A Study on Overestimating a Given Fraction Defective by an Imperfect Inspector

It has been believed that even an imperfect inspector with nonzero inspection errors could either overestimate or underestimate a given FD (fraction defective) with a 50 : 50 chance. What happens to the existing inspection plans, if an imperfect inspector overestimates a known FD, when it is very low? We deal with this fundamental question, by constructing four mathematical models, under the assumptions that an infinite sequence of items with a known FD is given to an imperfect inspector with nonzero inspection errors, which can be constant and/or randomly distributed with a uniform distribution. We derive four analytical formulas for computing the probability of overestimation (POE) and prove that an imperfect inspector overestimates a given FD with more than 50%, if the FD is less than a value termed as a critical FD. Our mathematical proof indicates that the POE approaches one when FD approaches zero under our assumptions. Hence, if a given FD is very low, commercial inspection plans should be revised with the POE concept in the near future, for the fairness of commercial trades.