Intervals & Tests
The following is an excerpt from The Quality Engineering Handbook by Thomas Pyzdek, © QA Publishing, LLC.
We have found that confidence limits may be determined so that the interval between these limits will cover a population parameter with a certain confidence, that is, a certain proportion of the time. Sometimes it is desirable to obtain an interval which will cover a fixed portion of the population distribution with a specified confidence. These intervals are called tolerance intervals, and the end points of such intervals are called tolerance limits. For example, a manufacturer may wish to estimate what proportion of product will have dimensions that meet the engineering requirement. In quality engineering, tolerance intervals are typically of the form , X-bar ± Ks, where K is determined, so that the interval will cover a proportion P of the population with confidence g. Confidence limits for m are also of the form X-bar ± Ks. However, we determine k so that the confidence interval would cover the population mean m a certain proportion of the time. It is obvious that the interval must be longer to cover a large portion of the distribution than to cover just the single value m. Table 11 in the Appendix gives K for P = 0.75, 0.90, 0.95, 0.99, 0.999 and g = 0.75, 0.90, 0.95, 0.99 and for many different sample sizes n.
Example of calculating a tolerance interval
Assume that a sample of n=20 from a stable process produced the following results: . We can estimate that the interval = 20 ± 3.615(1.5) = 20 ± 5.4225, or the interval from 14.5775 to 25.4225 will contain 99% of the population with confidence 95%. The K values in the table assume normally distributed populations.
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