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IE 361 Homework Set #1 Fall 20041. An IE 361 in-class measurement demo nstration (using a fairly nice micrometer) producedsize measurem ents for inexpensive b inder clips. (The raw data are on the IE 361 HandoutPage for anyon e w ho is interested .) The un its of m easu re were " mm above 32.00 mm"(sothat 31.67 was read as −.33).• Single measuremen ts on n =10different bind er clips had sample meany = −.050 mmand sample standard deviation sy= .0254 mm.• m =7repeat measuremen ts on an additional single binder clip had sample meany = −.051 mm and sample standard deviation s = .0035 mm.In what follow s, w e’ll analyze th e results of this simple measurement study using the methodsgiven on the "Elementary Statistical Consid eration s in Metrology" han dou t.a) In Stat 231 you learned how to make (t)confidence intervals for means and (χ2)confidence intervals for standard deviations. Use the results from the 2nd part of thestudy and make (6-degrees-of-freedom) 90% confidence in tervals of these two t y pes.Whattwoquantitiesdothoseintervalsintendtobracket/estimateinthiscontext?b) Now consider the first part of the study. If I applied the Stat 231 formula for a (t)confid ence interval for a mean, what would I be attempting to estimate?c) Combine results from the two par ts of the study and estimate σx, the standard devi-ation of the “real” sizes of the binder clips (not including measuremen t error). Thenmake make appro x im ate 90% confidence limits for σx. What do these indicate aboutthe precision with whic h one has pinned dow n σx? If this precision seems unaccept-ably low, what w ould you have to do in order to improv e the precision with whic h σxis known?2. Co nsider pr oblem 2.5 of the text.a) Do parts b) through g) of this problem using the range-based form u las.b) Redo a) using the ANOVA-based formulas on the "Some Additional Notes on ANOVA-based Gauge R&R Estimation" handout.c) Use ANOVA-based form ulas from the handout and find 90% con fidence limits forσrepea t abilityand approxim ate 90% confidence limits for σrep rod u c ib ilityand for σR&R.Thenuse your confide nce limits for σR&Rto ma ke app roxim ate 9 0% limits for the gagecapabilit y ratio (the measurement "precision to tolerance" ratio). Wha t is to be doneif a company findsthislastintervaltobeuninformative?3. Below is a data set from a real calibration study (taken from a paper by John Mandelof NBS/NIST). Unfortunately, the units are not given in the paper. For sak e of argumen t,suppose that they are ppm (parts per million) of some conta m inant. x is the “truth”/gold-standar d-m easu rem ent. y is the local laboratory measurem ent. All on n =14differen tspecimen s.1x y647 605728 6751039 9651095 9951116 10181194 11171557 14221594 14701896 17621983 17392136 19182192 19832224 20082244 2010a) Fit a simp le linear regression model to these data. For a fixed x/specimen, what do youestimate “σmeasurement” to be at the local laboratory? What “conv ersion formula” doyou recommend for translating “local lab measurem ents” to estimated “gold standardmeasurements”?b) A new specimen is measu red as y =2000at the local laboratory. Give an app roxim a te95% confidence in terval for the “true”/gold standard valu e for this specimen. Do thist wo way. First sim ply read one interval off a plot of 95% prediction intervals for anadditional y at various fixed values of x. Then use the approximate confidence in tervalform ula giv en in class and on the “Class Outline” handout.You can get help with using the JMP statistical package b y looking ath ttp://www.stat.iastate.edu/resources/jmp.h tmlor by using the statistical software primer written for Vardeman and Jobe’s Basic Engine eringData Colle ction and A nalysis av ailable both ath ttp://www.duxbury.com/default.h tm(under the “Book Com pa nio ns”) and in a local/developmen t v ersion ath ttp://www.public.iastate.edu/∼vardeman /book_site/index.htmlTo do the ANO VA part of Problem 2, you will need to en ter 3 columns of length 60 in tothe JM P w o rksheet. Th e first should giv e part n umbers, the second should give operatorn u mbers and the third should give the measur em e nts. After enterin g the data, click on the“Part” column heading, go to the “Cols” menu and choose “Column Info.” There m ake surethat the “Modeling Type” is “Nominal.” Do the sam e for the “Operator” column. Thenfrom the “A nalyze” men u choose “Fit M odel.” You’ll get a dialogue bo x. The “Measure-ment” variable gets entered into the “Y” part of the box . The “Pa rt” and “Operator”variables get entered in to the “Construct Model Effects” part of the box. Then b y high-lighting the “P ar t” variable in the “Select Column s” list and the “Operator” variable alread yin the “Construct Model Effects” part of the box and clickin g the “Cross” button, y ou can2add the in tera ction effects to the model. Then clickin g “Run Model” will get you a JMPanalysis. The “Effect Tests” part of the report con tains the necessary sums of squares anddegrees of freedom to make the necessary estimates.To do Problem 3, you will need to enter 2 columns of length 14 in to a new data table. Thefirst should give x and the second should giv e y. After en tering the data, click on “Fit Y b yX” under the “Analyze” menu. Put the x variable in the “Factor” part of the dialogue boxand the y variable in the “Response” part of the dialogue box and click on “OK .” This willbring up a scatterplot (tha t can be re-sized to improve resolution if y ou wish). Click the redtriangle on the bar abo ve the plot and bring up a men u. Select “Fit Line.” This will producea SLR analysis for these data (and plot the least squares line). If you then clic k on the redtriangle b y the “Linear Fit” bar below the plot, y o u can bring up a m enu that includes anen try “Confid Curv es Indiv.” Checking that option will put 95% prediction limits for anadditional y at the various x’s on the plot. You can then select the cross-hair tool fromthe JMP toolbar and read off values on these plots of prediction limits. Other values you ’llneed (like√MSE) can be read from the JM P report. Simple descriptiv e statistics for thecolum ns (likex and sx) can be gotten b y c hoosing “Distribution” from the “Ana lyze” menu.3IE 361 Homework Set #2 Fall 2004Do problems 3.18 and 3.22 of the


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