3.9 Summary

• The sample variance \(s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2\) is unbiased for \(\sigma^2\). The divisor is \(n-1\) because \(\bar{x}\) was estimated from the same data, leaving \(n-1\) free deviations. If \(\mu\) is known, the divisor is \(n\) and the deviations are taken from \(\mu\).
• A proportion is a mean — of a variable coded 1 and 0 — so every result about \(\bar{X}\) transfers to \(\hat{p} = x/n\) without new proof.
• \(\mathrm{se}(\hat{p}) = \sqrt{p(1-p)/n}\), largest at \(p = 0.5\). Evenly split populations are the hardest to estimate; near-universal ones are easy.
• A confidence interval is \(\bar{x} \pm z_{\alpha/2}\,\sigma/\sqrt{n}\) when \(\sigma\) is known, and \(\bar{x} \pm t_{n-1,\,\alpha/2}\,s/\sqrt{n}\) when it is not. For a proportion, \(\hat{p} \pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}\).
• The \(t\) distribution compensates for estimating \(\sigma\) from the same sample. It does not compensate for a non-normal population — two different assumptions, one correction.
• The confidence level is a property of the procedure: across repeated samples, that proportion of intervals contains the parameter. A particular interval either contains it or does not.
• Higher confidence means a wider interval. Halving the margin of error quadruples the required sample, and \(n = (z_{\alpha/2}\sigma/E)^2\) makes that explicit before any fieldwork is commissioned.
• What determines precision is how many people were asked, not what fraction of the population they are. Population size appears in none of these formulas.

Everything in this unit produced a range of values consistent with the data. The next unit asks a sharper question: given a specific claim about the population, is our sample consistent with it?