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# PROB&INFERNC, ENGRS STAT 231

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This 14 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 231 at Iowa State University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/214402/stat-231-iowa-state-university in Statistics at Iowa State University.

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Date Created: 09/26/15

IE 361 Module 18 Process Capability Analysis Part 2 Profs Stephen B Vardeman and Max D Morris Reading Section 53 Statistical Quality Assurance Methods for Engineers In this module we consider methods of characterizing process output that focus on the values of individual future process outcomes rather than on process sum mary measu res Inference for One More Value from the Process and for the Location of quotMostquot of the Distribution of Process Values A way of characterizing process output different from trying to pin down process summary parameters like the mean and standard deviation or even functions 2 of those parameters like 60019 and Cpk is to instead try to quantify what one has learned about future process outcomes from data in hand lfl KNOW process parameters making statements about future individual values generated by the process is a matter of simple probability calculation Suppose for example that l model individual values as normal with u 7 and a 1 Then doing simple normal distribution calculations both 0 there s a quot90 chance the next 90 is between 5355 and 8645 0 90 of the process distribution is between 5355 and 8645 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 3 of a process it is important to hedge statements in light of sample variabil ityuncertainty this can be done 0 for normal processes using E and s o in general using the sample minimum andor maXimum values Methods for Normal Processes We consider first methods for normal processes Just as we cautioned in Module 17 that the methods for estimating capabilities are completely unreliable 4 unless the data generating process is adequately described by a normal model so too does the effectiveness of the next 2 formulas depend critically on the normal assumption being appropriate For normal processes quotprediction limitsquot for a single additional individual l 1 fits 1 n Prediction limits are sometimes met in Stat 231 in the context of regression analysis but the simple one sample limits above are even more basic and un fortunately not always taught in an introductory course They are intended to capture a single additional observation from the process that generated E and are 8 Example 181 EDM drilling What do the n 50 measured angles in Table 57 tell us about additional angles drilled by the process One possible answer 5 can be phrased in terms of a 95 prediction interval for a single additional output Recall that for the hole angle data 50 441170 and s 9840 Using the fact that the upper 25 point of the 75 distribution for df I 50 1 49 is 2010 95 prediction limits for a single additional output are 1 44117 l 2010 984 1 44117 l 1998 One can in some sense be 95 sure that the next angle drilled will be at least 421190 and no more than 461150 This is a way of quantifying what the sample tells us about the angles drilled by the process different from confidence limits for quantities like ua60Cp and Cpk The 95 figure is a quotlifetime batting average that is associated with a long series of repetitions of the 6 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 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 quottolerance limitsquot for a large fraction p of all additional individuals are E l 73928 the values 7392 are special constants tabled in Table A9a of SQAME One sided limits are similar but use constants 7391 from Table A9b of SQAME Tolerance limits are not commonly taught in an introductory statistics course so most students will not have seen this idea Example 181 continued A second possible answer to the question quotWhat do the n 50 measured angles in Table 57 tell us 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 quot99 sure contains quot95 of all future values Reading directly in Table A9a of SQAME produces a multiplier of 7392 2 258 note that the table is set up in terms of sample size NOT degrees of freedom and therefore the two sided tolerance limits 44117 l 258 984 or 44117 0 l 254 O 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 7392 with a value 7391 from Table A9b of SQAME 8 A quotthought experiment illustrating the meaning of quotconfidencequot associated with a tolerance interval method involves 1 drawing multiple samples 2 for each one computing the limits E l 73928 3 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 4 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 The confidence level for the method is then the lifetime batting average of the method Prediction and Tolerance Intervals Based on Sam ple Minima and Maxima A second approach to making prediction and tolerance interval that doesn t depend upon normality of the data generating process for its 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 907 maxxi 10 as either a prediction interval or a tolerance interval Provided the quotrandom sampling from a fixed not necessarily normal universe model is sensible used as a prediction interval for one more observation this has confidence level n 1 n 1 and used as a tolerance interval for a fraction p of all future observations from the process it has associated confidence level n l 1 pn n1 pp Example 181 continued The smallest and largest angles among the n 50 in the data set of Table 57 are respectively 42017 and 46050 We consider the interval 42017 46050 11 for locating future observed angles As a prediction interval for the next one the appropriate confidence level is 50 1 961 961 50 1 And for example as a tolerance for 95 of EDM drilled angles the appropriate confidence level is 1 9550 50 05 9549 721 721 The interpretation of confidence 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 E and s The news here is that these levels are guaranteed for any continuous process distribution normal or not 12 Section Formula Summary Below is a table summarizing the formulas of Section 53 of SQAME 13 Two Sided Intervals One Sided Intervals E tslloogt Pl ltf ts l l Ets lgt 1 lt oo ts 1 5 Normal Process TI E 73928 E 73928 3600 PI min 907 max 90721 min 907 or 00 max confzdence Z 1 confzdence 2 Any Stable Process min 907 max min 907 00 or 00 max Tl confidence 2 confidence 2 1 p n1 pp 1 1 p 14

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