PSTAT 120C:
Solutions to Assignment # 1
April 14, 2009
1. We have 9 independent observations from a normal distribution with standard deviation 10,
and we want ttest
H
0
:
μ
= 150 vs.
H
A
:
μ
6
= 150
The GLRT with level
α
= 0
.
05 has the critical set
C
1
=
{
¯
x <
143
.
5 or ¯
x >
156
.
5
}
.
In all of these calculations ¯
x
∼ N
(
μ,
100
9
)
.
(a) The power of this test against the alternative
μ
= 151 is
P
{
¯
x <
143
.
5
}
+
P
{
¯
x >
156
.
5
}
=
P
±
¯
x

151
10
/
3
<
143
.
5

151
10
/
3
²
+
P
±
¯
x

151
10
/
3
>
156
.
5

151
10
/
3
²
=
P
{
Z <

2
.
25
}
+
P
{
Z >
1
.
65
}
=
.
0122 + 0
.
0495 = 0
.
0617
(b) The power of this test against the alternative
μ
= 123 is
P
{
¯
x <
143
.
5
}
+
P
{
¯
x >
156
.
5
}
=
P
±
¯
x

123
10
/
3
<
143
.
5

123
10
/
3
²
+
P
±
¯
x

123
10
/
3
>
156
.
5

123
10
/
3
²
=
P
{
Z <
6
.
15
}
+
P
{
Z >
10
.
05
}
≈
1
(c) Alternatively, consider the test with
C
2
=
{
¯
x <
144
.
5 or ¯
x >
160
.
9
}
.
i. The level of this second test is the probability when the mean is 150:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 EduardoMontoya
 Null hypothesis, #, Statistical hypothesis testing, 1 L, θy θ−1

Click to edit the document details