Question

Consider the following partially completed two-way ANOVA table. Suppose there are four levels of Factor A and three levels of Factor B. The number of replications per cell is 5. Use the 0.05significance level. (HINT: estimate the values from the F table)

A) Complete the ANOVA Table (Round MS to 1 decimal point, Round the other values to the nearest whole number.)

Source	SS	dF	MS	F
Factor A	75
Factor B	25
Interaction	300
Error	600
Total	1000

b) Find the critical values to test for equal means. "Critical values to test for equal means are Factor A(blank), Factor B(blank), And Interaction(blank) respectively."

c) Determine if there is a significant difference in Factor A means, Factor B means. "There is (difference/no difference) in Factor A and there is (difference/no difference) in Factor B means".

d) Determine if there is a significant difference in interaction means. "There is (interaction/ no interaction)"

Answer 1

Given:

Suppose there are four levels of Factor A i.e a = 4

three levels of Factor B. i.e. b = 3

The number of replications per cell is 5, r = 5

Degrees of freedom (df):

The degrees of freedom of factor A is, a-1 = 4-1 = 3

The degrees of freedom of factor B is, b-1 = 3-1 = 2

The degrees of freedom of Interaction effect is,(a-1)*(b - 1) = (4-1)*(3-1) = 3*2 = 6

degrees of freedom_within = ab(r-1) = 4*3*(5-1) = 4*3*4 = 48

degrees of freedom_Total =abr-1 = 4*3*5 = 60 -1 = 59

MS:

MS = SS / Corresponding degrees of freedom.

MS_A = SS_A / df_A = 75/3 = 25

MS_B = SS_B / df_B = 25/2 = 12.5

MS_Interaction = SS_Interaction / df_Interaction = 300 / 6 = 50

MS_error = SS_error / df_error = 600 / 48 = 12.5

F value:

F = MS / MS_Error

F_A = MS_A / MS_Error = 25 / 12.5 = 2

F_B = MS_B / MS_Error = 12.5 / 12.5 = 1

F_Interaction = MS_Interaction / MS_Error = 50 / 12.5 = 4

ANOVA TABLE:

Source	SS	df	MS	F
Factor A	75	= a-1 = 4-1 = 3	25	2
Factor B	25	=b-1 = 3-1 = 2	12.5	1
Interaction	300	=(a-1)*(b-1) = 3*2 = 6	50	4
Error	600	= ab(r-1) = 4*3*(5-1) = 4*3*4 = 48	12.5
Total	1000	=abr-1 = (4*3*5)- 1 = 59

b)

Critical values:

C)

F_Calculated_A < Critical value of factor A, i.e. 2 < 2.7981, i.e. there is No significant difference for Factor_A

That is, there is No difference in Factor A

F_Calculated_B < Critical value of factor B, i.e. 1 < 3.1907, i.e. there No significant difference for Factor_B

That is, there is No difference in Factor B

d)

F_Calculated_Interaction > Critical value for Interaction, i.e. 4 > 2.2946, i.e. there significant difference for Interaction

That is, there is Interaction.