-1-Stat/IE 531ExamIMarch 11, 1999Prof. Vardeman1. IE 361 students Grilliot, Chang and Wong worked with a company on the manufacture of somerubber hoses. In one part of their study, they monitored the lengths of hoses cut on an automatic saw.Below are some summary statistics taken from samples of hoses coming off a particular saw.#!8œ&(Units are inch above the target length.)"Î"'SampleSample.BVBV"Þ'"""!Þ!##"Þ!'"#!Þ!#$"Þ#""$Þ#"%•!Þ#$"%•!Þ%"&Þ'$"&•!Þ#$'Þ)$"'•!Þ%%(•!Þ#&"(Þ%")"Þ!#")•!'%*Þ#""*Þ##"!"Þ###!Þ#"For these hoses, and . A time series plot of the values follows.––––BœÞ#)Vœ#Þ%B 1.20+ * * B - - * * - * - 0.60+ * * - - * - * * * * - 0.00+ * * - - * * * - * * - -0.60+ * - +---------+---------+---------+---------+ 0 5 10 15 20 sample(a) Consider first the matter of assessing the behavior of the process short term standard deviation, .5Find retrospective control limits for , and say what they indicate about the behavior of the sawingVprocess over the period studied.(b) Consider now assessing the behavior of the process mean, . Find retrospective control limits for.B– and say what they indicate about the behavior of the hose cutting process over the period of the study.(c) Describe the plot of values. Do you see any indications of lack of process stability on the plot–B(that may or may not be detected by your limits from part (b))?-2-(d) Find an estimate of the process short term standard deviation based on . If one simply takes the–V"!! measurements that lead to the summary statistics given on the previous page and computes theirsample standard deviation, the value obtained is ( inch). How does this compare to your"Þ"$"Î"'estimate? Based on your answers to parts (a) through (c), argue that the relationship between thesefigures is about what one should expect.2. Suppose that in a context like that of Problem 1, one is going to CUSUM sample means . TheBideal or target value for is ...Bœ!(a) Apply a two-sided CUSUM scheme with , , and to the5œÞ&5œ•Þ&?œ@œ!2œ2œ%"#"#first values given in Problem 1. Are any alarms signaled in the first periods? Explain."!B"!Suppose for the rest of this problem that the "all-OK" process standard deviation is (in the units of5 Þ*Problem 1).(b) Among two-sided CUSUM schemes with head starts and the same on-target ARL as the scheme!from (a), the one from (a) is optimal for detecting what kind of a change in ?.(c) Approximately what is the "all-OK" ARL here? (Give parameters needed to enter a V&J table andsay which table. Don't waste time interpolating.)3. Consider the problem of monitoring integer-valued variables (we'll suppose that UßUßUßÞÞÞU"#$can take any integer value, positive, or negative). Define!2ÐBÑœ%•Bkkand consider the following definition of an alarm scheme: 1) alarm at time if , and3œ"U %kk" 2) for alarm at time if .3 #3U 2ÐUÑkk33•"Forinteger , let and suppose the are iid. Carefully describe how to find the ARL4;œTÒUœ4ÓU4"3for this situation. (You don't need to produce a formula, but you do need to set up an appropriate MCand tell me exactly/completely what to do with it in order to get the ARL.)4. Cylinders of (outside) diameter must fit in ring bearings of (inside) diameter , producingSMclearance . We would like to have some idea of the variability in actual clearances that willGœM•Sbe obtained by "random assembly" of cylinders produced on one production line with ring bearingsproduced on another.The gages used to measure and are (naturally enough) different.MSIn a study using a single gage to measure outside diameters of cylinders, different cylinders8œ"!Swere measured once each, producing a sample standard deviation inch. In a subsequent=œÞ!!"S-3-study, this same gage was used to measure the outside diameter of an additional cylinder 7œ&Stimes, producing a sample standard deviation inch.=œÞ!!!&SgageIn a study using a single gage to measure inside diameters of ring bearings, different inside8œ#!Mdiameters were measured once each, producing a sample standard deviation inch. In a=œÞ!!$Msubsequent study, this same gage was used to measure the inside diameter of another ring bearing7œ"!=œÞ!!"MM times, producing a sample standard deviation inch.gage(a) Give a sensible (point) estimate of the standard deviation of produced under random assembly.G(b) Find a sensible standard error for your estimate in (a). (No need to do the arithmetic necessary tosimplify.)5. The following is an ANOVA Table from a real Gage R&R study involving parts, operators and&%#‚ measurements for each partoperator combination.ANOVA TableSourcePartsOperatorsPartsOperatorsErrorTotalWW.0QW$!Þ)&!%(Þ("$!Þ#(&$!Þ!*#‚#Þ$&!"#!Þ"*'(Þ&!!#!!Þ$(&%!Þ*(&$*(a) Give point estimates of both "repeatability" and "reproducibility" standard deviations based on thetable above. Also give a standard error for the estimate of repeatability variation.(b) As a matter of fact, even a cursory look at the data summarized in the ANOVA table above leadsone to question the appropriateness of the standard -way random effects analysis in this situation. The#"measurements" are in fact (?rounded to?) integers. For example, the data for one of the parts were:Operator #1Operator #2Operator #3Operator #4"!"ß"!""!#ß"!!"!"ß"!""!!ß"!"In this part of the problem consider estimating the repeatability standard deviation based only the dataabove, treating them as rounded values from different normal distributions with possibly<œ%different means but a common standard deviation ..53i) Write out a prescription for a profile loglikelihood for based on the data above._55‡ÐÑß(Indicate appropriate maximization over the 's but don't try to carry that out.).3ii) Attached to this exam is a plot of . (The maximum value of is ,_5_5‡‡ÐÑÐÑ•"#Þ%$)attained at .) Based on this plot give both a point estimate and an approximate %5 œ"Þ"!)*!confidence interval for .5-4-Stat 531 Final ExamMay 3, 1999Prof. Vardeman1.