HP 50g Graphing Calculator - 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