Factorial ANOVA Problems
In a 2-Factor ANOVA, measuring the effects of 2 factors (A and B) on a response (y), there are 3 levels each for
factors A and B, and 4 replications per treatment combination. Give the values of of the F-statistic for the AB
interaction for which we will conclude the effects of Factor A levels depend on Factor B levels and vice versa
(=0.05):
1. The following partial ANOVA table was obtained from a 2-way ANOVA where three advertisements were being
compared among men and women. A total of 30 males and 30 females were sampled, and 10 of each were exposed
to ads 1, 2, and 3, respectively. A measure of attitude toward the brand was obtained for each subject.
Source
df
Sum of Squares
Ads
1000
Gender
500
Ads*Gender
1200
Error
54
Total
59
Mean Square
F
--5400
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a) Complete the ANOVA table
b) Test whether the ad effects differ among the genders (and vice versa) (=0.05).
i.
Test Statistic: _________________________
ii.
Conclude interaction exists if test statistic is ________________________
iii.
P-value is above or below 0.05 (circle one)
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2. The following partial ANOVA table was obtained from a 2-way ANOVA where 4 newspaper editorials regarding a
statewide referendum were to be compared. A total of 60 Republicans, 60 Democrats, and 60 Independents were
sampled, and 15 of each were exposed to editorials 1, 2, 3, and 4, respectively. A numeric measure of attitude
toward the referendum was obtained for each subject.
Source
df
Sum of Squares
Editorials
3600
Political Party
4000
Editorial*Party
7200
Mean Square
Error
Total
F
--179
23200
---
---
c) Complete the ANOVA table
d) Test whether the editorial effects differ among the parties (and vice versa) (=0.05).
iv.
Test Statistic: _________________________
v.
Conclude interaction exists if test statistic is ________________________
vi.
P-value is above or below 0.05 (circle one)
3. A study is conducted as a 2-Factor ANOVA, with each factor at 3 levels. The Mean Square error was computed to be
180, and each cell of the table is based on 10 replicates (n=10). Obtain the minimum significant difference for
comparing means for levels of Factor A (or, equivalently B), based on Bonferroni’s method with experiment-wise
error rate of 0.05, by completing the following parts (assume no interaction exists):
a) Number of comparisons among levels of Factor A
b) Critical t-value for simultaneous comparisons
c) Standard error of difference between means of 2 levels of Factor A:
d) Bonferroni minimum significant difference (part you’d add and subtract from estimated difference between level
means):
Q.6.A study was conducted to compare the energy efficiencies among a=3 clothing types b=4 dryer types. There were 3
replicates for each combination of clothing type and dryer type. The cell means and marginal means are given below.
Clothes\Dryer
1
2
3
Overall
1
1.179
1.439
1.292
1.303
2
1.248
1.437
1.346
1.344
3
2.180
2.302
2.283
2.255
4
1.557
1.669
1.536
1.587
Overall
1.541
1.712
1.614
1.622
p.5.a. Complete the following ANOVA table.
Source
Treatments
Clothes
Dryers
Clothes*Dryers
Error
Total
df
SS
5.442
0.176
5.230
5.601
MS
#N/A
F_obs
#N/A
F(0.05)
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
p.5.b. Use Bonferroni’s method to compare all pairs of Dryer types.
Q.2.: An experiment to measure the effects of 2 factors on dye color permanence in fabrics. The researcher is interested
in 3 varieties of cotton (Factor A) and 4 brands of detergent (Factor B). He bases his analysis on a 3 replicates for each
combination of cotton and detergent, and the response measured is the level of color in fabric specimen after 10 wash
cycles.
p.2.a. Complete the ANOVA table, and all tests at the  = 0.05 significance level.
Source
Fabric (A)
Detergent (B)
Interaction (AB)
Error
Total
p.2.b. i.
df
SS
800
1200
600
MS
F_obs
F(0.05)
5000
Is there a significant interaction between fabric and detergent? Yes or No
p.2.b. ii. Is there a significant main effect for fabric? Yes or No
p.2.b. iii. Is there a significant main effect for detergent? Yes or No
p.2.c. Compute Bonferroni’s Minimum Significant Differences based on an experiment-wise error rate of E = 0.05
(assuming no interaction):
p.2.c.i.: For Fabrics:
p.2.c.ii: For Detergents:
Q.3. A hotel is interested in studying the effects of washing machines and detergents on whiteness of bed sheets. The
hotel has 4 washing machines and 3 brands of detergent. They randomly assign n=4 sheets for each combination of
machine and detergent (each sheet is only observed for one combination of machine and detergent). After washing, the
sheets are measured for whiteness (high scores are better).
Washer\Detergent
1
2
3
4
Mean
Source
Machine
Detergent
M*D
Error
Total
1
25
20
25
30
25
2
30
35
45
50
40
df
3
35
65
50
70
55
Mean
30
40
40
50
40
SS
36000
57000
MS
F
F(0.05)
#N/A
#N/A
#N/A
#N/A
#N/A
p.3.a. Complete the ANOVA table.
p.3.b. Is there a significant interaction between Machine and Detergent on whiteness scores? Yes / No
p.3.c. Is there a significant main effect for Machines? Yes / No
p.3.d. Is there a significant main effect for Detergents? Yes / No
Q.3. A company productivity office is interested in the effects of lighting and music on employee productivity. Factor A
(lighting) had 2 levels (soft and bright) and Factor B (music) had 3 levels (jazz, easy listening, and pop). They selected 30
offices within the firm, and randomly assigned them so that n = 5 received each of the 6 combinations of lighting and
music. The response measured was the total output for each office. The summary data are given below.
Mean Productivity
Lighting\Music Jazz Easy Pop Overall
Soft
30
40
50
40
Bright
20
40
60
40
Overall
25
40
55
40
ANOVA
Source df
Lighting
Music
L*M
Error
Total
SS
MS
5000
48000
57500
F
#N/A
F(0.05)
#N/A
#N/A
#N/A
#N/A
p.3.a. Complete the ANOVA table.
p.3.c. Assuming the interaction is not significant, use Bonferroni’s method to compare all 3 pairs of music types. Which
types (if any) are significantly different?
A study is conducted to compare the effects of 4 types of feed for horses. The researcher also believes that the three
breeds she works with may differ in terms of weight gains. She samples 12 colts from each breed, randomly assigning 3 to
each diet. She measures weight gain on each colt over a 2 month period. Complete the following ANOVA table and
conduct the following tests (at 0.05 significance level)
Source
df
SS
Feeds
3000
Breeds
1000
Interaction
1000
MS
F
F(.05)
Error
Total
35
10000
H0: No Feed/Breed Interaction
H0: No Feed Main Effects
H0: No Breed Main Effects
HA: Interaction Present
HA: Feed Main Effects
HA: Breed Main Effects
Reject / Don’t Reject H0
Reject / Don’t Reject H0
Reject / Don’t Reject H0