Practice Problems
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)
For which values of the sample mean
X
should we reject
H
0
, if a 5% level of
significance will be used, and 50 automobiles will be tested (each automobile to
be tested a single time)?
b)
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).
c)
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 a random sample of size
n
= 16
is taken from a normal distribution with
σ
= 5
for the purpose of testing
H
0
:
μ
= 20
vs.
H
1
:
μ
≠
20
at a 5% significance
level.
a)
What is the power of this test when
μ
= 21?
b)
What is the pvalue of this test if the observed value of the sample mean is
x
= 22?
3.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
= 19
from the normal
distribution
N
(
μ
,
σ
2
)
.
a)
Find a rejection region of size
α
= 0.05
for testing
H
0
:
σ
2
= 30
vs.
H
1
:
σ
2
= 80.
For which values of the sample variance
s
2
should the null hypothesis be
rejected?
b)
What is the probability of Type II Error for the rejection region in part (a)?
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4.
Let
X
1
, X
2
, … , X
n
be a random sample from
N
(
0
,
σ
2
)
.
a)
Show that
{
(
x
1
,
x
2
, … ,
x
n
)
:
c
x
n
i
i
≥
∑
=
1
2
}
is the best rejection region for
testing
H
0
:
σ
2
= 4
vs.
H
1
:
σ
2
= 16.
b)
If
n
= 15,
find the value of
c
so that
α
= 0.05.
c)
If
n
= 15
and
c
is the value found in part (b), find the probability of Type II Error.
5.
The head coach of a successful college football program claims that over 40% of the
players graduate in four years, and wants to test
H
0
:
p
= 0.40
vs.
H
1
:
p
> 0.40
,
where
p
represents the proportion of players who graduate in four years.
In order to
do this, he obtains a random sample of
20
former players, and counts the number
S
of those who graduated in four years.
The coach plans to use the decision rule
"Reject
H
0
if
S
≥
11."
a)
What is the probability of Type I error associated with this decision rule?
b)
Compute the power of the test when
p
= 0.50?
c)
Suppose there are
S
= 9
players who graduated in four years in the sample of 20.
Suppose
p
= 0.40.
Will the coach make a correct decision?
What type of error may
have been made?
d)
Find the pvalue if
S
= 9.
e)
Find the decision rule with the probability of Type I error closest to 0.05.
6.
A state legislature says that it is going to decrease its funding of a state university
because, according to its sources, 40% of the university’s graduates move out of the
state within three years of graduation.
In an attempt to save the university’s funding,
you want to show that the proportion of graduates who move out of state is less than
0.40, and decide to test
H
0
:
p
= 0.40
vs.
H
a
:
p
< 0.40
.
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 Fall '08
 Monrad
 Normal Distribution, Statistical hypothesis testing, significance level, CDF, rejection region, GAMMA

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