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

386 CHAPTER 12 | Introduction to Analysis of Variance
their own questions. The authors note that generating questions took twice as much time
as simply answering questions but the extra time did not seem to have any effect on test
performance.
The data in Table 12.4 are from an independent-measures study attempting to replicate
the results from the Weinstein et al. study. We use an ANOVA is to determine whether there
are any significant differences among the three study strategies.
Before we begin the hypothesis test, note that we have already computed several summary
statistics for the data in Table 12.4. Specifically, the treatment totals (T) and SS values
are shown for each sample, and the grand total (G) as well as N and ΣX 2 are shown for
the entire set of data. Having these summary values simplifies the computations in the
hypothesis test, and we suggest that you always compute these summary statistics before
you begin an ANOVA.
TABLE 12.4
Test scores for students
using three different study
strategies.
Read and Reread
Read, then Answer
Prepared Questions
Read, then
Create and Answer
Questions
2 5 8
3 9 6 N = 18
8 10 12 G = 144
6 13 11 ΣX 2 = 1324
5 8 11
6 9 12
T = 30 T = 54 T = 60
M = 5 M = 9 M = 10
SS = 24 SS = 34 SS = 30
STEP 1
State the hypotheses and select an alpha level.
H 0
: μ 1
= μ 2
= μ 3
(There is no treatment effect.)
We will use α = .05.
H 1
: At least one of the treatment means is different.
STEP 2
Often it is easier to postpone
finding the critical
region until after Step 3,
where you compute
the df values as part of
the calculations for the
F-ratio.
Locate the critical region. We first must determine degrees of freedom for MS between treatments
and MS within treatments
(the numerator and denominator of the F-ratio), so we begin by analyzing
the degrees of freedom. For these data, the total degrees of freedom are
df total
= N – 1 = 18 – 1 = 17
Analyzing this total into two components, we obtain
df between
= k – 1 = 3 – 1 = 2
df within
= Σdf inside each treatment
= 5 + 5 + 5 = 15
The F-ratio for these data has df = 2, 15. The distribution of all the possible F-ratios with
df = 2, 15 is presented in Figure 12.8. Note that F-ratios larger than 3.68 are extremely rare
(p < .05) if H 0
is true and, therefore, form the critical region for the test.