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Stat 5102 Final Exam
May 15, 2003
Name
Student ID
The exam is open book and open handouts. You may also use three
8
1
2
×
11 sheets of paper with formulas, etc. You may use a calculator. Put
all of your work on this test form (use the back if necessary). Show your
work or give an explanation of your answer. No credit for numbers with no
indication of where they came from.
The points for the questions total to 200. There are 4 pages and 6 prob
lems.
1. [30 pts.] Suppose
X
is an Exponential(
θ
) random variable where
θ
is
an unknown parameter satisfying
θ >
0. Suppose we wish to test the
hypotheses
H
0
:
θ
= 1
H
1
:
θ >
1
The the sensible test statistic is just
X
, so we reject
H
0
when
X
≤
c
for some critical value
c
.
(a) Find the critical value
c
that gives a hypothesis test having signif
icance level
α
= 0
.
05.
(b) Suppose we observe
X
= 0
.
45, what is the
P
value of the test?
2. [30 pts.] Suppose
X
1
,
X
2
,
...
,
X
n
are independent and identically dis
tributed with distribution Normal(0
,
1
/θ
), that is, normal with mean
zero and variance 1
/θ
(as usual in Bayesian problems involving the nor
mal distribution, the socalled
precision
θ
is a more sensible parameter
than the variance), the prior distribution for
θ
is the Exp(1) distribu
tion, that is, exponential with rate parameter 1. Find the posterior
distribution of
θ
given data
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