Acceptance Sampling: An Application of the Hypergeometric Distribution
Acceptance sampling is an important field of statistical quality control that was popularized by
Dodge and Romig and originally applied by the U.S. military to the testing of bullets during World
War II. If every bullet was tested in advance, no bullets would be left to ship. If, on the other hand,
none were tested, malfunctions might occur in the field of battle, with potentially disastrous results.1
Dodge reasoned that a sample should be picked at random from the lot, and on the basis of
information that was yielded by the sample, a decision should be made regarding the disposition of
the lot. In general, the decision is either to accept or reject the lot. This process is called Lot
Acceptance Sampling or just Acceptance Sampling.
Acceptance sampling is "the middle of the road" approach between no inspection and 100%
inspection.
Acceptance sampling is employed when one or several of the following hold:



Testing is destructive
The cost of 100% inspection is very high
100% inspection takes too long
Example: An electronic component for a medical X-ray unit is produced in lots of size N = 25.
An acceptance testing procedure is used by the purchaser to protect against lots that contain too
many nonconforming components. The procedure consists of randomly selecting five components
from the lot and testing them. If none of the components is nonconforming, the lot is accepted.
a) Assume that the lot actually contains two nonconforming components, and let Y = no. of
nonconforming components in the sample. Then Y ~ Hypergeometric(N = 25, r = 2, n = 5).
What is the probability of lot acceptance? The probability that no nonconforming units will be
found in the sample is P(Y = 0) =
23
(2
0)( 5 )
(25)
5
= 0.6333.
b) Now assume that the lot actually contains 5 nonconforming components. What is the probability
of lot acceptance? The probability that no nonconforming components will be found in the sample
is P(Y = 0) =
(5)(20)
0 5
(25)
5
= 0.2918.
The greater the proportion of nonconforming components in the lot, the less likely it is that
acceptance sampling will lead to acceptance of the lot.
1
http://www.itl.nist.gov/div898/handbook/pmc/section2/pmc21.htm