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Confidence interval on the variance and standard deviation of a normal distribution

Confidence Interval on the Mean, Variance Unknown: if ¯x and s are the mean and
2
standard deviation of a random sample from a normal distribution with unknown variance σ , a
100(1 − α)% confidence interval on µ is given by
√ √
¯ x − t α/2,n−1 s/ n ≤ µ ≤ ¯x + t α/2,n−1 s/ n (2.31)

where t α/2,n−1 is the upper 100α/2 percentage point of the t-distribution with n − 1 degrees of
freedom.
One-sided confidence bounds on the mean of a normal distribution are also of interest and are
easy to find. Simply use only the appropriate lower or upper confidence limit from Equation (2.31)
and replace t α/2,n−1 by t α,n−1 .

Example 2.8. The load at specimen failure is as follows (in megapascals):
19.8 10.1 14.9 7.5 15.4 15.4
15.4 18.5 7.9 12.7 11.9 11.4
11.4 14.1 17.6 16.7 15.8
19.5 8.8 13.6 11.9 11.4
The sample mean is ¯x = 13.71, and the sample standard deviation is s = 3.55. We
want to find a 95% CI on µ. Since n = 22, we have n − 1 = 21 degrees of freedom for
t, so t 0.025,21 = 2.080. The resulting CI is
√ √
¯ x − t α/2,n−1 s/ n ≤ µ¯x + t α/2,n−1 s/ n,
√ √
13.71 − 2.080 · 13.55/ 22 ≤ µ ≤ 13.71 + 2.080 · 3.552/ 22,

13.71 − 1.57 ≤ µ ≤ 13.71 + 1.57,
12.14 ≤ µ ≤ 15.28

Practical Interpretation: The CI is fairly wide because there is a lot of
variability in the tensile adhesion test measurements. A larger sample size
would have led to a shorter interval. ,

It is not as easy to select a sample size n to obtain a specified length (or precision of estimation)
for this CI as it was in the known-σ case, because the length of the interval involves s (which is
unknown before the data are collected), n, and t α/2,n−1 . Note that the t-percentile depends on the
sample size n. Consequently, an appropriate n can only be obtained through trial and error. The
results of this will, of course, also depend on the reliability of our prior ”guess” for σ.

Confidence interval on the variance and standard deviation of a normal

distribution

Sometimes confidence intervals on the population variance or standard deviation are needed.
When the population is modeled by a normal distribution, the tests and intervals described in
this section are applicable. The following result provides the basis of constructing these
confidence intervals.

2
Theorem 2.1. (χ Distribution) Let X 1 , X 2 , . . . , X n be a random sample from a normal
2
2
distribution with mean µ and variance σ , and let S be the sample variance. Then the random
variable
(n − 1)S 2
2
X = (2.32)
σ 2
2
has a chi-square (χ ) distribution with n − 1 degrees of freedom. ⋆
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