HP 48gII - Manual (Page 604)

Page 18-33
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
∑
=
− ⋅
−
=
n
i
i
X X
n
S
1
2 2 , ) (
1
1
ˆ
is an unbiased estimator of the variance σ 2 .
The quantity
∑
=
− = ⋅ −
n
i
i
X X
S
n
1
2
2
2
, ) (
ˆ
) 1 (
σ
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.
The confidence interval for the population variance
σ 2 is therefore,
[(n-1) ⋅S 2 / χ 2
n-1, α /2
; (n-1) ⋅S 2 / χ 2
n-1,1- α /2
].
where
χ 2
n-1, α /2
, and χ 2
n-1,1- α /2
are the values that a χ 2 variable, with ν = n-1
degrees of freedom, exceeds with probabilities α/2 and 1- α /2, respectively.
The one-sided upper confidence limit for
σ 2 is defined as (n-1) ⋅S 2 / χ 2
n-1,1- α
.
Example 1 – Determine the 95% confidence interval for the population
variance
σ 2 based on the results from a sample of size n = 25 that indicates
that the sample variance is s 2 = 12.5.