Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 12.6 | More about ANOVA 399
variance that contributes to the denominator of the F-ratio. Finally, you should realize that
it is easy to see the mean difference between the two samples. The fact that there is a clear
mean difference between the two treatments is confirmed by computing the F-ratio for
experiment A.
F 5
between{treatments difference
within{treatments differences 5 MS between
MS within
5 56
0.667 5 83.96
An F-ratio of F = 83.96 is sufficient to reject the null hypothesis, so we conclude that
there is a significant difference between the two treatments.
Now consider the data from experiment B, which are shown in Figure 12.9(b) and
present a very different picture. This experiment has the same 4-point difference between
treatment means that we found in experiment A (M 1
= 8 and M 2
= 12). However, for these
data the scores in each treatment are scattered across the entire scale, indicating relatively
large variance inside each treatment. In this case, the large variance within treatments overwhelms
the relatively small mean difference between treatments. In the figure it is almost
impossible to see the mean difference between treatments. The within-treatments variance
appears in the bottom of the F-ratio and, for these data, the F-ratio confirms that there is no
clear mean difference between treatments.
F 5
between{treatments difference
within{treatments differences 5 MS between
MS within
5 56
40.33 5 1.39
For experiment B, the F-ratio is not large enough to reject the null hypothesis, so we conclude
that there is no significant difference between the two treatments. Once again, the
(a)
Experiment A
Between
treatments
Treatment 1
Treatment 2
Frequency
3
2
1
0 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20
FIGURE 12.9
A visual representation of
the between-treatments
variability and the withintreatments
variability that
form the numerator and
denominator, respectively, of
the F-ratio. In (a) the difference
between treatments is
relatively large and easy to
see. In (b) the same 4-point
difference between treatments
is relatively small and is
overwhelmed by the withintreatments
variability.
(b)
Frequency
3
2
1
Experiment B
M 1 5 8
SS 1 5 4
Between
treatments
M 2 5 12
SS 2 5 4
Treatment 1
Treatment 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
M 1 5 8
SS 1 5 242
M 2 5 12
SS 2 5 242