2. The structure of a two-factor study can be presented as a matrix with the levels of one factor
determining the rows and the levels of the second factor determining the columns. With this structure in mind, describe the mean differences that are evaluated by each of the three hypothesis tests that make up a two-factor ANOVA.
4. For the data inthe following matrix:
5. The following matrix presents the results from an independent-measures, two-factor study with
a sample of n = 10 participants in each treatment condition. Note that one treatment mean is missing.
No Treatment
Treatment
Male
M=5 M _=3
Overall M = 4
Female
M = 9 M =13
Overall M =11
overall M =7
overall M = 8
a. Which two means are compared to describe the treatment main effect?
b. Which two means are compared to describe the gender main effect?
c. Is there an interaction between gender and treatment? Explain your answer.
6. The following matrix presents the results of a two factor study with n = 10 scores in
each of the six treatment conditions. Note that one of the treatment means is missing.
Factor
B
B1 B2 B3
A1 M =10 M =20 M = 40
Factor A
A2 M =20 M =30
a. What value for the missing mean would result in no main effect for factor A?
b. What value for the missing mean would result in no interaction?
8. A researcher conducts an independent-measures, two-factor study using a separate sample of n
= 15 participants in each treatment condition. The results are evaluated using an ANOVA and the researcher reports
an F-ratio with df =1, 84 for factor A, and an F-ratio with df = 2, 84 for factor B.
a. How many levels of factor A were used in the study?
b. How many levels of factor B were used in the study?
c. What are the df values for the F-ratio evaluating the interaction?
10. The following results are from an independent-measures,two-factor study with n = 5
participants in each treatment condition.
Factor
B
B1 B2
T =40 T =10
M =4
M = 1
SS _=50
SS =30
T =50 T =20
M = 5
M = 2
SS =60
SS =40
N =40
G =120
X2 =640
Factor
A
A1
A2
a. Use a two-factor
ANOVA with sd=.05 to evaluate the main effects and the
interaction.
b. Test the simple
main effects using sd= .05 to evaluate the mean difference
between treatment
A1 and A2 for each level of
factor B.
12. Most sports injuries are immediate and obvious, like a broken leg. However, some can be
more subtle, like the neurological damage that may occur when soccer players repeatedly head a soccer ball. To examine long-term effects of repeated heading, Downs and Abwender (2002) examined two different age groups of soccer players and swimmers. The dependent variable was performance on a conceptual thinking task. Following are hypothetical data, similar to the research results.
a. Use a two-factor ANOVA with = .05 to evaluate the main effects and interaction.
b. Calculate the effects size (n2) for the main effects and the interaction.
c. Briefly describe the outcome of the study.
Factor
B:
Age
College
Older
n _ 20
n _ 20
M _ 9
M _ 4
T _ 180
T _ 80
SS _ 380
SS _ 390
n _ 20
n _ 20
M _ 9
M _ 8
T _ 180
T _ 160
SS _ 350
SS _ 400
X2 _ 6360
14. The following
table summarizes the results from a two-factor study with 2 levels of factor A
and 3 levels
of factor B using a
separate sample of n = 8 participants in each treatment condition. Fill
in the missing values. (Hint: Start with the df values.)
Source
SS df MS
Between treatments 60
Factor A 5 F _
Factor B F_
A _ B
Interaction
25 F _
Within treatments 2.5
Total
a. Use a two-factor
ANOVA with = .05 to evaluate the mean differences.
b. Briefly explain
how temperature and pouring influence the bubbles in champagne according to
this pattern of results.
16. The Preview
section for this chapter described a two factor study examining performance
under two audience
conditions (factor B) for
high and low self-esteem participants (factor A). The following summary
table presents possible results from the analysis of that study.
Assuming that the study used a
separate sample of n = 15 participants in each treatment condition (each
cell), fill in the missing values in the table.
(Hint: Start with the df
values.)
Source
SS df MS
Between treatments 67
Audience F =
Self-esteem 29 F =
Interaction F = 5.50
Within treatments 4
Total
18. The following
data are from a two-factor study examining the effects of two treatment
conditions on males and females.
a. Use an ANOVA with
= .05 for all
tests to evaluate the significance of the main effects and the interaction.
b. Compute n2
to
measure the size of the effect for each main effect and the interaction.
Treatments
I
II
3 2
8 8
9 7
Tmale = 48
4 7
M =6
M = 6
T =24
T = 24 N =16
SS =26
SS =22 G =96
0 12
X2 = 806
0 6
2 9
6 13
M = 2
M = 10 Tfemale = 48
T =8
T =40
SS =24
SS =30
TI = 32 TII = 64
20. Mathematics word
problems can be particularly difficult, especially for primary-grade children.
A recent study investigated a combination of techniques for teaching students
to master these problems (Fuchs,Fuchs, Craddock, Hollenbeck, Hamlett, &
Schatschneider, 2008). The study investigated the effectiveness of small-group
tutoring and the
effectiveness of a classroom
instruction technique known as “hot math.” The hot-math program teaches
students to recognize types or categories of problems so that they can
generalize skills from one problem to another. The following data are similar
to the results obtained in the study. The dependent variable is a math test
score for each student after 16 weeks in the study.
Traditional
Instruction
No Tutoring
With Tutoring
3
9
6
4
2
5
2
8
4
4
7
6
Hot-Math Instruction
7
8
7
12
2
9
6
13
8
9
6
9
a. Use a two-factor
ANOVA with = .05 to evaluate the significance of
the main effects and the
interaction.
b. Calculate the _2
values
to measure the effect size for the two main effects.
c. Describe the
pattern of results. (Is tutoring significantly better than no tutoring? Is
traditional classroom instruction significantly different from hot math? Does
the effect of tutoring depend on the type of classroom instruction?)
22. In Chapter 11, we
described a research study in which the color red appeared to increase men’s
attraction to women (Elliot & Niesta, 2008). The same researchers have
published other results showing that red also increases women’s attraction to
men but does not appear to affect judgments of same sex individuals (Elliot, et
al., 2010). Combining these results into one study produces a two-factor design
in which men judge photographs of both women and men, which are
shown on both red and white backgrounds.
The dependent variable is a rating of attractiveness for the person shown in
the photograph. The study uses a separate group of participants for each
condition. The following table presents data similar to the results from
previous research.
a.
Use a two-factor ANOVA with = .05 to evaluate
the main effects and the interaction.
Person Shown in Photograph
WhiteFemale Male
n
= 10 n = 10
M
= 4.5 M = 4.4
SS
=6 SS = 7
RedFemale Male
n
=10 n =10
M
=7.5 M =4.6
` SS
= 9 SS = 8
b.
Describe the effect of background color on judgments
of males and females.