Download ANOVA one way between

One-Way Between Groups ANOVA
Supplemental Problems
1. The accompanying table is a one-way, independent groups ANOVA summary table with part
of the material missing.
Source
Between groups
Within groups
Total
SS
1253.68
df
3
5016.40
39
Mean square
F
a. Fill in the missing values
b. How many groups are there in the experiment
c. Assuming an equal number of subjects in each group, how many subjects are there in
each group
d. What is the value of Fcrit using  = 0.05
e. Is there a significant effect
2. Find the critical F (Fcrit) for the following situations:
a. df(numerator) = 2, df(denominator) = 13,  = 0.05
b. df(numerator) = 3, df(denominator) = 38,  = 0.05
c. df(numerator) = 4, df(denominator) = 56,  = 0.01
3. For some reason you want to determine whether there is a difference in useful life among
the top-of-the-line car batteries produced by 3 manufacturers (A, B and C). To provide the
database for your assessment, you randomly sample 4 batteries from each manufacturer
and run them through laboratory tests that allow you to determine the useful life of each
battery. The following are the results given in months of useful battery life:
A
52
57
55
59
Battery Manufacturer
B
46
52
51
50
C
44
53
50
51
Use an ANOVA with  = 0.05 to determine whether there is a difference among the 3 brands of
batteries.
4. An investigator is interested in the effects of hormone X on sexual behavior (now were talking,
who cares about car batteries anyways?). The experiment involves 4 different
concentrations of the hormone. The concentrations are arranged in ascending order; that is
0 concentration is where there is zero amount of hormone X (this is the placebo group), and
concentration 3 represents the highest amount of the hormone:
0
5
6
3
4
7
8
6
5
4
8
Concentration of Hormone X
1
2
4
8
5
10
6
12
4
6
5
6
7
7
7
9
8
8
4
7
8
11
3
13
10
9
12
12
14
9
13
10
12
Use an ANOVA with  = 0.05 to determine whether hormone X effects sexual behavior.
5. If the null hypothesis is true, MSbet is expected to be approximately equal to:
a) 0
b) 1
c) MSw
d) F
6. Suppose that the F ratio calculated for a particular experiment is equal to .04. Which of the
following can be concluded?
a) A calculation error must have been made.
b) The null hypothesis cannot be rejected.
c) The null hypothesis can be rejected at the .05 level.
d) Nothing can be concluded without knowing the degrees of freedom.
7. When the null hypothesis is rejected in a one-way ANOVA, which of the following is implied?
a) Each population mean is different from all of the other population means.
b) Each population variance is different from all of the other population variances.
c) The population means are not all equal to each other.
d) All of the above.
8. Which of the following leads to a smaller critical F in a typical one-way ANOVA?
a) an increase in dfbet
b) an increase in dfw
c) an increase in the size of alpha
d) all of the above
9. Which of the following is assumed when performing the ordinary one-way ANOVA?
a) All of the population means are the same.
b) All of the population variances are the same.
c) All of the sample sizes are the same.
d) All of the sample variances are the same.
10. If dfw = 50 and dfbet = 4, how many subjects were in each sample?
a) 4 b) 5 c) 10 d) 11
11. If your experiment consists of five groups with seven subjects in each, what is the critical
value for F at the .01 level?
a) 2.69 b) 3.70 c) 4.02 d) 9.15
12. If your study consists of four groups, each with 15 subjects, and SSbet = 42 and SSw = 35,
what does your F ratio equal?
a) 4.2 b) 5.6 c) 16.8 d) 22.4
13. A neuropsychologist is studying patients with five types of brain damage on a maze learning
task. The mean number of trials to complete the task, the standard deviation, and the size
of each group are as follows: Mean1 = 8, s1= 2.5, n1 = 8; Mean2 = 4, s2= 2, n2 = 14; Mean3
= 6, s3= 1.8, n3 = 10; Mean4 = 9, s4= 2.5, n4 = 16; Mean5 = 8, s5= 2.2, n5 = 12. What is the
F ratio for this study?
a) 2.5 b) 5.6 c) 8.8 d) 11.1
14. A clinical psychologist is comparing (bipolar) manic-depressives with both unipolar manics
and unipolar depressives on their clinical response to a new type of psychoactive drug.
Clinical improvement scores for subjects in all three groups are given below:
Bipolar
3
4
3
2
Manics
4
4
5
3
4
3
Depressives
3
4
2
5
4
What is the F ratio for this study?
a) 1.02 b) 2.04 c) 3.06 d) 4.08
15. In a one-way ANOVA, eta-squared is:
a) an unbiased estimate of omega-squared
b) the proportion of variance in your samples accounted for by group membership
c) SSbet divided by SSw
d) all of the above
16. Assuming you are a nutritionist who has been asked to determine whether there is a difference
in sugar content among three leading brands of breakfast cereal (Sugar Bombs, Sugar Loops,
Sugar Flakes). To assess the amount of sugar in the cereals, you randomly sample 15
packages (4 for Sugar Bombs, 5 for Sugar Loops, 6 for Sugar Flakes) and chemically determine
their sugar content. The following grams of sugar were found:
Sugar Bombs Sugar Loops Sugar Flakes
12
76
54
47
52
43
35
31
40
39
60
73
47
89
51
a) Use an ANOVA with  = 0.05 to determine whether there is a difference in sugar content
among the 3 brands of cereal.
b) Calculate the strength of association between your variables and explain
17. A sleep researcher conducts an experiment to determine whether sleep loss affects memory.
17 individuals are assigned to the following three groups of subjects: group 1 (4 subjects),
which gets normal amount of sleep; group 2 (6 subjects), which is sleep-deprived for 24
hours; group 3 (7 subjects), which is sleep-deprived for 48 hours. All three groups are tested
on a simple item recognition task (identify previously seen items). The following number of
correct items were recalled were observed for each subject.
Normal sleep
14
13
15
17
24 hrs sleep- 48 hrs sleepdeprived
deprived
12
13
15
10
16
7
11
8
12
6
18
9
4
a) Use an ANOVA with  = 0.05 to determine whether sleep-deprivation has an effect on
memory.
b) Calculate the strength of association between your variables and explain.
c) Was the assumptions of homogeneity of variance violated in the above experiment?
18. A recent study examined the stats scores of 3 groups of stats students: those who drink
before class, those who drink during class, and those that drink after class. The following
results were obtained:
Number of subjects per group (n)
Mean
Variance
Before
14
71
80
During
15
85
46
a) What is the obtained F value? Is it significant at an alpha of .05?
b) What is the eta-squared value for the study?
After
23
69
190
ANSWERS
1.
Source
Between groups
Within groups
Total
b) 4
c) 10 d) 2.86
2. a) 3.81
b) 2.85
SS
1253.68
3762.72
5016.40
df
3
36
39
Mean square
417.89
104.52
F
4.00
e) yes the effect is significant
c) 3.67
3.
Source
Between groups
Within groups
Total
SS
100.167
92.50
192.667
df
2
9
11
Mean square
50.083
10.278
F
4.87
Since F obtained = 4.87 is greater than F critical = 4.26 you reject H0 at  = 0.05. The
batteries of at least 1 manufacturer differ regarding useful life.
4.
Source
Between groups
Within groups
Total
SS
221.6
116.8
338.4
df
3
36
39
Mean square
73.867
3.244
F
22.77
Since F obtained = 22.77 is greater than F critical = 2.86 you reject H0 at  = 0.05. The
hormone X effects sexual behavior.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
A
B
C
D
B
D
C
D
D
A
B
16.
Source
Between groups
Within groups
Total
SS
1590.05
3574.88
5164.93
df
2
12
14
Mean square
795.03
297.91
F
2.67
a) Since F obtained = 2.67 is smaller than F critical = 3.88 you fail to reject H0 at  = 0.05.
There is no difference in the sugar content of the different cereal brands.
b) The independent variable accounts for .31 (31%) of the changes observed in the dependent
variable.
17.
Source
Between groups
Within groups
Total
SS
157.45
97.61
255.06
df
2
14
16
Mean square
78.73
6.97
F
11.29
a) Since F obtained = 11.29 is greater than F critical = 3.74 you reject H0 at  = 0.05. Sleep
deprivation does effect memory.
b) The independent variable accounts for .62 (62%) of the changes observed in the dependent
variable.
c) No. Levene’s statistic returned an F obtained = .83, which is smaller than F critical = 3.74.
18.
Source
Between groups
Within groups
Total
SS
2514.77
5864
8378.77
df
2
49
51
Mean square
1257.39
119.67
a) Yes. F obtained = 10.51, is greater than F critical = 3.19.
b) .30 (30%)
F
10.51