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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:
Θ Reject H
o
if P-value < α
Θ Do not reject H
o
if P-value > α.
Example1
-- Consider two samples drawn from normal populations such that n
1
= 21, n
2
= 31, s
1
2
= 0.36, and s
2
2
= 0.25. We test the null hypothesis, H
o
:
σ
1
2
= σ
2
2
, at a significance level α = 0.05, against the alternative hypothesis,
H
1
: σ
1
2
≠σ
2
2
. For a two-sided hypothesis, we need to identify s
M
and s
m
, as
follows:
s
M
2
=max(s
1
2
,s
2
2
) = max(0.36,0.25) = 0.36 = s
1
2
s
m
2
=min(s
1
2
,s
2
2
) = min (0.36,0.25) = 0.25 = s
2
2
Also,
n
M
= n
1
= 21,
n
m
= n
2
= 31,
ν
N
= n
M
- 1= 21-1=20,
ν
D
= n
m
-1 = 31-1 =30.