Estimation of the quantity of a drug in a consignment from measurements on a sample

A consignment of individual packages is thought to contain illegal material, such as drugs, in some or all of the packages. A sample from the consignment is inspected and the quantity of drugs in each package of the sample is measured. It is desired to estimate the total quantity of drugs in the consignment. Sampling variation is present in the original measurements and it is not sufficient just to adjust the sample mean pro rata. An analysis is described which takes account of the uncertainty concerning the proportion of the packages that contain drugs and provides a probabilistic summary of the quantity of drugs in the consignment. In particular, a probabilistic lower bound for the quantity of drugs in the consignment is given, which is dependent on the required standard of proof. This is in contrast to the approach based on confidence intervals which assumes that in the long run, the interval will contain the correct quantity the appropriate proportion of the time, but gives no measure of uncertainty associated with the particular consignment under consideration.