Confidence interval

A point estimator gives a single value and says nothing about its precision. The confidence interval adds that margin: it is built with a procedure that, over repeated sampling, captures the parameter a fraction \(1-\alpha\) of the time. The usual levels are \(90\,\%\), \(95\,\%\) and \(99\,\%\), with \(z=1.645,\ 1.96\) and \(2.576\).

When \(\sigma\) is known and the sample mean is normal (exact if the population is, approximate by the central limit theorem if \(n\) is large), the interval for \(\mu\) is \(\bar x\pm z\,\sigma/\sqrt n\). The term \(z\,\sigma/\sqrt n\) is called the margin of error; the interval is symmetric about \(\bar x\) and its length is twice the margin.

The margin depends on three things: it grows with confidence (larger \(z\)), grows with the spread \(\sigma\) and shrinks with \(\sqrt n\). Hence, to fix a margin \(E\), it suffices to solve \(n\ge(z\sigma/E)^2\) and round up. With unknown \(\sigma\), replace it by \(S\) and \(z\) by a value from the \(t\) distribution.

The correct interpretation is the most misunderstood: we do not say that \(\mu\) has probability \(0.95\) of being in an already computed interval, because \(\mu\) is fixed. The \(95\,\%\) is a property of the method. This same reasoning, inverted, gives hypothesis tests.