Problem 294/12
(a)
The parameter of interest is
μ
=true average braking distance at 40 mph
for the new design
.
The hypothesis are
H
0
:
μ
= 120 vs
H
a
:
μ <
120.
(b)
E
(¯
x
)
(
= 120
,
if
H
0
is true,
<
120
,
if
H
a
is true.
Hence
R
2
should be used.
(c)
σ
¯
X
=
σ
√
n
=
10
6
= 1
.
6667, so
α
=
P
(
R
2
when
H
0
is true) =
P
(
¯
X
≤
155
.
20when
μ
= 120) =
P
(
¯
X

120
1
.
6667
≤
115
.
20

120
1
.
6667
) =
N
(

2
.
88; 0
,
1) = 0
.
002
.
To obtain
α
= 0
.
001, we need to replace 115.20 by a number
c
such that
P
(
¯
X
≤
c
when
μ
= 120) =
N
(
c
;
μ
= 120
, σ
¯
X
= 1
.
6667) = 0
.
001.
Hence, we use JMP to compute
c
= Normal Quantile(0.001,120,1.667) = 114
.
87.
(d)
P
(accepting
H
0
when
μ
= 115) =
P
(
¯
X
≥
115
.
20when
μ
= 115) =
P
(
¯
X

115
1
.
6667
≥
115
.
20

115
1
.
6667
) =
1

N
(0
.
12; 0
,
1) = 0
.
4522. For the first region,
α
=
P
(
Z
≤ 
2
.
33) =
N
(

2
.
33; 0
,
1) = 0
.
01.
For the second region,
α
=
P
(
Z
≤ 
2
.
88) = 0
.
002.
Problem 305/24
μ
= true average viscosity.
H
0
:
μ
= 3000 vs
H
a
:
μ
6
= 3000.
t
obs
=
¯
x

3000
s/
√
n
=
2887
.
6

3000
84
/
√
5
=

2
.
99.
At level
α
, the rejection region is

t
obs

= 2
.
99
> t
α
2
,
4
.
For example, using the table below, we can see that we reject at
α
= 0
.
05 but don’t reject
at
α
= 0
.
01. Hence, the requirement is not satisfied at
α
= 0
.
05. On the other hand, there’s
no evidence that indicates that the requirement is satisfied at
α
= 0
.
01. (For the homework,
it suffices to report the test for your favorite choice of
α
).
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Problem 311/44
p
= proportion of defectives, by robots.
H
0
:
p
= 0
.
035 vs
H
a
:
p <
0
.
035.
We use
z
obs
=
ˆ
p

0
.
035
√
(0
.
035)(0
.
965)
/
500
=
z
obs
=
15
500

0
.
035
√
(0
.
035)(0
.
965)
/
500
=

0
.
005
√
0
.
0082
=

0
.
61.
At
α
= 0
.
01, the rejection region is
z
obs
≤
z
α
=

2
.
33.
Since
z
obs
=

0
.
61
>

2
.
33,
H
0
is not rejected and robots have not demonstrated their
superiority.
Using JMP, we can construct the following table and check the appropriate test by doing
Analize¿Distribution¿Test probabilities. The results are exhibited below, with a
p

value
much larger than 0.01. This also shows that there is no evidence at all that the robots are
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '05
 STAFF
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, vs ha

Click to edit the document details