Stat 511 Final Exam May 5, 2004 Prof. Vardeman 1. In a study meant to determine the variability of diameters of widgets produced on a manufacturing line, an engineer measures 10m = widgets produced on the line once each. Then the engineer measures the diameter of an 11th widget 8n = times. Suppose one models a measured widget diameter, y, as yxε=+ where x is the true diameter of the particular widget and ε is measurement error, for ()2N0,xxσ∼ independent of ()2N0,εσ∼. With 2ys the sample variance of the measurements on the 10 widgets, 222Eyxsσσ=+, and with 2s the sample variance of the repeat measurements on the 11th widget, 22E sσ=. If the engineer observes .05 mmys = and .01 mms = , find approximate 90% confidence limits for xσ. (Hint: Cochran-Satterthwaite.) 2. Printout #1 concerns the analysis of data taken from an article of Chowdhury and Mitra on a study intended to reduce defects produced by a wave soldering machine in an electronics plant. Numbers of “shorts” and numbers of “dry solder” defects were counted on a standard unit of product, as levels of 10 two-level factors (that we here simply call A through J) were changed. The data on the printout concern counts for 32 (out of 102 1024= possible) combinations of levels of these and a GLM analysis using a Poisson model and the canonical link function ()()lnhµµ= . a) The first two models fit (to “shorts” and “dry solder”) include main effects for the 10 two-level factors A through J. Why would it be hopeless to try to fit a model with main effects and all 2 factor interactions? Based on the results for the first two models, the smaller 3rd and 4th models were fit. Use these smaller models to answer the rest of the questions about this scenario. b) Which factor seems to have the biggest effect on shorts? Which factor seems to have the biggest effect on dry solder defects? biggest effect on shorts:__________ biggest effect on dry solder defects:__________c) For the 3rd model (the smaller model fit to “shorts”) the first fitted/predicted value is 7.968834. Show how this is obtained from the vector of estimated coefficients β. (Show calculation of this value.) d) Some R code (and what is returned by the program) in this context is below. Use the result and show why a sensible se.fit for the first fitted value is 0.9388530. (Hints: “delta method” and “link function.”) > t(c(1,-1,1,-1))%*%vcov(glm.out3)%*%c(1,-1,1,-1) [,1] [1,] 0.01388052 e) As it turns out, only factors A through I were factors whose levels could be set on the wave soldering machine. (Factor J was a factor whose levels fluctuated beyond the control of the process operators.) What levels of Factors C,D,E, and H do you recommend for future operation of the machine, if one wishes to minimize the total of shorts and dry solder defects at the worst level of J? What do you estimate to be the mean total under the conditions you recommend? C level __________ D level __________ E level __________ H level __________ estimated mean shorts dry solder defects+ at the worst level of J ___________________________ 3. Several nominally identical bolts are used to hold face-plates on a model of transmission manufactured by an industrial concern. Some testing was done to determine the torque required to loosen bolts number 3 and 4 on 34 transmissions. Since the bolts are tightened simultaneously by two heads of a pneumatic wrench fed from a single compressed air line, it is natural to expect the torques to be correlated. Printout #2 concerns the estimation of the correlation based on these 34 pairs. a) Give a point estimate of the “population” correlation between Bolt 3 torque and Bolt 4 torque, a bootstrap standard error for that estimate, and a “bias-corrected” version of the estimate. estimate __________ standard error __________ bias-corrected estimate __________b) What are 90% bootstrap percentile confidence limits for the population correlation? (Report two numbers.) lower limit __________ upper limit __________ c) As it turns out, the “acceleration factor” for the bootstrap samples represented on the printout is ˆ.02290961a =− . The lower limit of a 90% aBC confidence interval for the population correlation is at approximately which percentile of the bootstrapped correlations? (Show your work.) 4. Printout #3 concerns the analysis of some very old data of Rumford taken to study the cooling of a large cannon barrel that had been heated to a uniform (high) temperature by friction. Newton’s Law of cooling predicted that when the source of heating was removed, the temperature would decline exponentially to the ambient temperature (that was apparently about 60 F on the day these data were collected). The model fit to the temperatures (in degrees F) and times (in minutes from the cessation of heating) is ()expiiiytαβθε=+−+ a) Give approximate 90% prediction limits for an unrecorded temperature at 45 minutes based on this model. (Plug numbers in everywhere, but you need not simplify.) b) Notice from the plot on the printout, that over the first 45 minutes or so of cooling the fitted curve does a nice job of tracking the plotted points. There is, however, clear statistical evidence that this trend can not continue. Explain carefully/statistically. (The real temperature series had to eventually decline to about 60 F … is that consistent with what has thus far been observed?)5. Printout #4 concerns the analysis of some data from a test of thermal battery lives. Batteries using 4 different Electrolytes were tested at 3 different Temperatures, in 9 different Batches/Tests, over a period of 3 Days (3 Batches per day). Each batch/test was run at a single temperature and included one battery of each electrolyte. Let ()()the life of the battery with electrolyte in test/batch the number of the day on which test/batch is runthe level of temperature at which test/batch is runijyijdj jtj j=== Consider fixed effects , , ,it itµητητ (say, satisfying the sum restrictions) where the 'sη represent Electrolyte main effects, the 'sτ represent Temperature main effects, and the 'sητ the Electrolyte by Temperature interactions. Suppose that 12 9,,,βββ… are iid ()2N0,βσ, independent of 123,,δδδ that are iid ()2N0,δσ,