3. Two-factor analysis of variance Emphasis on calculations W. Thomas Boyce, a p

Question

3. Two-factor analysis of variance Emphasis on calculations W. Thomas Boyce, a professar 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 enironmental factor, you choose two levels of oral hygiene. As the physiological factor, you choose three levels of cortisal value. The outcome is number of cavities, and the research participants are kindergartners. You conduct a two-factor ANOVA on the data. The two-factur ANOVA involves several hypothesis tests. Which of the following are null hypotheses that you could use this ANOVA to test? Check all that apply 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 Oral hygiene has no effect on number of cavities. There is no interaction between oral hygiene and cortisol value. The results of your study are summarized by the corresponding sarmple means below. Each cell reports the average number of cavities for 7 kindergartners. Factor B: Cortisol Value Low Medium ig M3.29 M 2.71 M 1.86 Low T=23 T= 13 = 55 SS = 1.4286 | SS = 1.4286 | SS 0.8571 Factor A: ×2 = 303 Oral Hygiene M-1.86 M-2.86 M = 3.00 High T = 13 T=20 T = 21 = 54 ss = 0.8571 | SS = 2.8571 = 36 = 34 10012 = 39 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 presen ted in the following ANOoVA table ANOVA Table Source df MS Between treatments 12.6905 0.0238 0.12 Factor A Factor B A X B interaction 0.4524 2.19 11.7619 Within treatments Total 20.1 191] 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 factar 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 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 A X B interaction, the within treatments mean square, and the F-ratio for the A X B interaction

Explanation / Answer

Hypotheses

There are three sets of hypothesis with the two-way ANOVA. The null hypotheses for each of the sets are given below.

So in this question we can say A, C & D option.

A. Cortisol value has no effect on number of cavities.

C. Oral hygiene has no effect on number of cavities

D. There is no interaction between oral hygiene and cortisol value

1 2 3 Total (A) 1 23.03 18.97 13.02 55.02 2 13.02 19.6 21 53.62 Total (B) 36.05 38.57 34.02 108.64 SS df mS A 2859.077 1 B 1299.603 2 AXB 11.7619 2 Within 36 Total 20.119 41

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