21
.
7
79
.
4
14
/
68
.
12
771
.
1
21
.
1
14
/
68
.
12
771
.
1
21
.
1
/
/
D
D
D
D
D
n
s
t
d
n
s
t
d
Notice that the confidence interval on µ
D
includes zero. This implies that, at the 90%
level of confidence, the data do not support
the claim that the two cars have different
mean parking times µ
1
and µ
2
. That is, the
value µ
D
= µ
1

µ
2
= 0 is not inconsistent
with the observed data.
Subject
1(
x
1
j
)
2(
x
2
j
)
(
d
j
)
1
37.0
17.8
19.2
2
25.8
20.2
5.6
3
16.2
16.8
0.6
4
24.2
41.4
17.2
5
22.0
21.4
0.6
6
33.4
38.4
5.0
7
23.8
16.8
7.0
8
58.2
32.2
26.0
9
33.6
27.8
5.8
10
24.4
23.2
1.2
11
23.4
29.6
6.2
12
21.2
20.6
0.6
13
36.2
32.2
4.0
14
29.8
53.8
24.0
Table 104
The
F
Distribution
Inferences on the Variances of Two Normal Populations
14
Let
W and Y
be independent chisquare random variables with
u
and
v
degrees of
freedom respectively. Then the ratio
(1028)
has the probability density function
(1029)
and is said to follow the distribution with
u
degrees of freedom in the numerator
and
v
degrees of freedom in the denominator. It is usually abbreviated as
F
u,v
.
v
Y
u
W
F
/
/
x
x
v
u
v
u
x
v
u
v
u
x
f
v
u
u
u
0
,
1
2
2
2
)
(
/2
1
/2)
(
/2
Sec 105 Inferences on the Variances of Two Normal Populations
Sec 105 Inferences on the Variances of Two Normal
Populations
15
Let
be a random sample from a normal
population with mean µ
1
and variance
, and let
be a random sample from a second normal
population with mean µ
2
and variance
. Assume that
both normal populations are independent. Let
and
be the sample variances. Then the ratio
has an
F
distribution with
n
1
1 numerator degrees of
freedom and
n
2
1 denominator degrees of freedom.
1
1
12
11
,
,
,
n
X
X
X
2
1
2
2
22
21
,
,
,
n
X
X
X
2
2
2
1
S
2
2
S
2
2
2
2
2
1
2
1
/
/
S
S
F
The
F
Distribution
Confidence Interval on the Ratio of Two Variances
If
and
are the sample variances of random samples of sizes
n
1
and
n
2
, respectively, from two independent normal populations with unknown
variances
and
, then a 100(1
)% confidence interval on the
ratio
is
(1033)
where
and
are the upper and lower
/2
percentage points of the
F
distribution with
n
2
–
1 numerator and
n
1
–
1
denominator degrees of freedom, respectively.
A confidence interval on the ratio of the standard deviations can be
obtained by taking square roots in Equation 1033.
2
1
s
2
2
s
2
1
2
2
2
2
2
1
/
1
,
1
,
/2
2
2
2
1
2
2
2
1
1
,
1
/2,
1
2
2
2
1
1
2
1
2
n
n
n
n
f
s
s
f
s
s
1
,
1
,
/2
1
2
n
n
f
2
1
/2,
1,
1
n
n
f
Example 1015
Surface Finish for Titanium Alloy
A company manufactures impellers for use in jetturbine engines. One of the
operations involves grinding a particular surface finish on a titanium alloy
component. Two different grinding processes can be used, and both processes
can produce parts at identical mean surface roughness. The manufacturing
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 Summer '20
 Normal Distribution, Standard Deviation, Variance