Biostatistics

6.9 CONFIDENCE INTERVAL FOR THE VARIANCE OF A NORMALLY DISTRIBUTED POPULATION 195
0.4
d.f. = 1
0.3
d.f. = 2
0.2
d.f. = 4
d.f. = 10
0.1
0.0
0 2 4 6 8 10 12 14
FIGURE 6.9.1 Chi-square distributions.
(Source: Gerald van Belle, Lloyd D. Fisher, Patrick J. Heagerty, and Thomas Lumley, Biostatistics: A
Methodology for the Health Sciences, 2nd Ed., # 2004 John Wiley & Sons, Inc. This material is reproduced
with permission of John Wiley & Sons, Inc.)
The Chi-Square Distribution Confidence intervals for s 2 are usually based on
the sampling distribution of ðn 1Þs 2 =s 2 . If samples of size n are drawn from a normally
distributed population, this quantity has a distribution known as the chi-square ðx 2 Þ
distribution with n 1 degrees of freedom. As we will say more about this distribution in
chapter 12, we only say here that it is the distribution that the quantity ðn 1Þs 2 =s 2 follows
and that it is useful in finding confidence intervals for s 2 when the assumption that the
population is normally distributed holds true.
Figure 6.9.1 shows chi-square distributions for several values of degrees of freedom.
Percentiles of the chi-square distribution are given in Appendix Table F. The column
headings give the values of x 2 to the left of which lies a proportion of the total area under
the curve equal to the subscript of x 2 . The row labels are the degrees of freedom.
To obtain a 100ð1
aÞ percent confidence interval for s 2 , we first obtain the
100ð1 aÞpercent confidence interval for ðn 1Þs 2 =s 2 . To do this, we select the values of
x 2 from Appendix Table F in such a way that a=2 is to the left of the smaller value and a=2
is to the right of the larger value. In other words, the two values of x 2 are selected in such a
way that a is divided equally between the two tails of the distribution. We may designate
these two values of x 2 as x 2 a=2 and x2 1 ða=2Þ , respectively. The 100 ð 1 a Þ percent
confidence interval for ðn 1Þs 2 =s 2 , then, is given by
x 2 a=2 < ðn
1Þs2
s 2 < x 2 1 ða=2
Þ