ESE 302
ADDITIONAL PRACTICE
QUESTIONS FOR EXAM 1
Spring, 2011
1.
Consider the problem of predicting gasoline consumption,
Y
, as a function of miles
traveled,
x
. Here it is reasonable to assume that zero miles traveled involves zero gas
consumption. So for a given set of observations,( ,
),
1,.
..,
ii
x
yi
n
, it is natural to fit this
relation in terms of a
nointercept
regression model of the form:
(1)
i
Yx
,
n
i
,...,
1
with
iid
errors,
2
~(
0
,)
i
N
,
.
,...,
1
n
i
Given this model:
(
a
) Determine the leastsquares estimator,
ˆ
, of
.
(
b
) Show that
ˆ
is an unbiased estimator of
.
(
c
) Determine the variance of
ˆ
(Show all work).
2.
A beer manufacturer is considering automating the final labeling and crating stages of
his bottling plant. Before making a decision, he wishes to estimate the probability,
p
,
that a randomly sampled bottle will be broken during this automated process.
(i)
If the Bernoulli random variables
1
B
and
2
B
denote the events that a
randomly sampled bottle breaks during the labeling stage and crating
stage, respectively, and if
11
Pr(
1)
pB
and
22
1
Pr(
1
0)
B
, then
show that probability,
p
, is given by:
(1)
121
2
ppp p
p
(ii)
A published study of automated labeling reported that
1
X
of
1
N
sampled
bottles were broken, and a similar study of automated crating reported that
2
X
of
2
N
sampled bottles were broken. Assuming that these studies were
independent, show that an unbiased estimator of
p
is given by
(2)
2
2
ˆ
X
XX
X
p
NNN
N
(iii)
Now suppose that a single study of the
combined
process was carried out
using
N
bottles. In this case,
1
N
above is simply the number of bottles,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentN
, starting the process, and
2
N
is the number of bottles “surviving” the
labeling stage, i.e.,
(3)
21
NN
X
In terms of the above Bernoulli random variables, it must also be true
that
2
Pr(
1)
Pr(
0,
BB
B
[since a bottle can only be broken in
crating if it “survives” the labeling stage]. Under these conditions,
verify that
2
Pr(
(1
)
B
pp
, and use this to determine the bias
(or unbiasedness) if the estimate in (2).
3.
A
US manufacturer of combustion engines is considering a new type of piston rod
produced in England. Before deciding, they need to determine whether these rods are
strong enough to function properly at the high cylinder temperatures of their engines. A
British research firm has done a study of this type by regressing the stressfailure loads
1
( ,.
.,
)
n
y
y
of the new rods at a range of cylinder (piston crown) temperature settings
1
( ,.
., )
n
tt
. Unfortunately their published beta estimates,
0
ˆ
5500
eng
b
=
and
1
ˆ
10.2
eng
b
=
,
are not directly applicable, since the stressfailure loads were measured in
kilograms
rather than
pounds
, and cylinder temperatures were measured in
Centigrade
rather than
Fahrenheit
. By obtaining the British data and performing the appropriate units
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 ese302
 unbiased estimator, Bernoulli random variables, randomly sampled bottle, cylinder temperatures, British research firm

Click to edit the document details