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STAT 409
Examples for 10/14/2011
(2)
Fall 2011
H
0
true
H
0
is
NOT
true
Do Not Reject
H
0
☺
Type II Error
Reject
H
0
Type I Error
☺
α
= significance level = P
(
Type I Error
)
= P
(
Reject
H
0

H
0
is true
)
β
= P
(
Type II Error
)
= P
(
Do Not Reject
H
0

H
0
is NOT true
)
Power
= 1 – P
(
Type II Error
)
= P
(
Reject
H
0

H
0
is NOT true
)
1.
A car manufacturer claims that, when driven at a speed of 50 miles per hour
on a highway, the mileage of a certain model follows a normal distribution with
mean
μ
0
= 30 miles per gallon and standard deviation
σ
= 4 miles per gallon.
A consumer advocate thinks that the manufacturer is overestimating average
mileage.
The advocate decides to test the null hypothesis
H
0
:
μ
= 30
against
the alternative hypothesis
H
1
:
μ
<
30.
0a)
Suppose the actual overall average mileage
μ
is indeed 30 miles per gallon.
What is
the probability that the sample mean is 29.4 miles per gallon or less, for a random
sample of
n
= 25 cars?
P(
X
≤
29.4
) =

≤
25
4
30
4
.
29
Z
P
= P(
Z
≤

0.75
) =
Φ
(

0.75
) =
0.2266
.
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View Full Document0b)
A random sample of 25 cars yields
x
= 29.4 miles per gallon.
Based on the answer
for part (a), is there a reason to believe that the actual overall average mileage is not
30 miles per gallon?
If
μ
= 30, it is not unusual to see the values of the sample mean
x
at 29.4 miles
per gallon or even lower.
It does not imply that
μ
= 30, but
we have no reason
to doubt the manufacturer’s claim.
0c)
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 Fall '11
 STEPHANOV

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