Lot acceptance and compliance testing based on the sample mean and minimum/maximum

A dual acceptance criterion in terms of the sample mean and an extremum (minimum or maximum) has been used in many inspection procedures in diverse industries. An approximation is given in Vangel (Technometrics, 2002, pp. 242–248) for the joint distribution of the sample mean and an extremum when the population is normally distributed. In this paper we obtain a simple expression that depends on the distribution of the sample mean and the truncated sample mean. This expression allows us to evaluate the joint distribution exactly, in two cases, or approximately, in more general cases, making the dual acceptance criterion easier to calculate in practice. We present a saddlepoint approximation for the joint tail probability, with the application to the dual acceptance criterion under the assumption of normality.