Question

Consider the following partially completed two-way ANOVA table. Suppose there are 4 levels of Factor A and 2 levels of Factor B. The number of replications per cell is 4. Use the 0.01 significance level. (Hint: estimate the values from the F table.)

Complete an ANOVA table. (Round MS and F to 2 decimal places.)

Sources: Factor A, Factor B, Interaction, Error, total.

SS: 70, 50, 210, 400, 730

Find the critical values to test for equal means. (Round your answers to 2 decimal places.)

Determine if there is a significant difference in Factor A means, Factor B means.

Determine if there is a significant difference in interaction means.

Answer 1

To complete the table above, we should first obtain the table of formulas needed.

n= 4 * 2 *4 = 32

ANOVA
Source	SS	df	MS	F
Factor A	Factor A	k-1	MSA= SSA/(k-1)	MSA/MSE
Factor B	Factor B	b-1	MSB = SSB/(b-1)	MSB/MSE
Interaction	SSI	(k-1)(b-1)	MSI=SSI/(k-1)(b-1)	MSI/MSE
Error	SSE	n-kb	MSE = SSE/(n-kb)
Total	SS Total	n-1

Therefore, the completed ANOVA table is as follows:

ANOVA
Source	SS	df	MS	F
Factor A	70	3	23.33	1.339
Factor B	50	1	50	2.99
Interaction	210	3	70	4.199
Error	400	24	16.667
Total	730	31

To test whether there is a significant difference in factor A means, we will use critical value from F table with 0.05 significance level, df numerator (k-1) = 3, and df denominator (n-kb) = 24. The critical value is 3.01. The computed value of factor A is smaller than the critical value, therefore, we fail to reject the null hypothesis. We can conclude that the factor A means are equal.

To test whether there is a significant difference in factor B means, we will use critical value from F table with 0.05 significance level, df numerator (b-1) = 1, and df denominator (n-kb) = 24. The critical value is 4.26. The computed value of factor B is smaller than the critical value, therefore, we fail to reject the null hypothesis. We can conclude that the factor B means are equal.

To test whether there is a significant effect from the interaction between factor A and factor B, we will use critical value from F table with 0.05 significance level, df numerator (k-1)(b-1) = 3, and df denominator (n-kb) = 24. The critical value is 3.01. The computed value of interaction is higher than the critical value, therefore, we reject the null hypothesis of no interaction. We can conclude that the combination between factor A and factor B has a significant effect on the response factor.