/2} + z_\beta)\sigma/\delta\right)^2$ (comparing processes).

"No difference found" needs power. A study with 30% power that finds nothing has shown almost nothing. Before accepting "the new supplier is no different", ask what shift the test could have detected.

Same idea elsewhere: the number of runs for a Monte Carlo estimate, $N = p(1-p)/\text{SE}^2$ (Monte Carlo tolerancing), and the α and β picture in hypotheses and errors.

Qualification plans (how many parts in a first-article inspection), reliability demonstration tests, trials of new production settings and clinical device studies are all sized this way before the first measurement. Customers and regulators routinely ask for the power calculation behind a "no difference found" claim, because a study too small to detect a meaningful shift can't vouch for anything.