STAT 409
Examples for 10/21/2009
Fall 2009
1.
An advertisement for a particular brand of automobile states that it accelerates from
0 to 60 mph in an average of 5.0 seconds.
Makers of a competing automobile feel
that the true average number of seconds it takes to reach 60 mph from zero is above
5.0.
Suppose the population standard deviation is believed to be 0.43 seconds.
We
wish to test
H
0
:
μ
≤
5.0
vs.
H
1
:
μ
> 5.0.
a)
Find the power of this test if the true average time is 5.15 seconds, a 5% level of
significance will be used, and 50 automobiles will be tested (each automobile to
be tested a single time).
b)
Fifty automobiles were tested (each automobile was tested a single time).
The sample
mean time to reach 60 mph from zero was 5.09 seconds.
Find the pvalue of this test.
2.
Suppose
n
= 49
observations are taken from a normal distribution where
σ
= 8.0
for the purpose of testing
H
0
:
μ
= 60
versus
H
1
:
μ
≠
60
at the
α
= 0.05
level
of significance.
What is the power of the appropriate test when
μ
= 59.2?
3.
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
( 29
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.
b)
Find the power of the test from part (a) at
σ
= 66.7.
c)
What is the probability of Type II Error if
σ
= 66.7?
The ChiSquare Distribution
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 Spring '11
 Stepanov
 Statistical hypothesis testing, 5%, Statistical significance, 5.0 seconds, 0.43 seconds

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