# Inferences Concerning Two Variances, - HP 48gII User Manual

Graphing calculator.

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