IE 361 Module 18Process Capability Analysis: Part 2Prof.s Stephen B. Vardeman and Max D . MorrisReading: Section 5.3 Statistical Quality Assurance Methods for Engineers1In this module we consider methods of characterizing process output that focuson the values of individual future process outcomes rather than on processsummary measures.InferenceforOneMoreValuefromtheProcessand for the Location of "Most" of the Distributionof Process ValuesA way of characterizing process output different from trying to pin down processsummary parameters like the mean and standard deviation (or even functions2of those parameters like 6σ, Cp, and Cpk) is to instead try to quantify whatone has learned about future process outcomes from data in hand. If I KNOWprocess parameters, making statements about future individual values generatedby the process is a matter of simple probability calculation. Suppose, forexample, that I model individual values as normal with μ =7and σ =1.Then doing simple normal distribution calculations, both• there’s a "90% chance" the next x is between 5.355 and 8.645• 90% of the process distribution is between 5.355 and 8.645But what if I only have a sample, and not the process parameters? Whatthen can I say? When one has to use a sample to get an approximate picture3of a process, it is important to hedge statements in light of sample variabil-ity/uncertainty . . . this can be done• for normal processes usingx and s• in general, using the sample minimum and/or maximum valuesMethods for Normal ProcessesWe consider first methods for normal processes. (Just as we cautioned inModule 17 that the methods for estimating capabilities are completely unreliable4unless the data-generating process is adequately described by a normal model,so too does the effectiveness of the next 2 formulas depend critically on thenormal assumption being approp riate.)For normal processes, "prediction limits" for a single additional individualarex ± tss1+1nPrediction limits are sometimes met in Stat 231 in the context of regressionanalysis, but the simple one-sample limits above are even more basic and un-fortunately not always taught in an introductory course. They are intended tocapture a single additional observation from the process that generatedx ands.Example 18-1 (ED M drilling) What do the n =50measured angles in Table5.7 tell us about additional angles drilled by the process? One possible answer5can be phrased in terms of a 95% prediction interval for a single additionaloutput. (Recall that for the hole angle data ¯x =44.117◦and s = .984◦.)Using the fact that the upper 2.5% point of the t distribution for df ν =50 − 1=49is 2.010, 95% prediction limits for a single additional output are44.117 ± 2.010 (.984)s1+150i.e.44.117 ± 1.998One can in some sense be 95% sure that the next angle drilled will be at least42.119◦and no more than 46.115◦. This is a way of quantifying what thesample tells us about the angles drilled by the process different from confidencelimits for quantities like μ, σ, 6σ.Cpand Cpk. The 95% figure is a "lifetimebatting average" that is associated with a long series of repetitions of the6whole business of selecting n, making the interval, selecting one more value,and checking for success. (In any given application of the method one is either100% right or 100% wrong.)Another way to identify what to expect from future process outcomes mightbe to locate not just a single outcome, but some large fraction (p)ofallfutureoutcomes (under the current process conditions). For a normal process, two-sided "tolerance limits" for a large fraction (p) of all additional individualsarex ± τ2s(the values τ2are special constants tabled in Table A.9.a of SQAME ). (One-sided limits are similar, but use constants τ1from Table A.9.b of SQAME.)Tolerance limits a re not commonly taught in an introductory statistics course,so most students will not have seen this idea.7Example 18-1 continued A second possible answer to the question "What dothe n =50measured angles in Table 5.7 tell us about additional angles drilledby 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 ofSQAME producesamultiplierofτ2=2.58 (note that the table is set up interms of sample size, NOT degrees of freedom) and therefore the two-sidedtolerance limits44.117 ± 2.58 (.984)or44.117◦± 2.54◦for the bulk of all future angles (assuming, of course, that current processconditions are maintained into the future). (A one-sided tolerance limit canbe had by replacing τ2with a value τ1from Table A.9.b of SQAME.)8A "thought experiment" illustrating the meaning of "confidence" associatedwith a tolerance interval method involves1. drawing multiple samples2. for each one computing the limitsx ± τ2s3. for each one using a normal distribution calculation based on the trueprocess parameters to ascertain the fraction of the population covered bythesampleinterval4. checking to see if the fraction in 3. is at least the desired value p ... if itis, the interval is a success, if it is not, the interval is a failure9The confidence level for the method is then the lifetime batting average of themethod.Prediction and Tolerance Intervals Based on Sam-ple Minima and M aximaA second approach to making prediction and tolerance interval (that doesn’tdepend upon normality of the data generating process for its validity, only onprocess stability) involves simply using the smallest and largest data values inhand to state limits on future individuals. That is, one may use the interval(min xi, max xi)10as either a prediction interval or a tolerance interval. Provided the "randomsampling from a fixed (not necessarily normal) universe" model is sensible, usedas a prediction interval for one more observation this has confidence leveln − 1n +1andusedasatolerance interval for a fraction p of all future observationsfrom the process it has associated con fidence level1 − pn− n (1 − p) pn−1Example 18-1 continued The smallest and largest angles among the n =50in the data set of Table 5.7 are respectively 42.017 and 46.050.Weconsiderthe interval(42.017, 46.050)11for locating future observed angles. As a prediction interval for the next one,the appropriate confidence level is50 − 150 + 1= .961 = 96.1%And, for example, as a tolerance for 95% of EDM