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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
, X
..., X
of independent normally-distributed variables with mean
1
2
n
µ, variance σ
2
, and sample mean X. The statistic
is an unbiased estimator of the variance σ
ˆ
S
(
n
) 1
The quantity
σ
distribution with ν = n-1 degrees of freedom. The (1-α)⋅100 % two-sided
confidence interval is found from
2
Pr[χ
n-1,1-
The confidence interval for the population variance σ
[(n-1)⋅S
where χ
2
, and χ
2
α
n-1,
/2
n-1,1-
degrees of freedom, exceeds with probabilities α/2 and 1- α /2, respectively.
The one-sided upper confidence limit for σ
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
1
n
ˆ
2
S
(
X
i
n
1
i
=
1
2
.
2
n
2
(
X
X
)
,
has a χ
i
2
i
=
1
2
2
< χ
2
< (n-1)⋅S
/σ
α
/2
n-1,
2
2
2
/ χ
/ χ
; (n-1)⋅S
α
n-1,
/2
are the values that a χ
α
/2
2
is defined as (n-1)⋅S
2
= 12.5.
2
X
)
,
2
(chi-square)
n-1
] = 1- α.
α
/2
2
is therefore,
2
].
α
n-1,1-
/2
2
variable, with ν = n-1
2
2
/ χ
n-1,1-
Page 18-33
.
α
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