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
no-intercept
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 least-squares 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,