Two sigma and Three sigma Control Limits
The textbook's abstract discussion of the business meaning of two sigma
and three sigma control limits on the bottom of page 336 is needlessly
vague. The concrete description of Western Jeans Company's
assumptions on the bottom of page 337 is simply wrong. (Western
Jeans Company's managers might think this, especially if it wasn't
explained to them any better than the textbook does, but they would be
in a state of confusion if they did.)
The real point is that, in an imperfect world, we have to balance two
risks: the "Type I" risk of a false alarm when the process is actually
under control, versus the "Type II" risk of failing to detect when the
process is really out of control.
The second kind of risk ("Type II error") is impossible to measure in
advance because it depends on how far out of control the process
actually is as well as the details of the situation..
Despite the inability to measure Type II risk, we know that if we
choose to accept a relatively high risk of a false alarm ("Type I
error") like 5 percent when the process is in control, we get in return
a lower risk of a type II error when the process is out of
control. If we insist on a very low risk of a false alarm like
1/4 of one percent (technically 0.26%) when the process is in
control, we must resign ourselves to a higher risk of a type II error
when the process is out of control.
To accept a 5% risk of Type I error to gain a lower risk of Type II
error, use "two sigma" control limits by setting z equal to 2 in the
equations in the middle of page 336. To insist on only 1/4 of one
percent risk of Type I error (thereby implicitly accepting a higher
risk of Type II error), use "three sigma" control limits by setting z
equal to 3 in the equations in the middle of page 336.
Western Jeans Company would be justified in choosing three sigma
control charts if they believed the business consequences of a
Type I error (false alarm) are much worse than the business
consequences of a Type II error. The justification given in the
book is actually meaningless.
TQM versus SPC:
Western Jeans Company is an example of Statistical Process Control
since it uses a p chart to detect deviations from a "standard" rate of
10% defective products. It is Not an example of Total Quality
Management because the goal there is to track characteristics of a
product in order to reduce the rate of actual defects is product of
service effectively to zero. A very simple example of the use of
p charts in TQM is the following:
The ButterYouUp company makes microwave popcorn; each bag in perfectly
filled if 95% of the kernels pop. If much more than that pop, the
bag overflows, which is not ideal for the customer, while if much less
than 95% pop the customer does not get a full bag. 25 samples of
100 kernels each are popped (1 per day for 26 days), with the following
number popping ineach sample:
91 91 96 98 99 95 95 92 96 99 100 88 94 97 95 91 98 95 92 97 97 94 97
98 93
Construct a p chart using two sigma control limits and discuss whether
or not this process is out of control.
Solution