Sampling - How big a sample?

It is thought that, in a consignment of discrete units, a certain proportio n of the units contain illegal material. A sample of the consignment is to be inspected. Various methods for the determination of the sample size are compared. The consignment will be considered as a random sample from some s uper-population of units, a certain proportion of which contain drugs. For large consignments, a probability distribution, known as the beta distr ibution, for the proportion of the consignment which contains illegal mater ial is obtained. This distribution is based on prior beliefs about the prop ortion. Under certain specific conditions the beta distribution gives the s ame numerical results as an approach based on the binomial distribution. Th e binomial distribution provides a probability for the number of units in a sample which contain illegal material, conditional on knowing the proporti on of the consignment which contains illegal material. This is in contrast to the beta distribution which provides probabilities for the proportion of a consignment which contains illegal material, conditional on knowing the number of units in the sample which contain illegal material. The interpret ation when the beta distribution is used is much more intuitively satisfact ory. It is also much more flexible in its ability to cater for prior belief s which may vary given the different circumstances of different crimes. For small consignments, a distribution, known as the beta-binomial distribu tion, for the number of units in the consignment which are found to contain illegal material, is obtained, based on prior beliefs about the number of units in the consignment which are thought to contain illegal material. As with the beta and binomial distributions for large samples, it is shown tha t, in certain specific conditions, the beta-binomial and hypergeometric dis tributions give the same numerical results. However, the beta-binomial dist ribution, as with the beta distribution, has a more intuitively satisfactor y interpretation and greater flexibility. The:beta and the beta-binomial di stributions provide methods for the determination of the minimum sample siz e to be taken from a consignment in order to satisfy a certain criterion. T he criterion requires the specification of a proportion and a probability.