Track 7 · Estimation and Confidence Intervals · Lesson 9 of 10

Planning sample size

Calculate the sample size needed for a target margin of error when estimating a mean or a proportion.

25 minIntermediate8 graded exercisesMembers

A city council wants to know the share of households that would use a recycling pickup service, to within 3 percentage points. Survey too few and the answer is too vague to act on; survey too many and money is wasted. A two-line calculation, done before any data are collected, gives the number to aim for.

The margin of error of an interval for a mean is $E = z_{\alpha/2}\,\sigma/\sqrt{n}$ . Planning reverses the question: choose the margin $E$ you need and solve for $n$ .

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Calculate the sample size for a mean from a planning σ and a target margin
Calculate the sample size for a proportion, using p = 0.5 when nothing is known
Always round up, and adjust for non-response and small populations
Explain why sample size controls only random error, not bias

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Formula sheet

Sample size for a mean (left) and for a proportion (right); always round **up**
$n = \left(\frac{z_{\alpha/2}\,\sigma}{E}\right)^2, \qquad n = \frac{z_{\alpha/2}^2\,p(1-p)}{E^2}$

Reference

Finite population correction (FPC)
When sampling without replacement a sizeable fraction of a finite population, the standard error is multiplied by the finite population correction √((N − n)/(N − 1)).
Sample size for a margin of error
Choose n so that a confidence interval for a mean or proportion has at most a desired margin of error E, rounding up and optionally applying the finite population correction.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.