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
XX
n
S
1
22
,)(
1
1
ˆ
is an unbiased estimator of the variance σ
2
.
The quantity
∑
=
−=⋅−
n
i
i
XX
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.
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