Sampling Inspection by Variables with an Additional Acceptance Criterion

Abstract

We deal with sampling inspection by variables, i.e. acceptance sampling procedures wherein the acceptability of a lot is statistically established from the measurement results of a specified continuous variable X obtained at the items in a sample from the lot. An item is qualified as nonconforming if its measured quality characteristic x is larger than a defined upper specification limit U. We accept the lot if $\bar{x} + k\sigma \leq U$ or $\bar{x} + \mathit{ks} \leq U$ in the case of known or unknown lot standard deviation σ, respectively, where $\bar{x}$ and s are mean and standard deviation of a sample of size n drawn at random from the lot and k is an acceptance constant given as a parameter of the sampling plan.In some cases the acceptance procedure is extended by an additional limit $U^{\star } = U+\varDelta$ with $\varDelta \in \mathbb{R}$ that must not be exceeded by any of the measurements $x_{1},x_{2},\ldots,x_{n}$, i.e. for acceptance of the lot the largest measurement result $x_{(n)} =\max (x_{1},x_{2},\ldots,x_{n})$ must be less or equal to $U^{\star }$, $x_{(n)} \leq U^{\star } = U +\varDelta.$ Of course, with this additional requirement for acceptance the probability of acceptance of the lot is smaller than without it for each fraction p of nonconforming items in the lot.Such extended sampling plans are, e.g., used for the evaluation of bacterial contamination in foods, the amount of active ingredient used in formulating drug products and the strength of concrete.The OC function of these extended sampling plans for inspection by variables is derived and the advantages/disadvantages in comparison with unextended sampling plans are discussed. It turns out that especially in the case of “known” σ plans the extended sampling plans protect against a true standard deviation that is larger than the value being used in the acceptance criterion $\bar{x} + k\sigma \leq U$.