Tolerance interval
A tolerance interval (TI) is a statistical interval within which, with some confidence level, a specified sampled proportion of a population falls. "More specifically, a 100×p%/100×(1−α) tolerance interval provides limits within which at least a certain proportion (p) of the population falls with a given level of confidence (1−α)." "A (p, 1−α) tolerance interval (TI) based on a sample is constructed so that it would include at least a proportion p of the sampled population with confidence 1−α; such a TI is usually referred to as p-content − (1−α) coverage TI." "A (p, 1−α) upper tolerance limit (TL) is simply a 1−α upper confidence limit for the 100 p percentile of the population." == Definition == Assume observations or random variates x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} as realization of independent random variables X = ( X 1 , … , X n ) {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})} which have a common distribution F θ {\displaystyle F_{\theta }} , with unknown parameter θ {\displaystyle \theta } .