## Acceptance Sampling When a customer buys a large quantity of a certain article
from a producer, it is customary to inspect a random sample before acceptance. If the
number of defective articles in the sample does not exceed the so-called The general form of this problem, posed by Pólya [Pólya, 1968] is known to anyone familiar with the basic principles of statistical analysis. Yet its solution takes a considerable amount of numerical work. Pólya remarks: "We shall not discuss the numerical work." So in this respect, our discussion starts where Pólya stops. First, we read in the appropriate standard package.
Then, we define the distribution function for the binomial distribution with parameters and .
The acceptance number is the 0.95-quantile of the binomial distribution with parameters and . It depends on the sample size .
Remember that we are looking for both the acceptance number and the sample size . If we had , we could also compute . Suppose now that . The probability of acceptance is then given by
and we are ready to look for the smallest such that this probability is less than . Here is a plot of for :
We zoom in between 260 and 360 to look for the smallest such that this probability is less than .
Using , we readily find the first index for which the probability is less than .
Thus the sample size is . The corresponding acceptance number is
and the problem is solved. In practical applications there are typically many
alternatives to the hypothesis. The probability of accepting the hypothesis, considered as
a function whose domain is the set consisting of the hypothesis and all the alternatives,
is called the We conclude with a graphical illustration of the operating characteristic relevant to our problem.
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