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Stat 511 Final Exam
May 5, 2004
Prof. Vardeman
-Z.?-Y~ In a study meant to determine the variability of diameters of widgets produced on a manufacturing line,
~ an engineer measures m = 10 widgets produced on the line once each. Then the engineer measuresthe
diameter of an 11thwidget n = 8 times. Supposeone models a measuredwidget diameter, y , as
y=x+F:
where x is the true diameter of the particular widget and F: is measurementerror, for x -N ( 0, 0"; )
independent of F:-N ( 0,0"2) .With
s~ the sample variance of the measurementson the 10 widgets,
Es~ = 0"; + 0"2, and with S2 the sample variance of the repeat measurementson the 11thwidget, E S2= 0"2. If
the engineer observes Sy= .05 mm and s = .01 mm, find approximate 90% confidence limits for 0"x. (Hint:
Cochran-Satterthwaite.)
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2. Printout #1 concerns the analysis ofdata taken from an article ofChowdhury 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 twolevel factors (that we here simply call A through J) were changed. The data on the printout concern counts
for 32 (out of 21°=1024 possible) combinations of Jevelsof these and a GLM analysis using a Poisson
model and the canonical link function h (Jl ) = In (Jl ) .
fJ?,!fSa) The first two models fit (to "shorts" and "dry solder") include main effects for the 10 two-Ievel factors A
~hrough
J. Why woul1 Jt be hopeless to try to fit a mpdell/i~ main effects and all 2 factor interactions?
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Based on the results for the first two models, the smaller 3rdand 4 models were fit. Use these smaller
models to answer the rest of the questions about this scenario.
,fJi:>~...b-)
Which factor seemsto have the biggest effect on shorts? Which factor seemsto have the biggest effect
Yon
dry solder defects?
biggest effect on shorts:~
biggest effect on dry solder defects:---1L
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.c)
For the 3rdmodel (the smaller model fit to "shorts") the first fitted/predicted value is 7.968834. Show
IJIiti, how this is obtained from the vector of estimated coefficients p .(Show calculation of this value.)
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-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 .ti t 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)
0.01388052
[1,)
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~ e) A~ it turns out, only factors A through I were factors whose levels could be set on the wave soldering
~machme.
(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? r;
t :t
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C level'
D level
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estimated mean shorts + dry
:
b
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3. Several nominally iden~1 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 ofa pneumatic wrench fed
from a single compressedair line, it is natural to expect the torques to be correlated. Printout #2 concerns
the estimation of the correlation based on these 34 pairs.
1 ,1:s a) Give a point estimate of the "population" correlation between Bolt 3 torque and Bolt 4 torque, a bootstrap
~
standarderror for that estimate, and a "bias-corrected" version of the estimate.
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standard error -( ~ IS
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bias-corrected estimate ~
b) What are 90% bootstrap percentile confidence limits for the population correlation?
~
(Report two
numbers.)
lower limit
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upper limit
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c) As it turns out, the "acceleration factor" for the bootstrap samples represented oil the printout is
a = -.02290961 .The lower limit of a 90% BCa confidence interval for the population correlation is at
approximately which percentile of the bootstrapped correlations?
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4. Printout #3 concerns the analysis of some very old data of Rumford taken to study t ~;;i...~
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 degreesF) and times (in minutes from the cessation of heating) is
~
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Yi =a+fJexp(-eti)+E:i
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 simp:lify.)
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Notice from the plot on the printout, that over the first 45 minutes or so of cooling the fitted curve does a
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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?)
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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 of3
Days (3 Batches per day). Each batch/test was run at a single temperature and included one battery of each
electrolyte. Let
Yii = the life of the battery with electrolyte i in test/batch}
d(})
= the number of the day on which test/batch}
is run
t ( } ) = the level of temperature at which test/batch} is run
Consider fixed effects 11,7]i'rl,7]ril (say, satisfying the sum restrictions) where the 7]'S represent Electrolyte
main effects, the r's represent Temperature main effects, and the 7]r's the Electrolyte by Temperature
interactions.
Suppose that .81,.82' ..., fJ9 are iid N ( 0, O"~) , independent of <51,
<52,<53that are iid N ( 0, 0"; ) ,
independent of e;s that are iid N( 0,0"2) .W,e will work under the model
Yii = 11+ 7]i + rl(J) + 7]rilW + fJi + <5
d(i) + eli
(In addition to the fixed effects, there are random day and batch/test effects, along with the basic
experimental error, e .)
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a) Give approximate 95% confidence intervals for 0" and 0"p .
for 0"
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for O"p
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b) H~w is it obvious from the printout that 0",} is poorly determined? In retrospect, why should that be no
surprIse?
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c) A bit of the printout is duplicated belov:' ..This gives 3 kind~ of predictio~s for the 'flrst case in the d.cita
set. Show how to get each of these 3 predIctlOns from other thmgs on the prIntout. (Show sums that gIve
these values.)
l'
> predict(battli~e,le,:el=0:2)
DayTest predJ.ct.fJ.xedp
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