Estimating Sample Sizes for Means

Determine the minimum sample size needed to estimate a population mean within a specified margin of error at a chosen confidence level. Use the formula $n = (z_{\alpha/2}\sigma/E)^2$ , estimate $\sigma$ from the range via the empirical rule when necessary, and analyze how $n$ scales with changes in precision and confidence.

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Tutorial

The Sample Size Formula

When estimating a population mean $\mu$ with a confidence interval, the margin of error is

E = z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}.

To guarantee a margin of error no larger than a target value $E$ , solve for $n$ :

n = \left(\dfrac{z_{\alpha/2}\,\sigma}{E}\right)^{\!2}.

Since $n$ must be a whole number, we always round up to the next integer. Rounding down would let the margin of error exceed $E$ .

The critical values for the most common confidence levels are:

Confidence	$z_{\alpha/2}$
$90\%$	$1.645$
$95\%$	$1.96$
$99\%$	$2.576$

For example, with $\sigma = 5$ , target $E = 1$ , and $95\%$ confidence:

n = \left(\dfrac{1.96 \cdot 5}{1}\right)^{\!2} = (9.8)^2 = 96.04 \;\longrightarrow\; n = 97.