Detectives at a crime scene look for something "out of place" -- that is something that you should not expect. Something unexpected is often connected with events that went wrong. This is true of crime, true of aircraft accidents and true during bolt manufacture.
I offer two Industrial examples where this kind of thinking is based on statistics.
Statistical process control is routinely used in automotive and aircraft part manufacture. First, production parts are statistically characterized. Then a sample is taken periodically, to make sure it is within the expected range. When a VERY UNLIKELY part is found, the production line is checked to make sure everything is still working correctly. A Gaussian Distribution (see chapter 4) is used for this kind of analysis.
The bolt example from Chapter 2 exactly fits the Binomial use criteria.
So, using methods from Chapter 3, we can compute the likelihood of 4 samples out of 12 being bad. If the probability is less than 1%, then it is unlikely we should find such a sample. We should suspect that something in the production machinery is out of adjustment.
If the bolt is not strong enough, we would check the cool down "times and temperatures" during the forging process. We would also check the certifications of the metal being used to make the bolt. If necessary, we would send the bolt to a metals lab to check the chemical composition and other properties.
If no problem is found with the manufacturing process, we would check the next 30 bolts to make sure they are good, before proceeding with "business as usual".
While this may seem like a lot of trouble, it is far cheaper than making 10,000 bolts that have to be thrown away.
Conclusion of Sidebar Discussion
Statistics is usually used to predict what we will find. What is unique about the bolt problem is that statistics is being used to DETECT a change in a process. The INDICATION to check the process is a higly improbable part. Statistics is thus being used in a reverse kind of way to say "This can't be happening. Something is wrong!"
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Paul F. Watson