On the exact tolerance intervals for univariate normal distribution

Abstract

Statistical tolerance interval is another type of interval estimator used for making statistical inference on an unknown population. Simply stated, it is an interval estimator, based on a sample from preliminary experiment, which can be asserted with confidence level 1−α (for example 0.95), to contain at least a speci- fied proportion, say 1−γ (for example 0.99), of the items in the population under consideration. The limits of a statistical tolerance interval are called statistical tolerance limits. The confidence level 1 − α is the probability that a statistical tolerance interval constructed in the prescribed manner (i.e. based on result of an experiment conducted under identical conditions) will contain at least a pro- portion 1 − γ of possibly infinite sequence of items coming from the considered (unknown) population (i.e. realizations of independent random variables from the given distribution). In contrast with other statistical intervals commonly used for statistical inference, like e.g. the confidence intervals for the parameters and/or the prediction intervals for future observation(s), the tolerance intervals are used relatively rarely. One reason is that the theoretical concept and compu- tational complexity is significantly more difficult, if compared with the commonly used confidence and prediction intervals. In this paper we briefly describe the theoretical background and computational approaches for computing the toler- ance factors and limits for statistical tolerance intervals based on samples from univariate normal (Gaussian) populations.