Sample-Size Determination for Two-Sided β-Expectation Tolerance Intervals for a Normal Distribution

Abstract

In statistical practice, tolerance limits are constructed to contain a specified proportion of a population. When only sample data are available, the actual proportion contained in the interval is random and unknown but controlled by a statistical criterion. In this article, we consider some properties of two-sided β-expectation tolerance intervals for a normal distribution based on the sample mean and the sample standard deviation S computed from a random sample of size n. The tolerance interval is given by ± kS, where k = (l/n + l)½t and t is an appropriate quantile of a Student-t distribution. In repeated sampling, such intervals will, on the average, contain l0O0β% of the sampled distribution. These intervals provide a useful description of a population if the spread in the actual proportion contained in the interval is controlled or evaluated. We consider the effect that sample size has on the proportion of the population contained in the interval, using two different criteria for measuring the variability of the coverage. We also give tables to enable the user to achieve the criterion specified.

KEY WORDS:

• Prediction intervals
• Tables
• Tolerance limits