corresponding to 40 positive 25 g samples in a lot of 1 tonne (400025g
samples), then the lot would be accepted 90% of the time if 10 samples
were tested on each occasion. When the number of samples taken at each
testing is increased then the chances of rejection are increased, so that
with 20 samples the lot would be accepted 82% of the time. If we went to
testing 100 samples, we would accept the same lot only 37% of the time.
A statistically equivalent situation would be the testing of an apperti-
zed food for the presence of surviving organisms capable of spoiling the
product, and where detection of one defective pack would mean rejection
of the lot. Using Table 11.1 again, we can see that if there was a failure
rate of one pack in a thousand (p¼0.001), even taking 100 packs for
microbiological testing we would only reject a lot on one out of ten
occasions. In fact we can calculate the number of packs it would be
necessary to take in order to have a 95% probability of finding one
defective sample (Prej¼0.95). This is done simply by substituting in
Equation (11.6) and solving forn. The answer, 2995, demonstrates why it
is necessary to have alternatives to microbiological testing to control the
quality of appertized foods.
The probabilities of acceptance or rejection associated with an at-
tributes sampling plan can be calculated from the binomial distribution.
For large batches of product these can also be represented graphically by
what is known as an operating characteristic (OC) curve of the type shown
in Figure 11.2. For each level of defectives in the lot, the probability of its
acceptance or rejection using that plan can be read off the curve. Figure
11.3 demonstrates that asnincreases for a given value ofc, the stringency
of the plan increases since a lot’s overall quality must increase withnfor it
to have the same chance of being passed. Ifcis increased for a given value
ofn(Figure 11.4), so the plan becomes more lenient as lot quality can
decrease but still retain the same chance of being accepted.
Figure 11.2 An operating characteristic curve
Chapter 11 401