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

SUMMARY 403
3. An independent-measures t test produced a t statistic with df = 20. If the same data
had been evaluated with an analysis of variance, what would be the df values for the
F-ratio?
a. 1, 19
b. 1, 20
c. 2, 19
d. 2, 20
ANSWERS
1. B, 2. C, 3. B
SUMMARY
1. Analysis of variance (ANOVA) is a statistical technique
that is used to test for mean differences among two or
more treatment conditions. The null hypothesis for this
test states that in the general population there are no
mean differences among the treatments. The alternative
states that at least one mean is different from another.
2. The test statistic for ANOVA is a ratio of two variances
called an F-ratio. The variances in the F-ratio are called
mean squares, or MS values. Each MS is computed by
MS 5 SS
df
3. For the independent-measures ANOVA, the F-ratio is
F 5 MS between
MS within
The MS between
measures differences between the treatments
by computing the variability of the treatment
means or totals. These differences are assumed to be
produced by
a. treatment effects (if they exist) or
b. differences resulting from chance.
The MS within
measures variability inside each of the
treatment conditions. Because individuals inside a
treatment condition are all treated exactly the same,
any differences within treatments cannot be caused by
treatment effects. Thus, the within-treatments MS is
produced only by differences caused by chance. With
these factors in mind, the F-ratio has the following
structure:
F 5
treatment effect 1 differences due to chance
differences due to chance
When there is no treatment effect (H 0
is true), the
numerator and the denominator of the F-ratio are measuring
the same variance, and the obtained ratio should
be near 1.00. If there is a significant treatment effect,
the numerator of the ratio should be larger than the
denominator, and the obtained F value should be much
greater than 1.00.
4. The formulas for computing each SS, df, and MS value
are presented in Figure 12.11, which also shows the
general structure for the ANOVA.
5. The F-ratio has two values for degrees of freedom,
one associated with the MS in the numerator and one
associated with the MS in the denominator. These df
values are used to find the critical value for the F-ratio
in the F distribution table.
6. Effect size for the independent-measures ANOVA is
measured by computing eta squared, the percentage of
variance accounted for by the treatment effect.
SS between
2 5
SS between
1 SS within
5 SS between
SS total
7. When the decision from an ANOVA is to reject the
null hypothesis and when the experiment has more
than two treatment conditions, it is necessary to continue
the analysis with a post hoc test, such as Tukey’s
HSD test or the Scheffé test. The purpose of these
tests is to determine exactly which treatments are
significantly different and which are not.