in physiology and differences in experience in determining health and well-being. Dr. Boyce found that some children are more sensitive to their environments. They do exceptionally well when the environment is supportive but are much more likely to have mental and physical health problems when the environment has challenges. You decide to do a similar study, conducting a factorial experiment to test the effectiveness of one environmental factor and one physiological factor on a physical health outcome. As the environmental factor, you choose two levels of oral hygiene. As the physiological factor, you choose three levels of cortisol value. The outcome is number of cavities, and the research participants are kindergartners. You conduct a two-factor ANOVA on the data. The two-factor ANOVA involves several hypothesis tests. Which of the following are null hypotheses that you could use this ANOVA to test? Check all that apply. There is no interaction between oral hygiene and cortisol value. Oral hygiene has no effect on number of cavities.
in physiology and differences in experience in determining health and well-being. Dr. Boyce found that some children are more sensitive to their environments. They do exceptionally well when the environment is supportive but are much more likely to have mental and physical health problems when the environment has challenges. You decide to do a similar study, conducting a factorial experiment to test the effectiveness of one environmental factor and one physiological factor on a physical health outcome. As the environmental factor, you choose two levels of oral hygiene. As the physiological factor, you choose three levels of cortisol value. The outcome is number of cavities, and the research participants are kindergartners. You conduct a two-factor ANOVA on the data. The two-factor ANOVA involves several hypothesis tests. Which of the following are null hypotheses that you could use this ANOVA to test? Check all that apply. There is no interaction between oral hygiene and cortisol value. Oral hygiene has no effect on number of cavities.
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Transcribed Image Text:2. Two-factor ANOVA - Emphasis on calculations
W. Thomas Boyce, a professor and pediatrician at the University of British Columbia, Vancouver, has studied interactions between individual differences
in physiology and differences in experience in determining health and well-being. Dr. Boyce found that some children are more sensitive to their
environments. They do exceptionally well when the environment is supportive but are much more likely to have mental and physical health problems
when the environment has challenges.
You decide to do a similar study, conducting a factorial experiment to test the effectiveness of one environmental factor and one physiological factor
on a physical health outcome. As the environmental factor, you choose two levels of oral hygiene. As the physiological factor, you choose three levels
of cortisol value. The outcome is number of cavities, and the research participants are kindergartners.
You conduct a two-factor ANOVA on the data. The two-factor ANOVA involves several hypothesis tests. Which of the following are null hypotheses that
you could use this ANOVA to test? Check all that apply.
There is no interaction between oral hygiene and cortisol value.
□ Oral hygiene has no effect on number of cavities.
Cortisol value has no effect on number of cavities.
The effect of oral hygiene on number of cavities is no different from the effect of cortisol value.
The results of your study are summarized by the corresponding sample means below. Each cell reports the average number of cavities for 11
kindergartners.
Factor A: Oral Hygiene
Low
High
Source
Between treatments
Factor A
Factor B
AX B interaction
Within treatments
Total
Factor B: Cortisol Value
Low
Medium
M = 1.36
T = 15
SS = 2.5455
M = 1.55
T = 17
SS=2.7273
M = 1.73
T = 19
SS=4.1818
TOOLI = 34
ANOVA Table
SS
df
6.2575
20
0.0908
22.4394
M = 2.09
T = 23
SS = 2.9091
TOOL2 = 40
65
You perform an ANOVA to test that there are no main effects of factor A, no main effects of factor B, and no interaction between factors A and B.
Some of the results are presented in the following ANOVA table.
MS
High
M = 1.82
T = 20
SS = 1.6364
3.4091
1.3788
M = 2.27
T = 25
SS = 2.1818
Toota = 45
F
TROWI = 52
12.64
5.11
TROW2 = 67
ΣΧΖ = 297
Work through the following steps to complete the preceding ANOVA table.
1. The main effect for factor A evaluates the mean differences between the levels of factor A. The main effect for factor B evaluates the mean
differences between the levels of factor B. Select the correct values for the sums of squares for factors A and B in the ANOVA table.
Transcribed Image Text:2. Select the correct value for the within-treatments sum of squares in the ANOVA table.
3. Select the correct degrees of freedom for all the sums of squares in the ANOVA table.
4. Select the correct values for the mean square due to AX B interaction, the within treatments mean square, and the F-ratio for the AX B interaction.
5. Use the results from the completed ANOVA table and the F distribution table (click on the following dropdown menu to access the table) to make
the following conclusions.
The F Distribution: df Denominators (20-36)
Degrees of Freedom: Denominator
20
21
22
23
24
25
26
27
28
29
30
32
34
36
The F Distribution: df Denominators (38-100)
Degrees of Freedom: Numerator
4
2
3.49
5.85
3
3.10
2.87
4.94
4.43
3.47 3.07 2.84
5.78
4.87
4.37
3.44
3.05
2.82
5.72
4.82
4.31
3.42 3.03 2.80
5.66
4.76
4.26
3.40
3.01
2.78
7.82
5.61
4.72
4.22
4.24
3.38
2.99
2.76
7.77
5.57
4.68
4.18
4.22 3.37
2.98
2.74
7.72
5.53
4.64
4.14
4.21
3.35 2.96 2.73
7.68
5.49
4.60
4.11
4.20 3.34 2.95 2.71
5.45 4.57
4.07
3.33 2.93 2.70
5.42 4.54
4.04
3.32
2.92
2.69
5.39 4.51
4.02
3.30
2.90 2.67
5.34 4.46
3.97
3.28
2.88
2.65
3.93
2.86 2.63
4.38 3.89
1
4.35
8.10
4.32
8.02
4.30
7.94
4.28
7.88
4.26
7.64
4.18
7.60
4.17
7.56
4.15
7.50
4.13
7.44
4.11
7.40
5.29 4.42
3.26
5.25
Table entries in lightface type are critical values for the .05 level of significance. Boldface type values are for the .01 level of significance.
At the significance level a = 0.05, the main effect due to factor Ais
the interaction effect between the two factors is
, the main effect due to factor B is
and
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