Extreme-order-statistic tolerance interval ($n\approx \vartheta /\epsilon ,(1+\vartheta )e^{-\vartheta }=\delta$)

The sample minimum and maximum enclose random population content given by the uniformized sample range. The exact failure probability for content at least $1-\epsilon$ is $(1-\epsilon {)}^{n-1}[1+(n-1)\epsilon ]$. For small $\epsilon$, setting $n\epsilon \approx \vartheta$ gives the displayed sample-size approximation at confidence $1-\delta$. The positive solution is unique for $0<\delta <1$, since $(1+\vartheta )e^{-\vartheta }$ decreases from one to zero. A tolerance interval concerns population content, rather than a confidence interval for a fixed parameter; the exact inequality should select the final integer size.