Stat 231 Final Exam
Fall 2010
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1. In a 1992 Journal of Quality Technology paper, Eibl, Kess, and Pukelsheim studied a number of
different set-ups for a painting process and their impacts on the variable
y  measured painting coat thickness (mm)
(several pieces of material were painted and evaluated for each set-up considered). Some summary
statistics for pieces from each of 5 particular set-ups included in the study are below.
Set-up #1
n1  4
y1  .980
s1  .146
Set-up #2
n2  4
y2  1.525
s2  .100
Set-up #3
n3  4
y3  1.130
s3  .207
Set-up #4
n4  4
y4  1.735
s4  .078
Set-up #5
n5  4
y5  1.490
s5  .080
Initially, consider ONLY the information from the study about Set-up #1, and assume that for
this set-up measured paint thickness is normally distributed.
5 pts
a) Give two-sided 95% prediction limits for measured paint thickness on a single additional item
painted under Set-up #1. (Plug in completely, but you need not simplify.)
5 pts
b) Give endpoints of a two-sided interval that you are 95% sure contains at least 99% of all
thicknesses for items painted under Set-up #1. (Plug in completely, but you need not simplify.)
Now consider ONLY Set-ups #1 and #2, under the assumption that thicknesses of paint on
items painted under either set-up are normally distributed.
5 pts
c) Give endpoints of a two-sided 90% confidence interval that compares how consistent Set-ups #1
and #2 are in terms of measured paint thickness produced. (Plug in completely, but you need not
simplify.)
2
5 pts
d) Give 95% two-sided confidence limits for the difference in mean measured paint thicknesses
produced by Set-ups #1 and #2. (Plug in completely, but you need not simplify.)
Now consider the results from ALL of Set-ups #1 through #5.
5 pts
e) For yij  measured thickness for piece j from Set-up i , below is a normal plot of all 20 values

3
 yi  /  sPooled
 . Say what it indicates to you about the appropriateness of analyzing these
4


experimental results based on the one-way normal model.
y
ij
As it turns out, in this problem SSTr  1.525 and SSE  .260 .
5 pts
f) Give the values of sPooled , and of an F statistic for testing H 0 :1  2  3  4  5 along with its
associated degrees of freedom.
sPooled  ____________
F  __________
df  _______ , _______
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Henceforth, if you couldn't do part f), you may use the incorrect value of sPooled  .20 .)
5 pts
g) Give two-sided 95% confidence limits for the difference in mean measured paint thicknesses
produced by Set-ups #1 and #2 under the one-way model assumptions. (Plug in completely, but you
need not simplify.)
5 pts
h) Set-ups #1 and #3 involve a high belt speed while the others involve a low belt speed. So it is
possible that the linear combination of means
1
1
1
1
1
2
3
2
3
3
L  1  2  3  4  5
might be of interest. What is an appropriate margin of error for estimating this with 95%
confidence? (Plug in completely, but you need not simplify.)
2. Below is a pdf, f  x  , for a simple continuous distribution.
 x if  1  x  1
f  x  
 0 otherwise
Suppose that a random variable X has pdf f  x  given above.
5 pts
a) Evaluate P  .5  X  .5 .
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5 pts
b) What are the values of EX and VarX ?
EX  ____________
5 pts
VarX  _____________
c) Suppose that X is the sample mean of n  25 independent variables, each with pdf f  x  .
Approximate P  .5  X  .5 . (If you were unable to do part b) you may use the incorrect value
VarX  .7 here.)
3. Nine one-inch holes are to be drilled in a steel hydraulic cylinder barrel. This part is fixtured
once on a CNC machining center and all nine holes are drilled one after another. Of interest is the
random variable
W  the number of holes that fail to meet engineering tolerances for radial position
5 pts
a) What feature of the "Bernoulli trials" model seems least appropriate here? Explain.
For the next part of the question, ignore any misgivings raised in part a) about the usefulness
of a Bernoulli trials model here.
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5 pts
b) If one judges that the chance that any single one of the nines holes on a barrel fails to meet
engineering tolerances for radial position is 5% , find P W  2 based on a Bernoulli trials model.
5 pts
c) Suppose that in fact production records show that among the last 100 barrels inspected, 7 had at
least one hole failing to meet engineering tolerances for radial position. Give 95% two-sided
confidence limits for the fraction of all barrels produced on this machine that have at least one hole
failing to meet specification for radial position. (Plug in completely, but you need not simplify.)
4. Attached at the end of this exam are some pages of JMP reports useful in the analysis of some
data of Heinz, Peterson, Johnson, and Kerk concerning body dimensions measured on n  145
males ages 18-30. We'll suppose here that one would like to quantify how
y  subject weight (kg)
varies with the 22 other variables (all measured in cm) for such men.
5 pts
a) What single predictor variable is most effective at explaining the observed variation in y ?
Explain. What is the sample correlation between y and this predictor? (Make a reasonable
assumption about the sign of this correlation.)
Most effective single predictor:
Why:
r  ____________
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There is a Fit Y by X output for inference based on the model
y   0  1  waist.girth  
included in the JMP reports. Use it as you answer the questions b) through d).
5 pts
b) Give 95% confidence limits for the standard deviation of weights (in kg) of males 18-30 that
have a particular waist girth. (Plug in completely, but you need not simplify.)
5 pts
c) Remember that in the original data, the units of weight are kg and length has units cm. There are
roughly 2.54 cm per inch and 2.205 lbs force per kg force. Give 95% confidence limits for the
increase in mean subject weight in lbs that accompanies a 1 inch increase in waist girth for males
18-30 years of age.
5 pts
d) On the plot on the Fit Y by X report, there are several  y, waist.girth  points that plot
outside the dotted lines. Is this a concern? If so, why, and if not why not? (Circle the correct
response below and explain.)
This IS a concern.
This IS NOT a concern.
Why:
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There are two Fit Model outputs included in the JMP reports. Use them as appropriate as you
answer the questions e) and f).
5 pts
e) The two models represented on the JMP reports involved respectively 2 and 4 predictor
variables. They have the largest R 2 values for models of their respective numbers of predictors. If
it is possible to judge whether the increase in R 2 for the second compared to the first is statistically
significant using an F test, find the value of statistic and give degrees of freedom. If it is not
possible to use an F test based on the given information, very carefully say why.
5 pts
f) If you were going to drop one predictor from the 2nd model (to produce a model using 3
predictors) which one would it be, and why? If you did this, would you have the 3-predictor model
that has the highest R 2 among all possible 3-predictor models (including those using other
predictors)? Explain.
Predictor to drop:
Why:
Circle one of the following:
This IS the best model of size k  3 .
This IS NOT the best Model of size k  3 .
Explain:
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