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This represents
5% of the area.
4.75
Figure 4.2
F
distribution showing the 95 percent and 5 percent areas for 1
df
and
12
df.
we can assert that the mean difference between the groups is statistically
significant at a .05 alpha level.
In Figure 4.2, we have marked the 5 percent boundary for the
F
distri-
bution applicable to our room color study. That boundary occurs at an
F
value of 4.75 given that we have 1
df
and 12
df
(1
df
for between-groups
variance and 12
df
for within-groups variance). The area under the curve
to the left of the
F
value of 4.75 in Figure 4.2 is shown by a shaded region
and represents 95 percent of the area. The region to the right represents
5 percent of the area.
F
ratios based on the given degrees of freedom in our
example whose values are equal to or greater than 4.75 fall in that 5 percent
area and thus may be said to represent a statistically significant amount
of between-groups variance in the study, that is, to represent a statistically
significant difference among the means of the groups in the study.
4.4.2 PROBABILITY OF
F
Before the days of computer-based statistical analysis, researchers would
look up the value of the
F
ratio they calculated in a table similar to the
Table of Critical
F
Values presented in Appendix C. Such tables display
F
values that correspond to particular benchmarks (probability breaks) in
the
F
distribution (e.g., .05, .01).
To read the table in Appendix C, we first recall that in computing the
F
ratio, the between-groups mean square (
MS
A
) was in the numerator
and the within-groups mean square (
MS
S/A
) was in the denominator. As
can be seen from the summary table (Table 3.1) for our worked example,
we have 1
df
associated with between groups (
df
A
)and12
df
associated
with within groups (
df
S
/
A
). In the Table of Critical
F
Values, the columns
indicate the
F
ratio numerator degrees of freedom and the rows indicate
the
F
ratio denominator degrees of freedom. We therefore focus on the first
column (for 1
df
) and the twelfth row (12
df
). The intersection of these
two degrees of freedom within the Critical Values of the
F
Distribution
table reveals the critical value of
F
at the .05 level to be 4.75 (this is the