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A modified cluster sampling method for estimating the bacterial density in Xanthan gum
Authors:CJ Park  KS Kang
Institution:San Diego State University and KELCO, Division of Merck & Co., Inc., California, USA
Abstract:Suppose particles are randomly distributed in a certain medium, powder or liquid, which is conceptually divided into N cells. Let pi denote the probability that a particle falls in the ith cell and Yi denote the number of particles in the ith cell. Assume that the joint probability function of the Yi follows a multinomial distribution with cell probabilities pi respectively. Take n (≤N) cells at random without replacement and put each of the cells separately through a mixing mechanism of dilution and swirl. These n cells constitute the first stage samples and the number of particles in these cells are not observable. Now conceptually divide each of n cells into M subcells of equal size and let Xij denote the number of particles in the jth subcell of the ith cell selected in the first stage; i=1,2,…,N and j=1,2,…,M. Consequently assume that the conditional joint probability function of the Xij given Yi=yi follows a multinomial distribution with equal cell probabilities. Now take m (≤M) subcells at random from each of the cells selected in the first stage sample. Assume that the numbers of particles in M×N subcells are observable. The properties of the estimator of the particle density per sample unit are investigated under the modified two-stage cluster sampling method. A laboratory experiment for Xanthan Gum Products is analyzed in order to examine the appropriateness of the model assumed in this paper.
Keywords:62D05  Two-stage cluster sampling  Multinomial distribution  Bacterial density estimate  Coefficient of variations
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