Page 117 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

96 3 Estimating Data Parameters

establishing a confidence interval for the variance only in the case that the
population distribution follows a normal law. Then, the sampling distribution of
the variance follows a chi-square law, namely (see Property 4 of section B.2.7):

( −n ) 1 v ~ χ 2 3.18
σ 2 n 1 −

The chi-square distribution is asymmetrical; therefore, in order to establish a
two-sided confidence interval, we have to use two different values for the lower
and upper percentiles. For the 95% confidence interval and df = n −1, we have:

df × v
2
2
χ df . 0 , 025 ≤ ≤ χ df . 0 , 975 , 3.19
σ 2

2
where χ df ,α means the α percentile of the chi-square distribution with df degrees
of freedom. Therefore:

df × v ≤ σ ≤ df × v . 3.20
2
2
χ df . 0 , 975 χ df 2 . 0 , 025

Example 3.7
Q: Consider the distribution of the average perimeter of defects, variable PRM, of
class 2 in the Cork Stoppers’ dataset. Compute the 95% confidence interval
of its standard deviation.

A: The assumption of normality for the PRM variable is acceptable, as will be
explained in Chapter 5. There are, in class 2, n = 50 cases with sample standard
variance v = 0.7168. The chi-square percentiles are:

χ 2 49 . 0 , 025 = 31 . 56 ; χ 49 . 0 , 975 = 70 . 22 .
2

Therefore:

49×v ≤ σ 2 ≤ 49×v ⇒ . 0 50 ≤ σ 2 ≤ . 1 11 ⇒ . 0 71≤ σ ≤ . 1 06 .
70 . 22 31 . 56

Confidence intervals for the variance are computed by SPSS, STATISTICA,
MATLAB and R as part of hypothesis tests presented in the following chapter.
They can be computed, however, either using Tools.xls or, in the case of the
variance alone, using the MATLAB command normfit mentioned in section 3.2.
We also provide the MATLAB and R function civar(v,n,alpha) for
computing confidence intervals of a variance (see Appendix F).