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Confidence Intervals

A point estimate without a range is close to useless. "The strategy returns 8% a year" means one thing if the interval is 7% to 9% and something entirely different if it is −15%-15\% to +31%+31\%.

A confidence interval
xˉ±z∗σnorxˉ±t∗sn\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \qquad \text{or} \qquad \bar{x} \pm t^* \frac{s}{\sqrt{n}}

Use z when sigma is known or n is large, and t when sigma is estimated from a small sample.

The critical values worth memorising are 1.961.96 for 95% and 2.5762.576 for 99%, both two-tailed.

Everything in the formula is an instance of the central limit theorem: the estimate is approximately normal, and its standard error is σn\frac{\sigma}{\sqrt{n}}.

What 95% actually refers to

The interval is random; the parameter is fixed. So the correct statement is about the procedure, not about any particular interval:

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