Information Technology Reference
In-Depth Information
1
3
4
5
2
F
Ratio
Figure 4.1
Sampling distribution of
F
.
larger denominators, we approach a barrier of zero. Thus, the distribution
“compresses” or “bunches up” as it approaches zero.
If the larger variance is in the numerator, many of the resulting ratios
will be large numbers well in excess of one. The curve actually extends
to infinity, never quite touching the
x
axis. Although we cannot carry the
curve out all the way to infinity, we can extend the
x
axis to values in
the range of
F
values of 5, 6, or 7 because the distribution is noticeably
above the
x
axis in this range of
F
.
If the null hypothesis is true, then most of the values of the variance
ratio turn out to be, approximately, between 0.5 and 2.0. Here is why.
Because the two samples were drawn from the same population and are
composed of equal numbers of scores, we would expect that their variances
would be of roughly the same magnitude; thus, their ratio should be
someplace in the general region of 1.0 much of the time.
4.2.3 THE EXPECTED VALUE OF
F
Most researchers may not need to do so very often, but it is possible
to calculate the value of the mean of a given sampling distribution. We
present this for those who might be interested.
The mean is the
expected value
of
F
based on drawing an infinite num-
ber of samples. It is a function of the within-groups degrees of freedom
(Hays, 1981, p. 314-15). When there are more than 2
df
for this source
of variance, as there will invariably be in the research that readers will
conduct, the expected value of
F
(the mean of the sampling distribution
of
F
)iscomputedasfollows:
df
Within Groups
df
Within Groups
−
expected
F
value
=
2
.
(4.2)
In our example and in Table 3.1, we see that there are 12
df
associated with
the within-groups source of variance; thus, the sampling distribution of
