STAT 409
Examples for 10/19/2011
Fall 2011
1.
Let
X
1
, X
2
, … , X
16
be a random sample of size
n
= 16
from a
N
(
μ
,
σ
2
)
distribution.
We are interested in testing
H
0
:
σ
= 39
vs.
H
1
:
σ
> 39.
Recall:
If
X
1
, X
2
, … , X
n
are
i.i.d.
N
(
μ
,
σ
2
)
,
then
(
)
2
2
σ
S
1
n
⋅

is
χ
2
(
n
– 1
)
.
a)
Find the “best” critical
(
rejection
)
region with the significance level
α
= 0.05.
Test Statistic:
(
)
2
2
2
0
2
2
39
15
1
s
σ
s
χ
⋅
=
⋅
=

n
.
Reject
H
0
if
2
2
α
>
χ
χ
(
n
– 1
)
=
2
0.05
χ
(
15
)
=
25.00
.
2
2
39
15
s
⋅
> 25.00
⇔
s
2
>
2535
.
b)
Find the power of the test from part (a) at
σ
= 66.7.
Power
=
P
(
Reject
H
0

H
0
is not true
)
=
P
(
S
2
> 2535

σ
= 66.7
)
=
P
(
(
)
2
2
σ
S
1
n
⋅

>
2
7
.
66
2535
15
⋅

σ
= 66.7
)
=
P
(
χ
2
(
15
)
>
8.547
)
=
0.90
.
c)
What is the probability of Type II Error if
σ
= 66.7?
P
(
Type II Error
)
=
1 – Power
=
0.10
.
The ChiSquare Distribution
P
(
X
≤
x
)
0.010
0.025
0.050
0.100
0.900
0.950
0.975
0.990
r
( )
r
2
99
.
0
χ
( )
r
2
975
.
0
χ
( )
r
2
95
.
0
χ
( )
r
2
90
.
0
χ
( )
r
2
10
.
0
χ
( )
r
2
05
.
0
χ
( )
r
2
025
.
0
χ
( )
r
2
01
.
0
χ
15
5.229
6.262
7.261
8.547
22.31
25.00
27.49
30.58
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2.
A scientist wishes to test if a new treatment has a better cure rate than the traditional
treatment which cures only 60% of the patients.
In order to test whether the new
treatment is more effective or not, a test group of 20 patients were given the new
treatment.
Assume that each personal result is independent of the others.
Let
X
denote the number of patients in the test group who were cured.
H
0
:
p
= 0.60
vs.
H
1
:
p
> 0.60.
Righttailed.
n
= 20.
a)
Suppose we decided to use the rejection region “Reject
H
0
if
X
≥
15.”
Find the significance level
α
associated with this Rejection Region.
α
= P
(
Type I error
) = P
(
Reject
H
0

H
0
true
) = P
(
X
≥
15

p
= 0.60
)
= 1 – CDF
(
14

p
= 0.60
) = 1 – 0.874 =
0.126
.
b)
Find the “best” rejection region with the significance level
α
closest to 0.05.
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 Fall '11
 STEPHANOV
 Statistical hypothesis testing, Statistical significance, significance level, CDF

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