Introduction to the Mathematical and Statistical Foundations of Econometrics
Quality Control
1.2.1. Sampling without Replacement
As a second example, consider the following case. Suppose you are in charge of quality control in a light bulb factory. Each day N light bulbs are produced. But before they are shipped out to the retailers, the bulbs need to meet a minimum quality standard such as not allowing more than R out of N bulbs to be defective. The only way to verify this exactly is to try all the N bulbs out, but that will be too costly. Therefore, the way quality control is conducted in practice is to randomly draw n bulbs without replacement and to check how many bulbs in this sample are defective.
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As in the Texas lotto case, the number M of different samples Sj of size n you can draw out of a set of N elements without replacement is
Each sample Sj is characterized by a number kj of defective bulbs in the sample involved. Let K be the actual number of defective bulbs. Then kj є {0, 1,..., min(n, K)}.
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Let ^ = {0,1,..., n} and let the a-algebra & be the collection of all subsets of ^. The number of samples Sj with kj = к < min(n, K) defective bulbs is
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because there are “K choose k” ways to draw к unordered numbers out of K numbers without replacement and “N - K choose n - k” ways to draw n - к unordered numbers out of N - K numbers without replacement. Of course, in the case that n > K the number of samples Sj with kj = к > min(n, K) defective bulbs is zero. Therefore, let
P({k}) = 0 elsewhere,
and for each set A = {k,...,km }є &, let P (A) = J™= P ({kj}). (Exercise: Verify that this function P satisfies all the requirements of a probability measure.) The triple {^, Ж, P} is now the probability space corresponding to this statistical experiment.
The probabilities (1.11) are known as the hypergeometric (N, K, n) probabilities.
