Question

A factorial experiment involving two levels of factor A and three levels of factor B resulted in the following data.

		Factor B
		Level 1	Level 2	Level 3
Factor A	Level 1	135	89	76
		165	65	94
	Level 2	125	128	119
		95	106	135

Test for any significant main effects and any interaction. Use α = 0.05.

Find the value of the test statistic for factor A. (Round your answer to two decimal places.)

Find the p-value for factor A. (Round your answer to three decimal places.)

p-value =  

State your conclusion about factor A.

Because the p-value > α = 0.05, factor A is significant.Because the p-value ≤ α = 0.05, factor A is not significant.    Because the p-value > α = 0.05, factor A is not significant.Because the p-value ≤ α = 0.05, factor A is significant.

Find the value of the test statistic for factor B. (Round your answer to two decimal places.)

Find the p-value for factor B. (Round your answer to three decimal places.)

p-value =

State your conclusion about factor B.

Because the p-value ≤ α = 0.05, factor B is significant.Because the p-value ≤ α = 0.05, factor B is not significant.    Because the p-value > α = 0.05, factor B is significant.Because the p-value > α = 0.05, factor B is not significant.

Find the value of the test statistic for the interaction between factors A and B. (Round your answer to two decimal places.)

Find the p-value for the interaction between factors A and B. (Round your answer to three decimal places.)

p-value =

State your conclusion about the interaction between factors A and B.

Because the p-value ≤ α = 0.05, the interaction between factors A and B is not significant.Because the p-value > α = 0.05, the interaction between factors A and B is not significant.    Because the p-value ≤ α = 0.05, the interaction between factors A and B is significant.Because the p-value > α = 0.05, the interaction between factors A and B is significant.

Answer 1

Let us construct an ANOVA to test various effectes:

Teh given data is

		Factor B
		Level1	Level2	Level3
Factor A	Level1	135	89	76
165	65	94
Total	300	154	170
Level2	125	128	119
95	106	135
Total	220	234	254

We have a 2X3 factorial experiments with 2 replivations each.

We shall foolow the following steps:

Let us denote by as the observation from the replicate of the   level of afctor A and level of Factor B.

1. Calculate the correction factor:

  

2. The total sum of squares :

We shall form a table of squares and add them to get the sum of squares:

18225	7921	5776
27225	4225	8836
15625	16384	14161
9025	11236	18225	156864

3. Sum of squares due to factor A:   

4. Sum of squares due to Factor B:

5. Sum of squares due to table

	Level1	Level2	Level3	Total
Level1	300	154	170	624
Level2	220	234	254	708
Total	520	388	424	1332

  

	Level1	Level2	Level3	Total
Level1	90000	23716	28900	142616
Level2	48400	54756	64516	167672
Total	138400	78472	93416	310288

6. The interaction SS

7. Error SS

ANOVA Table:

Source	df	SS	MS	F	Critical F
Main Effect A	1	588	588	2.0512	5.9874
Main Effect B	2	2328	1164	4.0605	5.1433
Interaction AXB	2	4376	2188	7.6325	5.1433
Error	6	1720	286.6667
Total	11

Find the value of the test statistic for factor A. 2.05

Find the p-value for factor A. (Round your answer to three decimal places.)

p-value = 0.202

State your conclusion about factor A.

Because the p-value > α = 0.05, factor A is not significant.

Find the value of the test statistic for factor B. 4.06

Find the p-value for factor B. (Round your answer to three decimal places.)

p-value =0.077

State your conclusion about factor B.

Because the p-value > α = 0.05, factor B is not significant.

Find the value of the test statistic for the interaction between factors A and B. 7.63

Find the p-value for the interaction between factors A and B.

p-value =0.023

State your conclusion about the interaction between factors A and B.

Because the p-value ≤ α = 0.05, the interaction between factors A and B is significant.