Question

A researcher is interested in studying the effect that the amount of fat in the diet and amount of exercise has on the mental acuity of middle-aged women. The researcher used three different treatment levels for the diet and two levels for the exercise. The results of the acuity test for the subjects in the different treatment levels are shown below.

	Diet
Exercise	<30% fat	30% - 60% fat	>60% fat
<60 minutes	4	3	2
	4	1	2
	2	2	2
	4	2	2
	3	3	1
60 minutes	6	8	5
or more	5	8	7
	4	7	5
	4	8	5
	5	6	6

1. Perform a Two-way analysis of variance (ANOVA) and report the results using correct APA style; report whether significance was found for Factor A, Factor B, and/or an interaction between Factors A and B was found.

2. If the test statistic is significant, run a post hoc test to determine between what groups significance was found.

3. Report an effect size for all significant results.

Answer 1

(a)

The output is:

			Factor 2
	Means:
		< 30 % fat	30% - 60% fat	> 60% fat
	< 60 minutes	3.4	2.2	1.8	2.5
Factor 1	60 minutes or more	4.8	7.4	5.6	5.9
		4.1	4.8	3.7	4.2
	5	replications per cell
ANOVA table
Source	SS	df	MS	F	p-value
Factor 1	90.13	1	90.133	135.20	2.39E-11
Factor 2	6.20	2	3.100	4.65	.0196
Interaction	18.47	2	9.233	13.85	.0001
Error	16.00	24	0.667
Total	130.80	29

The hypothesis being tested is:

H0: There is no main effect of Factor A

Ha: There is a main effect of Factor A

The p-value from the output is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is a main effect of Factor A.

The hypothesis being tested is:

H0: There is no main effect of Factor B

Ha: There is a main effect of Factor B

The p-value from the output is 0.0196.

Since the p-value (0.0196) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is a main effect of Factor B.

The hypothesis being tested is:

H0: There is no interaction effect of Factor A and B

Ha: There is an interaction effect of Factor A and B

The p-value from the output is 0.0001.

Since the p-value (0.0001) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is an interaction effect of Factor A and B.

(b)

The output is:

Post hoc analysis
p-values for pairwise t-tests for Factor 2
		> 60% fat	< 30 % fat	30% - 60% fat
		3.7	4.1	4.8
> 60% fat	3.7
< 30 % fat	4.1	.2842
30% - 60% fat	4.8	.0060	.0672
Tukey simultaneous comparison t-values (d.f. = 24)
		> 60% fat	< 30 % fat	30% - 60% fat
		3.7	4.1	4.8
> 60% fat	3.7
< 30 % fat	4.1	1.10
30% - 60% fat	4.8	3.01	1.92
critical values for experimentwise error rate:
		0.05	2.50
		0.01	3.21

There is a significant 30% - 60% fat & > 60% fat.

(c)

Source	effect size
Factor 1	0.689
Factor 2	0.047
Interaction	0.141