Question

Question A. The below data correspond to a two-factor ANOVA. (Independent-measures)

1. Using the following data below, determine whether the data is sufficient to conclude that there was a significant difference between your conditions. Show all of your work, including all of the steps in your hypothesis test, including a summary table.
   1. State hypotheses for all 3 questions that should be asked about this dataset.
   2. Perform all calculations (must show all work to get full credit!) and construct an ANOVA table. In ALL calculations/formulas, keep 4 decimals. In the table, round to 2 decimals.
   3. State criteria (locate the critical region) for rejection of the null for EACH of the 3 questions (alpha = .05). State whether or not you reject the null hypothesis for each.
2. Compute effect size, η², for both factors and the interaction. If your data are not significant, write “N/A – not significant” for this question.
3. If an interaction exists, compute the simple main effects for the data below. (Show all your work)

   Factor B
   Factor A		Level 1	Level 2
   	Level 1	M = 15.8 T = 79 SS = 18.8 n = 5	M=7 T = 35 SS = 10 n = 5	Mrow1 = Trow1 = nrow1 = 10	N = G = ΣX2 = 3781 k = 4
   	Level 2	M = 20.8 T = 104 SS = 8.8 n = 5	M = 3.8 T = 19 SS = 14.8 n = 5	Mrow2 = Trow2 = nrow2 = 10
   		M­col1 = Tcol1 = ncol1 = 10	Mcol2= Tcol2 = ncol2 = 10

Answer 1

Factor B
Factor A		Level 1	Level 2
	Level 1	M = 15.8	M=7	Mrow1 = 11.4	N = 40
		T = 79	T = 35	Trow1 = 114	G = 237
		SS = 18.8	SS = 10	nrow1 = 10	ΣX2 = 3781
		n = 5	n = 5		k = 4
	Level 2	M = 20.8	M = 3.8	Mrow2 = 12.3
		T = 104	T = 19	Trow2 = 123
		SS = 8.8	SS = 14.8	nrow2 = 10
		n = 5	n = 5
		M­col1 = 18.3	Mcol2= 5.4
		Tcol1 = 183	Tcol2 = 54
		ncol1 = 10	ncol2 = 10

a) Null and alternative hypothesis for Factor A:

Ho: There is no main effect due to factor A.

Ho: There is a main effect due to factor A.

Null and alternative hypothesis for Factor B:

Ho: There is no main effect due to factor B.

Ho: There is a main effect due to factor B.

Null and alternative hypothesis for interaction:

Ho: There is no interaction effect due to factor A and B.

Ho: There is an interaction effect due to factor A and B.

b)

N = 40

Replications, r =10

ΣX = 237

(ΣX)² =56169

ΣX² = 3781

SSA = Σ((ΣXⱼ)²/nⱼ) - (ΣX)²/N = (114²/20 + 123²/20) - 56169/40 = 2.0250

SSB = Σ((ΣXᵢ)²/nᵢ) - (ΣX)²/N = (183²/20 + 54²/20) - 56169/40 = 416.0250

SSBN = Σ((ΣX)²/n) - (ΣX)²/N = 460.0750

SSAxB = SSBN - SSA - SSB = 42.0250

SSW = SST - SSA - SSB - SSAxB = 1916.7000

SST = ΣX² - (ΣX)²/N = 3781 - 56169/40 = 2376.7750

dfA = a - 1 = 1

dfB = b-1 = 1

dfAxB = (a-1)*(b-1) = 1

dfW = ab(r-1) = 36

dfT = N-1 = 39

MSA = SSA/dfA = 2.025/1 = 2.0250

MSB = SSB/dfB = 416.025/1 = 416.0250

MSAxB = SSAxB/dfAxB = 42.025/1 = 42.0250

MSW = SSW/dfW = 1916.7/36 = 53.2417

F for Factor A = MSA/MSW = 0.0380

p-value for Factor A = F.DIST.RT(0.038, 1, 36) = 0.8465

Critical value for Factor A = F.INV.RT(0.05, 1, 36) = 4.1132

F for Factor B = MSB/MSW = 7.8139

p-value for Factor B = F.DIST.RT(7.8139, 1, 36) = 0.0083

Critical value for Factor B = F.INV.RT(0.05, 1, 36) = 4.1132

F for interaction = MSAxB/MSW = 0.7893

p-value for Interaction = F.DIST.RT(0.7893, 1, 36) = 0.3802

Critical value for Interaction = F.INV.RT(0.05, 1, 36) = 4.1132

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Treatment	460.0750	3
Factor A	2.0250	1	2.0250	0.0380	0.8465	4.1132
Factor B	416.0250	1	416.0250	7.8139	0.0083	4.1132
Interaction	42.0250	1	42.0250	0.7893	0.3802	4.1132
Within	1916.7000	36	53.2417
Total	2376.7750	39

The critical F value for factor A is 4.11. Therefore, the main effect due to factor A is insignifcant because calculated value of F = 0.04 < Fc = 4.11.

The critical F value for factor B is 4.11. Therefore, the main effect due to factor B is significant because calculated value of F = 7.81 > Fc = 4.11.

The critical F value for interaction of factor A and B is 4.11. Therefore, the effect due to interaction of factor A and B is insignifcant because calculated value of F = 0.79 < Fc = 4.11.

c) η² for factor A:

η²A = N/A

η² for factor B:

η²B = SSB /(SST - SSA - SSAxB) = 416.025/(2376.775 - 2.025 - 42.025) = 0.1783 = 17.83%

η² for Interaction:

η²AxB = N/A