STAT 409
Examples for 11/02/2011
(2)
Fall 2011
1.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( )
(
)
(
)
θ
2
X
X
ln
1
θ
θ
;
x
x
x
f
x
f
⋅

=
=
,
x
> 1,
θ
> 1.
a)
Find the form of the uniformly most powerful rejection region for testing
H
0
:
θ
= 2
vs.
H
1
:
θ
> 2.
b)
Suppose
n
= 5.
Find the rejection region with the significance level
α
= 0.05.
c)
Find the power function of the test from part (b).
d)
Suppose
n
= 5.
Find the significance level
α
for the rejection region
“Reject
H
0
if
∑
=
n
i
i
x
1
ln
≤
5
.”
e)
Find the power of the test from part (d) when
θ
= 3 and
θ
= 4.
f)
Suppose
n
= 5.
Find the significance level
α
for the rejection region
“Reject
H
0
if
∑
=
n
i
i
x
1
ln
≤
6
.”
g)
Find the power of the test from part (f) when
θ
= 3 and
θ
= 4.
h)
Suppose
n
= 5,
and
x
1
= 5,
x
2
= 1.2,
x
3
= 2,
x
4
= 12,
x
5
= 1.5.
Find the pvalue.
State your decision
(
Reject
H
0
or
Do NOT Reject
H
0
)
at
α
= 0.05.
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Answers:
1.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( )
(
)
(
)
θ
2
X
X
ln
1
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 Fall '11
 STEPHANOV
 Probability, Probability theory, Statistical hypothesis testing, Statistical significance, GAMMA

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