SECTION 6.13
159
3.
increase. If the level increases, the probability of rejecting
H
0
increases, so in particular, the probability of
rejecting
H
0
when it is false increases.
5. (a)
H
0
:
μ
50 000 versus
H
1
:
μ
50 000.
H
1
is true, since the true value of
μ
is 49,500.
(b) The level is the probability of rejecting
H
0
when it is true.
Under
H
0
,
X
is
approximately
normally
distributed
with
mean
50,000
and
standard
deviation
σ
X
5000
100
500.
The probability of rejecting
H
0
is
P
X
49 400 .
Under
H
0
, the
z
score of 49,400 is
z
49 400
50 000
500
1 20.
The level of the test is the area under the normal curve to the left of
z
1 20.
Therefore the level is 0.1151.
The power is the probability of rejecting
H
0
when
μ
49 500.
X
is approximately normally distributed with mean 49,500 and standard deviation
σ
X
5000
100
500.
The probability of rejecting
H
0
is
P
X
49 400 .
The
z
score of 49,400 is
z
49 400
49 500
500
0 20.
The power of the test is thus the area under the normal curve to the left of
z
0 20.
Therefore the power is 0.4207.
(c) Since the alternate hypothesis is of the form
μ
μ
0
, the 5% rejection region will be the region
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 Fall '11
 MattGilg
 Statistics, Normal Distribution, Probability, Standard Deviation, rejection region

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