Stat 231 Final Exam
Fall 2011
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1. An experiment was run to compare the fracture toughness of high purity 18% Ni-maraging steel
to that of commercial purity steel of the same type. Summary statistics for the measured toughness
of several specimens of each type are below. (Units are MPa m .)
High Purity
n1 = 10 specimens tested
Commercial Purity
n2 = 8 specimens tested
x1 = 65.6
s1 = .8
x2 = 59.8
s2 = 1.1
10 pts
a) Give a one-sided lower 95% confidence bound for the ratio of standard deviations of toughness
for specimens of this type, σ 1 / σ 2 . (Plug in completely, but you need not simplify.)
15 pts
b) Because it is expensive, an engineer wants to specify High Purity steel for use in a project only
if its mean fracture toughness exceeds that of Commercial Purity steel by more than 4 MPa m .
With α = .10 do the data summarized above force the conclusion that High Purity steel should be
specified? (Show the whole 7-step hypothesis testing format/process.)
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10 pts
c) Give 95% two-sided confidence limits for the difference in mean fracture toughness for
specimens of these two types of steel (High Purity minus Commercial Purity). (Plug in completely,
but you need not simplify.)
10 pts
d) Assuming that fracture toughness measurements of specimens of Commercial Purity steel are
normally distributed, give two-sided limits that you are 95% sure contain 99% of such
measurements. (Plug in completely, but you need not simplify.)
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2. Company officials at a meat processor are concerned with the fraction of packages of sausage
filled by a particular machine that contain more than 16.1 oz of sausage. They plan to weigh the net
contents of a sample of packages filled by the machine.
10 pts
a) If 15 packages are to be weighed and roughly 30% of all packages actually contain more than
16.1 oz of sausage, find the probability that 8 or more in the sample hold more than 16.1 oz of
sausage.
7 pts b) Find a sample size that would guarantee that a 95% confidence interval for the fraction of all
packages filled to more than 16.1 oz will have width no more than .02. (The interval will be of
form p̂ ± Δ for Δ no larger than .01.)
8 pts c) Company officials eventually settle on a sample size of 400 and find 150 (of 400) packages
inspected contain more than 16.1 oz of sausage. Give 95% two-sided confidence limits for the
fraction of all packages filled by this machine that contain more than 16.1 oz of meat (assuming that
it is behaving in a physically stable manner).
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3. One measure of air pollution is the amount of beta radioactivity in the air (e.g. measured in
μ Ci / m3 ). Below are summary statistics from 5 pollution measurement stations (all located within
10 feet of each other at a given site) at 4 different Ames locations on a single summer day.
Location 1
n1 = 5
x1 = 3.1
s1 = .1
Location 2
n1 = 5
x2 = 2.9
s2 = .3
Location 3
n3 = 5
x3 = 3.4
s3 = .1
Location 4
n4 = 5
x4 = 3.0
s4 = .2
13 pts
a) Give 2-sided 95% confidence limits for the standard deviation of pollution measurements at a
fixed location in Ames, based on the values above (and the assumption that measurements at a
given location are normal with a standard deviation that is common across locations). Use
information from ALL 4 locations in the calculation.
12 pts
b) Locations 1 and 2 are at schools in residential areas, and locations 3 and 4 are in commercial
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districts. What is a "margin of error" for ( x3 + x4 ) − ( x1 + x2 ) based on 95% confidence limits
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for the quantity ( μ3 + μ4 ) − ( μ1 + μ2 ) (that might be taken as a measure of increase in pollution
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in commercial areas over that in residential ones)?
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4. A QC inspector periodically inspects 9 in × 9 in floor tiles produced by a particular production
line in the person's facility, looking for pock marks above a critical size. The inspector has come to
the conclusion that the number of such blemishes on a single tile can be modeled as Poisson with
mean λ = .5 blemishes.
10 pts
a) Evaluate the probability that a given tile has at least one pock mark that is above the critical size.
10 pts
b) Approximate the probability that 100 tiles have from 45 to 55 pock marks total using the central
limit theorem. (This is the probability of an average of .45 to .55 pock marks per tile.)
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5. Attached at the end of this exam is some JMP output that concerns the analysis of a classical data
set (of N.H. Prater) concerned with the yield of gasoline in the refining of crude oil. The response
variable is
y = yield expressed as a percent of crude oil converted to gasoline
after distillation and fractionation
and predictors include the process variable
x1 = the "endpoint" or temperature (°F) at which all the gasoline is vaporized
and crude oil properties
x2 = 10% point ASTM (the temperature at which 10% of the crude oil
has become vapor)
x3 = the crude oil gravity (degrees API), and
x4 = the crude oil vapor pressure (lbf/in 2 )
First consider a simple linear regression analysis of y based on x1 alone.
5 pts
a) What fraction of the raw variability in y is accounted for by a linear equation in x1 ? (Report a
number.)
5 pts
b) What are 95% confidence limits for the increase in mean yield that accompanies a 1°F increase
in "endpoint" in this model? (Report numerical limits.)
7 pts
c) What are 95% prediction limits for the next yield of this process run at an endpoint of 300°F ?
(Plug in completely, but you need not simplify.)
Now consider analyses of y as potentially depending upon x1 , x2 , x3 , and x4 represented in the
JMP reports.
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10 pts
d) After accounting for endpoint ( x1 ), do the crude oil properties ( x2 , x3 , and x4 ) add detectable
ability to explain/model/predict the yield ( y )? Provide an observed value of an F test statistic,
degrees of freedom, and an answer for α = .05 .
F = ____________
5 pts
d . f . = ______ , ______
Yes/No (circle the correct one)
e) If you had to choose a model for y based on some or all of the predictors using only the
information provided on the JMP reports, which predictor(s) would you employ? WHY? (Explain
your choice!)
predictor(s) you would employ: ____________________________
Explanation:
5 pts
f) In a model including all predictors, give 95% two-sided confidence limits for the increase in
mean yield that accompanies a 1°F increase in "endpoint" with properties of the crude oil held
fixed. (Plug in completely, but you need not simplify.)
8 pts
g) In a model including all predictors, a standard error of prediction for the conditions of the first
set of predictors in the data is 1.193 . Give 95% confidence limits for the mean yield under that set
of conditions. (Plug in completely, but you need not simplify.)
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6. Bert and Ernie are both applying for summer work at the Large Corporation. There are 3
positions open and a total of 10 applicants including Bert and Ernie. Bert is not eligible for job A as
his uncle is the supervisor for that position. Otherwise, all applicants are eligible for all 3 jobs.
7 pts
a) In how many different ways can the 3 jobs A,B, and C be filled by the 10 applicants?
7 pts
b) Suppose that all applicants are equally qualified, so that hiring is done essentially at random
(except for the restriction that Bert cannot have job A). What is the probability that neither Bert nor
Ernie is hired?
6 pts
7. Events A, B are independent. Suppose P ( A ) = .3 and P ( B ) = .4 . What is P ( not ( A or B ) ) . (In
(
)
set notation this is P ( A  B ) .)
c
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8. A manufacturing process produces axels whose diameters may be modeled as continuous
random variables. The diameter (in inches) of a single axel, X , may be described by a distribution
with probability density (pictured below)
 1
for .99 < x < 1.01

f ( x ) =  .02
 0 otherwise
  X 2 
Axels are (exactly) 20 inches long, so the volume of an axel is V = 20  π    = 5π X 2 .
 2 


10 pts
a) Find EV , the expected volume of the axel.
10 pts
b) Find VarV , the variance of the volume of the axel. You must set up completely a correct
expression in terms of definite integrals, but you need not evaluate. (Hint: Var V = E V 2 − ( E V ) .)
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JMP Reports for Problem 5
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