Confidence Intervals for the Variance of a Normal
Population
Example (a variant of Problem 62, Ch5)
The total time for manufacturing a certain component is known to have a
normal distribution. However, the mean µ and variance σ 2 for the normal
distribution are unknown. After an experiment in which we manufactured
10 components, we recorded the sample time which is given as follows:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
with X = 64.95, s = 2.42
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
What is a 95% confidence for the population variance σ 2 ?
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
1/5
Confidence Intervals for the Variance of a Normal
Population
Theorem
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean µ
and variance σ 2 . Then the random variable
P
(n − 1)S 2
(Xi − X )2
=
σ2
σ2
has s chi-squared (χ2 ) probability distribution with n − 1 degrees of
freedom (df).
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
2/5
Confidence Intervals for the Variance of a Normal
Population
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
3/5
Confidence Intervals for the Variance of a Normal
Population
Notation
Let χ2α,ν , called a chi-squared critical value, denote the number on the
measurement axis such that α of the area under the chi-squared curve
with ν df lies to the right of χ2α,ν .
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
4/5
Confidence Intervals for the Variance of a Normal
Population
Proposition
A 100(1 − α)% confidence interval for the variance σ 2 of a normal
population has lower limit
(n − 1)s 2 /χ2α ,n−1
2
and upper limit
(n − 1)s 2 /χ21− α ,n−1
2
A confidence interval for σ has lower and upper limits that are the
square roots of the corresponding limits in the interval for σ 2 .
Liang Zhang (UofU)
Applied Statistics I
July 17, 2008
5/5