A Distribution-Free Confidence Interval for the Mean

Consider a random sample of N observations x₁, x₂, , xN, from a universe of mean and variance ². Let m and s² be the sample mean and variance respectively: equation*1 m = 1N N₈=₁ xᵢ, s² = 1N N₈=₁ (xᵢ - m) ². equation* It is shown that the following conservative confidence interval holds for: equation*2 Prob\ (m -) ² s²/ (N - 1) + ²2/N (N - 1) \ > 1 - ^-2, equation* where is any positive constant. Inequality (2) also holds if, in the braces, is replaced by ² - 1, with 1. Inequality (2) is much more efficient on the average than Tchebychef's inequality for the mean, namely, equation*3 Prob \ (m -) ² ²²/N\ > 1 - ^-2, equation* yet (2) and (3) are both distribution-free, requiring only knowledge about ². At the 1 - ^-2 =. 99 level of confidence, the expected value of the right member in the braces of (2) is only about 1/6 the corresponding member of (3) ; at the. 999 level of confidence the ratio is about 1/20. A more general inequality than (2) is developed, also involving only the single parameter ². *