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Unformatted text preview: IE 361 Module 18
Process Capability Analysis: Part 2 Reading: Section 5.3, Statistical Quality Assurance Methods for Engineers Prof. Steve Vardeman and Prof. Max Morris
Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 18 1 / 12 "Capability" and Future Values The measures 6, Cp , and Cpk considered in Module 17 attempt the summarize process "capability" in terms of a function of process parameters (and where appropriate, speci...cations for individual outcomes). In this module we consider methods of characterizing process output that focus what can be said about the values of individual future process outcomes based on data in hand (rather than about process summary measures). Vardeman and Morris (Iowa State University) IE 361 Module 18 2 / 12 "One More Value" or "Most of the Process Distribution"
If I KNOW process parameters, making statements about future individual values generated by the process is a matter of simple probability calculation. Suppose, for example, that I model individual values as normal with = 7 and = 1. there' a "90% chance" the next x is between 5.355 and 8.645 s 90% of the process distribution is between 5.355 and 8.645 But what if I only have a sample, and not the process parameters? What then can I say? When one has to use a sample to get an approximate picture of a process, it is important to hedge statements in light of sample variability/uncertainty . . . this can be done for normal processes using x and s in general, using the sample minimum and/or maximum values
Vardeman and Morris (Iowa State University) IE 361 Module 18 3 / 12 Methods for Normal Processes
Prediction Limits We consider ...rst methods for normal processes. (Just as we cautioned in Module 17 that the methods for estimating capabilities are completely unreliable unless the data-generating process is adequately described by a normal model, so too does the eectiveness of the next 2 formulas depend critically on the normal assumption being appropriate.) For normal processes, "prediction limits" for a single additional individual are r 1 x ts 1 + n Prediction limits are sometimes met in the context of regression analysis, but the simple one-sample limits above are even more basic and unfortunately not always taught in an introductory course. They are intended to capture a single additional observation from the process that generated x and s.
Vardeman and Morris (Iowa State University) IE 361 Module 18 4 / 12 Methods for Normal Processes
Example 18-1 (EDM drilling) What do the n = 50 angles in Table 5.7 say about additional angles drilled by the process? (Recall that here x = 44.117 and s = .984 .) One answer can be phrased in terms of a 95% prediction interval for a single additional output. Using the fact that the upper 2.5% point of the t distribution for df = 50 1 = 49 is 2.010, 95% prediction limits for a single additional output are r 1 i.e. 44.117 1.998 44.117 2.010 (.984) 1 + 50 One can in some sense be 95% sure that the next angle drilled will be between 42.119 and 46.115 . The 95% con...dence value is a "lifetime batting average" associated with a long series of repetitions of the whole business of selecting n, making the interval, selecting one more value, and checking for success. (In any given application of the method one is either 100% right or 100% wrong.)
Vardeman and Morris (Iowa State University) IE 361 Module 18 5 / 12 Methods for Normal Processes
Tolerance Limits Another way to identify what to expect from future process outcomes might be to locate not just a single outcome, but some large fraction (p) of all future outcomes (under the current process conditions). For a normal process, two-sided "tolerance limits" for a large fraction (p) of all additional individuals are x 2 s (the values 2 are special constants tabled in Table A.9.a of SQAME ). (One-sided limits are similar, but use constants 1 from Table A.9.b of SQAME.) Tolerance limits are not always taught in an introductory statistics course, so some students will not have seen this idea before. Vardeman and Morris (Iowa State University) IE 361 Module 18 6 / 12 Methods for Normal Processes
Example 18-1 continued A second answer to the question "What do the n = 50 angles in Table 5.7 say about additional angles drilled by the process?" can be phrased in terms of a 99% tolerance interval for 95% of all values from the process. (This would be an interval that one is "99% sure" contains "95% of all future values.") Reading directly in Table A.9.a of SQAME produces a multiplier of 2 = 2.58 (note that the table is set up in terms of sample size, NOT degrees of freedom) and therefore the two-sided tolerance limits 44.117 2.58 (.984) or 44.117 2.54 for the bulk of all future angles (assuming, of course, that current process conditions are maintained into the future). (A one-sided tolerance limit can be had by replacing 2 with a value 1 from Table A.9.b of SQAME.) Vardeman and Morris (Iowa State University) IE 361 Module 18 7 / 12 Interpretation of Tolerance Limits A "thought experiment" illustrating the meaning of "con...dence" associated with a tolerance interval method involves
1 2 3 drawing multiple samples, for each one computing the limits x 2 s, for each one using a normal distribution calculation based on the true process parameters to ascertain the fraction of the population covered by the sample interval, and checking to see if the fraction in 3 is at least the desired value p ... if it is, the interval is a success, if it is not, the interval is a failure 4 The con...dence level for the method is then the lifetime batting average of the method. Vardeman and Morris (Iowa State University) IE 361 Module 18 8 / 12 PI' and TI' Based on Sample Minima and Maxima s s
A second approach to making prediction and tolerance intervals (that doesn' depend upon normality of the data-generating process for its t validity, only on process stability) involves simply using the smallest and largest data values in hand to state limits on future individuals. That is, one may use the interval (min xi , max xi ) as either a prediction interval or a tolerance interval. Provided the "random sampling from a ...xed universe" model is sensible, used as a prediction interval for one more observation this has con...dence level n 1 n+1 used as a tolerance interval for a fraction p of all future observations from the process it has associated con...dence level 1
Vardeman and Morris (Iowa State University) pn n (1 p ) pn 1
9 / 12 IE 361 Module 18 PI' and TI' Based on Sample Minima and Maxima s s
Example 18-1 continued The smallest and largest angles among the n = 50 in the data set of Table 5.7 are respectively 42.017 and 46.050. We consider the interval (42.017, 46.050)
for locating future observed angles. As a prediction interval for the next one, the appropriate con...dence level is 50 1 = .961 = 96.1% 50 + 1 And, for example, as a tolerance for 95% of EDM drilled angles, the appropriate con...dence level is 1 (.95)50 50 (.05) (.95)49 = .721 = 72.1%
IE 361 Module 18 10 / 12 Vardeman and Morris (Iowa State University) PI' and TI' Based on Sample Minima and Maxima s s
Interpretation The interpretation of con...dence level associated with these intervals based on sample minimum and maximum values is exactly the same as for the normal distribution prediction and tolerance intervals based on x and s. The news here is that these levels are guaranteed for any (continuous) process distribution ... normal or not. We should also note that it is possible to employ only one of the values min xi and max xi to make one-sided prediction and tolerance intervals. The associated con...dence levels are given in the summary table on panel 12, and are (of course) larger than those for the (smaller) two-sided intervals. Vardeman and Morris (Iowa State University) IE 361 Module 18 11 / 12 Module Formula Summary Figure: Prediction and Tolerance Interval Summary
Vardeman and Morris (Iowa State University) IE 361 Module 18 12 / 12 ...
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This document was uploaded on 02/11/2012.
- Fall '09