HP 50G, User Manual

Page: 600 / 887

Page 18-33
These results assume that the values s
1
and s
2
are the population standard
deviations. If these values actually represent the samples’ standard deviations,
you should enter the same values as before, but with the option
_pooled
selected. The results now become:
Confidence intervals for the variance
To develop a formula for the confidence interval for the variance, first we
introduce the sampling distribution of the variance: Consider a random sample
X
1
, X
2
..., X
n
of independent normally-distributed variables with mean
μ
,
variance
σ
2
, and sample mean
⎯
X. The statistic
is an unbiased estimator of the variance
σ
2
.
The quantity has a
χ
n-1
2
(chi-square)
distribution with
ν
= n-1 degrees of freedom. The (1-
α
)
⋅
100 % two-sided
confidence interval is found from
Pr[
χ
2
n-1,1-
α
/2
< (n-1)
⋅
S
2
/
σ
2
<
χ
2
n-1,
α
/2
] = 1-
α
.
∑
=
− ⋅
−
=
n
i
i
X X
n
S
1
2 2
, ) (
1
1
ˆ
∑
=
− = ⋅ −
n
i
i
X X
S
n
1
2
2
2
, ) (
ˆ
) 1 (
σ