PSTAT 120C:
Assignment 2
Due April 16, 2009
1.
This is the problem from last week.
(Exercise 10.105 on page 554 in
WMS
.)
Let
Y
1
, . . . , Y
n
denote a random sample from a normal distribution with mean
μ
(unknown)
and variance
σ
2
.
For testing
H
0
:
σ
2
=
σ
2
0
against
H
A
:
σ
2
> σ
2
0
, show that the likelihood
ratio test is equivalent to the
χ
2
test with critical region
C
=
S
2
(
n

1)
σ
2
0
> χ
2
α
where
χ
2
α
is the
α
critical value for a
χ
2
distribution with
n

1 degrees of freedom.
2. Suppose that we have two independent binomial random variables
X
∼
Binomial(
n, p
x
) and
Y
∼
Binomial(
m, p
y
).
(a) Find the MLE for
p
under the assumption that
p
=
p
x
=
p
y
.
(b) Find the likelihood ratio,
λ
, for testing
H
0
:
p
x
=
p
y
vs.
H
a
:
p
x
6
=
p
y
.
(c) Evaluate the value of this statistic if
n
= 378,
X
= 95,
m
= 432, and
Y
= 123.
3. A survey of voters was seeking to find out if their was any difference between the economic
characteristics of Democrats and Republicans. The 58 Democrats in the survey had an average
annual income of $53,000 with a standard deviation of about $25,600. On the other hand, of
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 Spring '08
 EduardoMontoya
 Standard Deviation, Southern California, Los Angeles County, California, San Bernardino County, California, California counties

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