Equitable quality level and error-areas under the operating characteristic curves of normal single sampling inspection plans (with σ known)

Summary

This article assumes a continuous prior distribution for lot quality in the case of a normal single sampling inspection plan with known standard deviation. The definition of Equitable Quality Level (EQL) as given in this paper ensures that the proportion of lots of quality better than the EQL—as obtained under the prior distribution—is equal to the average probability of acceptance. The first kind of error-area at the quality levelp′ is the joint probability of producing a lot of quality equal to or better thanp′ and getting such a lot rejected by the plan whereas the second kind of error-area is the joint probability of producing a lot of quality worse thanp′ and getting it accepted. Certain measures of producer's and consumer's risks can therefore be defined in terms of error-areas. It is noted that the OC can be viewed as the upper cumulative distribution function of a hypothetical random variableY. It is shown that the EQL and the error-areas can be expressed in terms of the derivatives of the prior distribution and the moments ofY. The latter do not depend on the prior distribution. It is hinted how this technique can be used to construct plans having certain optimum properties and also to obtain approximations to compound distribution. The case of a normal prior distribution is fully dealt with.