Confidence Intervals
An interval estimate for a parameter is a calculated range within which it is deemed likely to fall. Given $$\alpha$$, the set of intervals from infinitely repeated random samples of size $$n$$ will contain the parameter $$100(1-\alpha )$$% of the time: each interval is a $$100(1-\alpha )$$% confidence interval.
Confidence interval for a population mean - $$\sigma^2$$ unknown
If $$X$$ has mean $$\mu$$ and variance $$\sigma^2$$, with $$n>30$$ an approximate $$100(1-\alpha)$$% confidence interval for $$\mu$$ is \[\bar{x}~-~\frac{t_{\alpha/2} s}{\sqrt{n}}~~\mbox{ to }~~ \bar{x}~+~\frac{t_{\alpha/2} s}{\sqrt{n}}.\] If $$X\sim N(\mu,\sigma^2)$$ the interval is exact for all $$n$$.