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In this example, we have a lot of 400 items. We take a sample of 100 items from the lot. We accept the entire lot as being good if the sample has two or less defective items.
We use the hypergeometric distribution (@PHG) to determine the exact producer risk (probability of rejecting a good lot) and the exact consumer risk (probability of accepting a bad lot). Now, in the days before computers were widely available, statisticians had to rely on published tables of the probability distributions to compute probabilities such as these. Because the hypergeometric distribution is specified by four parameters, it would have been unrealistic to carry around hypergeometric tables that covered all possible scenarios. Instead, statisticians routinely used distributions of fewer parameters to approximate the hypergeometric. So, in deference to the good old days, we make use of the binomial, Poisson, and normal approximations to the hypergeometric to compute these same risk probabilities. The interested reader can compare the accuracy of the various approximations.