STAT 409
Examples for 10/10/2011
(1)
Fall 2011
2.
Let
X
have a Binomial distribution with the number of trials
n
= 15.
To test
H
0
:
p
= 0.30
vs.
H
1
:
p
> 0.30,
we will use the rejection
(
critical
) region
rejection
(
critical
) region
“Reject
H
0
if
X
≥
9”.
a)
Find the significance level of this test.
α
= P
(
Type I error
) = P
(
Reject
H
0

H
0
true
) = P
(
X
≥
9

p
= 0.30
)
= 1 – CDF
(
8

p
= 0.30
) = 1 – 0.985 =
0.015
.
b)
Suppose we observe
X = 10.
Find the pvalue of this test.
pvalue = P
(
value of
X
as extreme or more extreme than
X = 10

H
0
true
)
= P
(
X
≥
10

p
= 0.30
) = 1 – CDF
(
9

p
= 0.30
) = 1 – 0.996 =
0.004
.
c)
Find the “best” rejection region with the significance level
α
closest to 0.05.
Rejection Region for a Right – tailed test:
Find
b
such that P
(
X
≤
b
– 1
)
≈
1 –
α
.
Then the Rejection Region is “Reject
H
0
if
X
≥
b
.”
Want
P
(
Type I error
) = 0.05.
P
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 Fall '11
 STEPHANOV
 Statistics, Binomial, Statistical hypothesis testing, Statistical significance, significance level, CDF

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