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

SECTION 12.1 | Introduction (An Overview of Analysis of Variance) 371
If you compute the variance for the three numbers in each set, then the variance is
s 2 = 58.33 for set 1 and the variance is s 2 = 2.33 for set 2 is. Notice that the two variances
provide an accurate representation of the size of the differences. In set 1 there are relatively
large differences between sample means and the variance is relatively large. In set 2 the
mean differences are small and the variance is small.
Thus, we can use variance to measure sample mean differences when there are two or
more samples. The test statistic for ANOVA uses this fact to compute an F-ratio with the
following structure:
variance sdifferencesd between sample means
F 5
variance sdifferencesd expected with no treatment effect
Note that the F-ratio has the same basic structure as the t statistic but is based on
variance instead of sample mean difference. The variance in the numerator of the F-ratio
provides a single number that measures the differences among all of the sample means.
The variance in the denominator of the F-ratio, like the standard error in the denominator
of the t statistic, measures the mean differences that would be expected if there is no
treatment effect. Thus, the t statistic and the F-ratio provide the same basic information.
In each case, a large value for the test statistic provides evidence that the sample mean differences
(numerator) are larger than would be expected if there were no treatment effects
(denominator).
LEARNING CHECK
1. Which of the following is the correct description for a research study comparing
problem-solving scores obtained for 3 different age groups of children?
a. single-factor design
b. two-factor design
c. three-factor design
d. factorial design
2. ANOVA is necessary to evaluate mean differences among three or more treatments,
in order to ____________________ .
a. minimize risk of Type I error
b. maximize risk of Type I error
c. minimize risk of Type II error
d. maximize risk of Type II error
3. The numerator of the F-ratio measures the size of the sample mean differences.
How is this number obtained, especially when there are more than two sample
means?
a. by computing the sum of the sample mean differences
b. by computing the average of the sample mean differences
c. by the largest of the sample mean differences.
d. by computing the variance for the sample means
ANSWERS
1. A, 2. A, 3. D