A Distribution-Free Confidence Interval for the Mean

Consider a random sample of N observations x₁, x₂, ⋯, xN, from a universe of mean μ and variance σ². Let m and s² be the sample mean and variance respectively: {equation*}{1} m = 1/N ∑^Nᵢ₌₁ x_i, s^2 = 1/N ∑^Nᵢ₌₁ (x_i - m)^2.{equation*} It is shown that the following conservative confidence interval holds for μ: {equation*}{2} Prob\{ (m - μ)^2 s^2/(N - 1) + λσ^2√2/N(N - 1)\} > 1 - λ⁻²,{equation*} where λ is any positive constant. Inequality (2) also holds if, in the braces, λ is replaced by √λ² - 1, with λ 1. Inequality (2) is much more efficient on the average than Tchebychef's inequality for the mean, namely, {equation*}{3} Prob \{(m - μ)^2 λ^2σ^2/N\} > 1 - λ⁻²,{equation*} yet (2) and (3) are both distribution-free, requiring only knowledge about σ². At the 1 - λ⁻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 σ².*