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The following is an excerpt from The Quality Engineering Handbook by Thomas Pyzdek, © QA Publishing, LLC.
A machine is supposed to produce parts in the range of 0.500 inches plus or minus 0.006 inches. Based on this, your statistician computes that the absolute worst standard deviation tolerable is 0.002 inches. In looking over your capability charts you find that the best machine in the shop has a stand-ard deviation of 0.0022, based on a sample of 25 units. In discussing the situation with the statistician and management, it is agreed that the machine will be used if a one-sided 95% confidence interval on sigma includes 0.002.
The correct statistic for comparing a sample standard deviation with a
standard value is the chi-square statistic. For our data we have s=0.0022, n=25, and s0=0.002. The c2 statistic has n-1 = 24 degrees of freedom. Thus,
Table 7 gives, in the 0.95 column (since we are constructing a one-sided confidence interval) and the df = 24 row, the critical value c2 = 36.42. Since our computed value of c2 is less than 36.42, we use the machine. The reader should recognize that all of these exercises involved a number of assumptions. E.g., that we "know" that the best machine has a standard deviation of 0.0022. In reality, this knowledge must be confirmed by a stable control chart.
Intervals & Tests