IE 361 Exam 1. b) Give *&% confidence limits for the bias of this viscometer. (No need to simplify.)

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1 October 9, 00 IE 6 Exam Prof. Vardeman. The viscosity of paint is measured with a "viscometer" in units of "Krebs." First, a standard iquid of "known" viscosity *# Krebs is tested with a company viscometer (by a singe empoyee). a) "! measured vaues have a mean of *$Þ' Krebs and standard deviation of Þ& Krebs. What do the *&% "" > confidence imits *$Þ' #Þ#'#ÐÞ&Î È"!Ñ then estimate? (Use 0 words or ess.) b) Give *&% confidence imits for the bias of this viscometer. (No need to simpify.) c) Give 95% confidence imits for the "repeatabiity" standard deviation of this viscometer. (No need to simpify.) d) Suppose that the "reproducibiity" standard deviation for use of the viscometer by company empoyees is about Þ) Krebs. What woud you expect for a standard deviation of measured viscosities if many different empoyees measure the viscosity of the standard iquid once each? & sampes of paint dipped from different ocations in singe vat have measured viscosities with mean 9.6 Krebs and standard deviation. Krebs. (A singe empoyee does a measuring.) e) Using both sets of measurements, estimate a standard deviation of actua viscosities (not incuding measurement error) of such sampes dipped from this vat of paint.

2 f) A standard error for the estimate in e) is approximatey Þ%% Krebs. What does this figure indicate about the reiabiity of your answer to e)? (Use 0 words or ess.). Kumaa, Nithang, Pramadi and Simpson worked with a company on improving a paperwork process. They deveoped a method of scoring the correctness of empoyee handing of a particuar kind of transaction on a scae from! to "!!. A arge number of transactions were scored by this method and initia standards of. œ(! and 5 œ## described the strongy eft-skewed distribution of scores at the beginning of their invovement. a) What probabiity fact from Stat suggests that (if the sampe size is not too sma) despite the fact that the distribution of scores was skewed, "usua" contro imits may be used for B? (Use 0 words or ess.) b) Suppose that each month, 8œ&! of these transactions wi be samped and scored. Find standards given contro imits for the sampe mean score. PGP œ YGP œ B c) Management ceary wanted high scores, and in particuar individua scores beow *! were considered "bunders." Your ower contro imit in b) shoud be beow *! and you shoud have an upper contro imit that is ess than "!!. BRIEFLY say why such imits are not incompatibe with management's goas for transaction scores. B

3 . Oberding, Pausen, Schreiner and Wiiams worked with a company on the miing of a high precision meta part. Beow are means and ranges for a critica dimension of this part from periodic sampes of size 8œ%. The units are Þ!!!" inchabove Þ*%*! inch. sampe B V " # $ % & ' ( ) * "! ! Bœ*(Þ#&! Vœ"!) a) Find retrospective contro imits for both sampe means and ranges. For B: For VÀ b) Is there any evidence of process instabiity in the vaues given in the tabe? Expain. c) Supposing the process to be stabe over the period of study, give estimates of the process mean and standard deviation. d) Engineering specifications on the dimension being monitored were to (in the units above). Use this "! $! fact, the estimates from c) and a norma distribution assumption to approximate the fraction of parts meeting engineering specifications.

4 November, 00 IE 6 Exam II Prof. Vardeman. Bar stock of height L and depth H is crosscut at an ange ) to the front face as pictured beow. The area of the face produced by the cut is Eœ LH sin) Suppose that L, H and ) vary with means. L œ"þ!! in,. H œ"þ!! in and.) œ # radians and standard deviations 5L œþ!" in, 5H œþ!" in and 5 ) œ.0 radians. Find an approximate mean and standard deviation for E (assuming that L, H and ) are independent). (Pug in competey, but no need to simpify.). In a rea appication of contro charting with attributes data, a company monitored the number of customer compaints received on their "customer compaint hotine" for each of its food products. a) The "standard" compaint rate for a particuar product is $ per week. What then are sensibe contro imits for the number of compaints on this product received in a "$ week "quarter"? 4

5 b) A company executive argues that "the correct" way to monitor compaints is in terms of "compaints per unit sod" rather than "compaints per unit time." Last year, roughy "ß!!!ß!!! units of a product were sod and &!! compaints received. Using the executive's idea, what are sensibe contro imits for the number of compaints in a month where "!!ß!!! units of this product are sod?. IE 6 students Oberding, Pausen, Schreiner and Wiiams studied the performance of a CNC machining process. A sampe of 8œ%! parts was produced and a particuar critica dimension measured on each. Summary statistics for these (in units of Þ!!!" inches above Þ#!)! inch) were Bœ)!Þ!& and =œ%þ'&. In these units, engineering specifications on this dimension were )! #!. a) Give imits that you are *&% sure contain **% of such dimensions. (No need to simpify.) b) Give a *&% ower confidence bound for a process capabiity index that measures current performance (not potentia). (Pug in, but again you need not simpify.) c) Here is a norma pot of the students' data. Say what it indicates about the practica usefuness of the answers to a) and b) above and use it to expain the "process capabiity" to a non-quantitative person. 5

6 4. A (rea) US government (FDA) draft guideine prescribing the monitoring of the concentration of white bood ces in certain fitered bood products cas for the weeky testing of 8œ& units. "Lack of contro" is to be decared if any of these units has more than a specified concentration of white ces. Supposedy, a company is in compiance with government guideines if &% or ess of a units produced have more than the specified concentration of white bood ces. A certain producer of these bood products is in fact consistenty producing ony &% of units that are unacceptabe in terms of the white ce concentration (and is therefore actuay "in compiance" with FDA guideines). What is the mean number of weeks the suppier wi test unti first having an "out of contro" resut? 5. Ideay, two varaibes B" and B# measured on a widget have means." œ& and.# œ(, standard deviations 5" œ# and 5# œ# and correation "# œ Þ#&. Process monitoring wi be done based on sampes of ony 8œ" widget. a) How do you suggest montioring B " aone? (Say exacty what you woud pot and what imits you woud use.) b) How do you suggest monitoring both B" and B# if the reationship between the two is physicay important? (Say exacty what you woud pot and what imits you woud use.) 6

7 December 6, 00 IE 6 Exam Prof. Vardeman. (Motivated by Reduction of Testing Time of a Chemica Parameter Using Design of Experiments by Rao and Roshan, Quaity Engineering, 4(), ) In the production of a rayon-grade pup in the poy-fiber industry, measuring pup soubiity in 7.4% caustic is important. Two steps (A and B) in the standard measurement process take respectivey 0 minutes and 80 minutes. An experiment is done to see if these times can be reduced by 0 minutes each without ruining measurement consistency. 9 pup sampes are prepared and spit. Soubiity of one part is measured via the standard method, and in a second measurement one or both of times for A and B are reduced. For each (spit) pup sampe the response measured soubiity under the reduced-time regimen y = measured soubiity under the standard regimen is recorded. Beow are summary statistics for the study. Reduced A Time Ony Reduced B Time Ony Reduced A and B Times n = n = n = y =.9765 y =.960 s =.00 s =.087 s =.044 Begin by assuming the reevance of the one-way norma mode for the r = aternative measurement regimens. a) Give an estimate of σ in this mode and say what it represents (interpret it!). y =.909 b) Note that µ = woud mean that the Reduced B Time Ony regimen produces vaues ike the standard regimen. Based on appropriate 95% confidence imits, is this pausibe? c) Do the first and third regimens produce detectaby different mean vaues of y? (Refer to appropriate 95% imits.) d) The summary statistics suggest that the usua one way mode may not be appropriate. More importanty, what do these suggest about the possibiity of measuring soubiity with one of the reduced-time methods and then correcting back to the standard regimen by dividing by an estimate of µ? (Which of the r = aternatives seems best for this? Why?) i

8 . Assume that each of the interaction pots above represents a different factoria study. Fi in each empty ce in the tabe beow with the number of a singe pot that has the indicated effects. A main effects present? B main effects present? AB interactions present? Pot Number No No No Yes No No No Yes No Yes Yes No No No Yes Yes No Yes No Yes Yes Yes Yes Yes. Now consider pairs of interaction pots: first # and #, then # and #4. Suppose that the first pot in each pair represents system response at the ow eve of factor C and the second represents system response at the high eve of factor C. Do the pairs represent situations where there is non-zero ABC -factor interaction? # and #? Yes or No (circe one) # and #4? Yes or No (circe one) 4. In a fu factoria study, four sampes of size and four of size produce a pooed sampe standard deviation of s P =.. How arge does a fitted effect from the Yates agorithm have to be (in absoute vaue) in order to be judged statisticay detectabe? (You may use 95% confidence.)

9 5. Beow are 8 sampe means isted in Yates standard order for factors A, B and C. a) Find the grand mean and fitted effects for the a high treatment combination ( a, b, ab, c, ac, bc, abc ). Mean b) If the means above come from a compete factoria study and ony effects arger in magnitude than.5 are big enough to be considered to be statisticay detectabe, what do you estimate as a mean response for a situation where A is high, B is ow, and C is high? 6 c) Suppose now that in fact the means above come from a fractiona factoria with generators D AB, E BC and F AC. What is the simpest possibe interpretation of the resuts of your cacuations in a) in ight of the criterion in b) that fitted effects arger in magnitude than.5 represent detectabe (sums of) effects? d) Write out the whoe defining reation for the fractiona factoria pan referred to in part c). I