Biostatistics

6.10 CONFIDENCE INTERVAL FOR THE RATIO OF THE VARIANCES OF TWO NORMALLY DISTRIBUTED POPULATIONS 199
f (x)
1.0
0.8
0.6
0.4
(10; ∞)
(10; 50)
(10; 10)
(10; 4)
0.2
0.0
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
F
FIGURE 6.10.1 The F distribution for various degrees of freedom.
(From Documenta Geigy, Scientific Tables, Seventh Edition, 1970. Courtesy of Ciba-Geigy Limited, Basel,
Switzerland.)
examined. If the confidence interval for the ratio of two population variances includes 1, we
conclude that the two population variances may, in fact, be equal. Again, since this is a form
of inference, we must rely on some sampling distribution, and this time the distribution of
s 2 1 1 =s2 = s
2
2 =s2 2 is utilized provided certain assumptions are met. The assumptions are
that s 2 1 and s2 2 are computed from independent samples of size n 1 and n 2 respectively, drawn
from two normally distributed populations. We use s 2 1 to designate the larger of the two
sample variances.
The F Distribution If the assumptions are met, s 2 1 1 =s2 = s
2
2 =s2 2 follows a
distribution known as the F distribution. We defer a more complete discussion of this
distribution until chapter 8, but note that this distribution depends on two-degrees-offreedom
values, one corresponding to the value of n 1 1 used in computing s 2 1 and the
other corresponding to the value of n 2 1 used in computing s 2 2 . These are usually referred
to as the numerator degrees of freedom and the denominator degrees of freedom.
Figure 6.10.1 shows some F distributions for several numerator and denominator
degrees-of-freedom combinations. Appendix Table G contains, for specified combinations
of degrees of freedom and values of a; F values to the right of which lies a=2 of the area
under the curve of F.
A Confidence Interval for s 2 1 =s2 2
To find the 100ð1 aÞ percent confidence
interval for s 2 1 =s2 2 , we begin with the expression
F a=2 < s2 1 =s2 1
s 2 2 =s2 2
< F 1 ða=2Þ
where F a=2 and F 1 ða=2Þ
are the values from the F table to the left and right of which,
respectively, lies a=2 of the area under the curve. The middle term of this expression may
be rewritten so that the entire expression is
F a=2 < s2 1
s 2 s2 2
2
s 2 1
< F 1 ða=2Þ