Page 18-48
Inferences concerning two variances
The null hypothesis to be tested is , H
o
:
σ
1
2
=
σ
2
2
, at a level of confidence (1-
α
)100%, or significance level
α
, using two samples of sizes, n
1
and n
2
, and
variances s
1
2
and s
2
2
. The test statistic to be used is an F test statistic defined
as
2
2
D
N
o
s
s
F
=
where s
N
2
and s
D
2
represent the numerator and denominator of the F statistic,
respectively. Selection of the numerator and denominator depends on the
alternative hypothesis being tested, as shown below. The corresponding F
distribution has degrees of freedom,
ν
N
= n
N
-1, and
ν
D
= n
D
-1, where n
N
and
n
D
, are the sample sizes corresponding to the variances s
N
2
and s
D
2
,
respectively.
The following table shows how to select the numerator and denominator for F
o
depending on the alternative hypothesis chosen:
____________________________________________________________________
Alternative
Test
Degrees
hypothesis
statistic
of
freedom
____________________________________________________________________
H
1
:
σ
1
2
<
σ
2
2
(one-sided)
F
o
= s
2
2
/s
1
2
ν
N
= n
2
-1,
ν
D
= n
1
-1
H
1
:
σ
1
2
>
σ
2
2
(one-sided)
F
o
= s
1
2
/s
2
2
ν
N
= n
1
-1,
ν
D
= n
2
-1
H
1
:
σ
1
2
≠σ
2
2
(two-sided)
F
o
= s
M
2
/s
m
2
ν
N
= n
M
-1,
ν
D
= n
m
-1
s
M
2
=max(s
1
2
,s
2
2
), s
m
2
=min(s
1
2
,s
2
2
)
___________________________________________________________________
(*) n
M
is the value of n corresponding to the s
M
, and n
m
is the value of n
corresponding to s
m
.
____________________________________________________________________
The P-value is calculated, in all cases, as: P-value = P(F>F
o
) = UTPF(
ν
N
,
ν
D
,F
o
)
The test criteria are:
•
Reject H
o
if P-value <
α
•
Do not reject H
o
if P-value >
α
.