Basic Statistics

TESTING WHETHER TWO VARIANCES ARE EQUAL: F TEST 1 19

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Figure 9.1 The F distribution.

variances, and a preliminary test of that assumption is sometimes made. This option
is often found in statistical programs.
Under the assumptions given below, a test of HO : of = o; can be made using
a distribution known as the F distribution. With two independent simple random
samples of sizes n1 and n2, respectively, we calculate their sample variances s: and
s:. If the two populations from which we have sampled are both normal with variances
oy and 022, it can be shown mathematically that the quantity (sf/of)/(s;/o;) follows
an F distribution. The exact shape of the distribution of F depends on both d.f.’s
n1 - 1 and n2 - 1, but it looks something like the curve in Figure 9.1. The F
distribution is available in Table A.5.
As an example of testing the equality of two variances, we return to Table 8.2
and wish to decide whether the variances are the same or different in the populations
of hemoglobin levels for children with cyanotic and acyanotic heart disease. We are
asking whether or not o! is different from o; and the null hypothesis is, Ho : o: = u;.
A two-sided test is appropriate here. We have n1 = 19, n2 = 12. sf = 1.0167, and
s; = 1.9898. The test statistic we use is given by

Under the null hypothesis Ho : of = a;, the of and 02” can be canceled and the F
statistic becomes
F=7j sf
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Because the areas less than .95 for F are not given in many tables, the usual procedure
is to reverse the subscripts on the populations if necessary so that sy 2 s;. Hence, we
will make s: = 1.9898 and sz = 1.0167, and n1 = 12 and n2 = 19. The computed
F value is
= 1.96
F=- 1.9898
1.0167