Exact tolerance interval with specified tail probabilities and a control chart for the sample variance

Abstract

A 𝛽-content tolerance interval (TI) is a statistical interval which contains at least some fraction (proportion) 𝛽 of the population with a given confidence level. When we are interested in the precision of a quality characteristic, a TI for the sample variance is useful. In this paper, we consider an exact two-sided 𝛽-content TI for the sample variance from a normal distribution with a specified ratio of the tail probabilities. The proposed tolerance interval allows the practitioner more control over how the probabilities in the tails are distributed, which may be useful in certain applications. A comparison with an existing two-sided 𝛽-content TI shows that the proposed TI is better on the basis of expected coverage and standard deviation of the coverage. In addition, the proposed TI is shown to require fewer subgroups to achieve a specific accuracy level. Moreover, a phase II control chart with guaranteed performance is obtained from the proposed TI. Finally, a real and a simulated data are used for illustration.

Tolerance limits for mixture‐of‐normal distributions with application to COVID‐19 data

“…There may be a need to weigh the two‐tail probabilities of a TI when the cost of failing to detect larger (smaller) values is higher than the cost of failing to detect smaller (larger) values. Two‐sided TIs with equal or unequal tail probabilities, also known as balanced or unbalanced TIs, could be chosen depending on the objective at hand, and both might have their place in practice (Alqurashi et al, 2023).…”