# Essay on Home Work Solution of Stat-528

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Chapter 4 Exercise Solutions

Several exercises in this chapter differ from those in the 4th edition. An “*” following the exercise number indicates that the description has changed. New exercises are denoted with an “(”. A second exercise number in parentheses indicates that the exercise number has changed.

4-1.
“Chance” or “common” causes of variability represent the inherent, natural variability of a process - its background noise. Variation resulting from “assignable” or “special” causes represents generally large, unsatisfactory disturbances to the usual process performance. Assignable cause variation can usually be traced, perhaps to a change in material, equipment, or operator method.

A Shewhart control chart can be used to monitor a process and to identify occurrences of assignable causes. There is a high probability that an assignable cause has occurred when a plot point is outside the chart's control limits. By promptly identifying these occurrences and acting to permanently remove their causes from the process, we can reduce process variability in the long run.

4-2.
The control chart is mathematically equivalent to a series of statistical hypothesis tests. If a plot point is within control limits, say for the average [pic], the null hypothesis that the mean is some value is not rejected. However, if the plot point is outside the control limits, then the hypothesis that the process mean is at some level is rejected. A control chart shows, graphically, the results of many sequential hypothesis tests.

NOTE TO INSTRUCTOR FROM THE AUTHOR (D.C. Montgomery):
There has been some debate as to whether a control chart is really equivalent to hypothesis testing. Deming (see Out of the Crisis, MIT Center for Advanced Engineering Study, Cambridge, MA, pp. 369) writes that: “Some books teach that use of a control chart is test of hypothesis: the process is in control, or it is not. Such errors may derail self-study”.

Deming also warns against using statistical theory to study control chart behavior (false-alarm probability, OC-curves, average run lengths, and normal curve probabilities. Wheeler (see “Shewhart’s Charts: Myths, Facts, and Competitors”, ASQC Quality Congress Transactions (1992), Milwaukee, WI, pp. 533–538) also shares some of these concerns: “While one may mathematically model the control chart, and while such a model may be useful in comparing different statistical procedures on a theoretical basis, these models do not justify any procedure in practice, and their exact probabilities, risks, and power curves do not actually apply in practice.”

4-2 continued

On the other hand, Shewhart, the inventor of the control chart, did not share these views in total. From Shewhart (Statistical Method from the Viewpoint of Quality Control (1939), U.S. Department of Agriculture Graduate School, Washington DC, p. 40, 46): “As a background for the development of the operation of statistical control, the formal mathematical theory of testing a statistical hypothesis is of outstanding importance, but it would seem that we must continually keep in mind the fundamental difference between the formal theory of testing a statistical hypothesis and the empirical theory of testing a hypothesis employed in the operation of statistical control. In the latter, one must also test the hypothesis that the sample of data was obtained under conditions that may be considered random. … The mathematical theory of distribution characterizing the formal and mathematical concept of a state of statistical control constitutes an unlimited storehouse of helpful suggestions from which practical criteria of control must be chosen, and the general theory of testing statistical hypotheses must serve as a background to guide the choice of methods of making a running quality report that will give the maximum service as time goes on.”

Thus Shewhart does not discount the role of hypothesis