Confidence intervals for the variance, Confidence intervals for the variance ,18-33 – HP 50g Graphing Calculator User Manual
Page 600
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