Determining the Weight for Obtaining a Large Number of Items

摘要

A simple procedure is proposed to determine a sample size for estimating the mean weight of items in a problem of obtaining a batch of a large number of items. Suppose it is desired to obtain a large number N (sub s) of items for which individual counting is impractical, but one can demand a batch to weigh at least w units and hope that the number of items in the batch is close to the desired number N(sub s). If the items have mean weight theta, it is reasonable to have w equal to theta N sub s when theta is known. When theta is unknown, one can take a sample of size n, not bigger than N(sub s), estimate theta by a good estimator theta sub n and set w equal to theta sub n/sub s. The proposed procedure determines the sample size to be the integer closest to rhe CN sub s, where C is a function of the cost coefficients if the coefficient of variation rhe is known. It is shown to be optimal in some sense. If rho is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given theta and theta has a prior inverted gamma distribution, the optimal sample size in some sense can be found to be the nonnegative integer closest to rhe CN sub s + rho squared A(Rho C-1), where A is a known constant given in the prior distribution.