Stat 231 Final Exam
Spring 2010
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1. ISU Students Kotlers, MacFarland, and Tominson studied strength properties of wood joints.
Stress at failure, y , was measured for joints of each of four types (in units of psi). Some summary
statistics like those from the students' study are below.
Type #1
n1 5
y1 712.6
s1 164.8
Type #2
n2 1
y2 1169.0
Type #3
n3 2
y3 929.5
Type #4
n4 4
y4 1428.0
s3
s4
99.7
188.1
Initially, consider only the information from the study about joints of Types #3 and #4, and
assume strengths of such joints are normally distributed.
5 pts
a) Give two-sided 95% confidence limits for the difference in mean-stress-at-failure values for the
two joint types. (Plug in completely, but you need not simplify.)
5 pts
b) Give two-sided 95% confidence limits for the ratio of standard deviations of measured stress at
failure for the two joint types. (Plug in completely, but you need not simplify.)
Now, consider all four joint types, under the assumption that strengths of joints of any fixed
type are normally distributed and the variability of strengths is the same for each type.
5 pts
c) Give an estimate of the standard deviation of measured strength for any fixed joint type.
2
5 pts
d) Joint Types #1 and #2 are Butt joints and Types #3 and #4 are Lap joints. So, the quantity
1
1
is potentially of interest. Give 95% two-sided confidence limits for this.
1
2
3
4
2
2
(No need to simplify.)
5 pts
e) It turns out (you need NOT show this) that SSTr 1,176,581 . Give the value of an F statistic
and corresponding degrees of freedom for testing the hypothesis that all four mean-stress-at-failure
values are the same.
F
5 pts
____________
df
__________ , __________
f) Suppose that in a follow-up study, 10 out of 200 tests of joints of Type #1, produce failures at
stresses below 500 psi. Give 95% confidence limits for the long run fraction of such joints that
would fail below 500 psi. (Plug in completely, but you need not simplify.)
3
2. A particular 1-dimensional continuous distribution has pdf
3 x 2 if 0 x 1
f x
0 otherwise
8 pts
a) Model X as having pdf f x and evaluate P X
P X
7 pts
.75
____________
EX
.75 and EX .
____________
b) Model a pair of random variables X 1 and X 2 as independent, each with marginal pdf f x .
Completely set up (but do not evaluate) an expression for P X 1
5 pts
c) Suppose that X 1 , X 2 ,
X2 1 .
, X 10 are independent random variables, each with pdf f x . It is
possible to show (you need not do so) that for any one of these variables P X
Y the number of variables X 1 , X 2 ,
evaluate EY and VarY .
EY
____________
VarY
.5
.125 . For
, X 10 that are less than .5
____________
4
5 pts
3. Find P A and C if three events A, B, and C are such that
P A or B or C
.9
A, B are mutually exclusive
P B .6
P B and C
P C
.5
P A
.2
P A and C
5 pts
.3
___________
4. A very small (fake) data set consists of the 5 observations 3,4,6,7,9. A Normal plot for this data
set is a plot of 5 ordered pairs. Below, fill in numerical values for the 2nd coordinates of 5 points
that make up such a Normal plot.
3, __________
4, __________
6, __________
7, __________
9, __________
5. Attached at the end of this exam are some pages of .14 reports useful in the analysis of some
data taken from Statistics by McClave and Sincich. The data concern the selling prices, y (in
), of n 25 Minneapolis apartment buildings. Predictor variables were
$
x1
building area (sq ft)
x2
number of apartments
x3
building age (yrs)
3 pts
x4 number of parking spaces
a) What fraction of the raw variability in selling price is accounted for using building area as a
predictor variable?
5 pts
b) Modeling selling price in terms of building area alone, what are 95% confidence limits for the
increase in mean selling price associated with each additional square foot of building area?
5
3 pts
c) A Minneapolis landlord gets notice of valuation for a 35,000 sq ft apartment building (that is
supposed to be based on fair market value) of $950, 00 . The landlord wants to challenge the
valuation. Does the person have a case? EXPLAIN. (More is required here than simply noting that
ŷ is less than $950, 000 . BUT YOU NEED NOT SHOW ANY CALCULATIONS.)
4 pts
d) What does the parameter "
" measure in the two models
y
0
1 x1
and
y
?
0
1 x1
2 x2
3 x3
4 x4
Why is it reasonable to expect an estimate of this parameter to be smaller for the second model?
What:
Why:
5 pts
e) Give the value of an F statistic and degrees of freedom for testing whether any of the 4
predictors provides detectable explanatory power for predicting y .
F
5 pts
____________
df
__________ , __________
f) Give the value of an F statistic and degrees of freedom for testing whether after accounting for
building size, the other 3 predictors add detectably to one's ability to predict y .
F
____________
df
__________ , __________
6
6. Widgets produced by Company V are tested at the end of a production line. A measurement X
is produced an automated tester. Good widgets produce X that is Normal with G 10 and
1,
while Bad widgets produce Normal X with B 8 and
1 . Company V gets to set its "test
limit" TL to apply to measurements obtained from the tester, scrapping widgets when X TL .
5 pts
a) If Company V uses TL 8.5 , what are Type 1 and Type 2 error probabilities if the "null
hypothesis" is that a tested widget is a good widget (and scrapping is "rejecting that null
hypothesis")?
____________
__________
5 pts
b) Suppose that 80% of the widgets that Company V makes are in fact Good, and that the company
employs TL 8.5 . What long run faction of the scrapped widgets are in fact Good ones?
5 pts
c) Here are two possible interpretations of the modeling in this problem. (Others are possible.)
Measurement noise is negligible.
1 represents what is entirely real widget-to-widget
variation in the "Good" and "Bad" populations. (Test results for a given widget are
consistent and while suggestive of its quality, "Good" or "Bad" means more a passing or not
passing test result.)
Widget variation is negligible within the "Good" and "Bad" populations.
1 represents
what is entirely measurement variation. (So retesting a given widget will produce a new
random test result with mean either 10 or 8.)
In which of these scenarios (first, second, both, or neither) is it possible to improve the screening of
widgets by applying some (potentially new) test limit to X derived from repeat tests of a widget?
EXPLAIN.
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