Question

The results of a​ two-way ANOVA using the accompanying data and hypothesis tests for interaction between Factor A and Factor B and for each factor are provided below. Using these data and​ results, determine which means are different using α= 0.01when warranted.

Factor A
FactorB	Level 1	Level 2	Level 3
Level 1	66	88	1010
	77	1212	3232
	1010	1919	3030
Level 2	2828	3030	3030
	1111	2020	3838
	1616	2020	3939
Level 3	3535	4040	4242
	3131	2929	4141
	2121	4040	3232

Are the means for Factor​ A, Level 1 and Factor​ A, Level 2 significantly​ different?

A.Yes

B. No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ A, Level 1 and Factor​ A, Level 3 significantly​ different?

A.Yes

B. No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ A, Level 2 and Factor​ A, Level 3 significantly​ different?

A.Yes

B.No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 1 and Factor​ B, Level 2 significantly​ different?

A.Yes

B.No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 1 and Factor​ B, Level 3 significantly​ different?

A.No

B.Yes

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 2 and Factor​ B, Level 3 significantly​ different?

A.No

B.Yes

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Answer 1

The results of a​ two-way ANOVA using the accompanying data and hypothesis tests for interaction between Factor A and Factor B and for each factor are provided below. Using these data and​ results, determine which means are different using α= 0.01when warranted.

MINITAB used.

General Linear Model: Data versus A, B

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
A	Fixed	3	A1, A2, A3
B	Fixed	3	B1, B2, B3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
A	2	11325157	5662578	9.66	0.001
B	2	21976136	10988068	18.75	0.000
A*B	4	1484102	371026	0.63	0.645
Error	18	10548520	586029
Total	26	45333915

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
765.525	76.73%	66.39%	47.65%

Comparisons for Data

Tukey Pairwise Comparisons: A

Grouping Information Using the Tukey Method and 99% Confidence

A	N	Mean	Grouping
A3	9	3299.33	A
A2	9	2366.44	A	B
A1	9	1721.67		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of A Levels	Difference of Means	SE of Difference	Simultaneous 99% CI	T-Value	Adjusted P-Value
A2 - A1	645	361	(-555, 1844)	1.79	0.202
A3 - A1	1578	361	(378, 2777)	4.37	0.001
A3 - A2	933	361	(-266, 2132)	2.59	0.047

Individual confidence level = 99.62%

Tukey Pairwise Comparisons: B

Grouping Information Using the Tukey Method and 99% Confidence

B	N	Mean	Grouping
B3	9	3490.11	A
B2	9	2603.56	A
B1	9	1293.78		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of B Levels	Difference of Means	SE of Difference	Simultaneous 99% CI	T-Value	Adjusted P-Value
B2 - B1	1310	361	(110, 2509)	3.63	0.005
B3 - B1	2196	361	(997, 3396)	6.09	0.000
B3 - B2	887	361	(-313, 2086)	2.46	0.060

Individual confidence level = 99.62%

Are the means for Factor​ A, Level 1 and Factor​ A, Level 2 significantly​ different?

A.Yes

Answer: B. No   ( p value 0.202 which is > 0.01 level)

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ A, Level 1 and Factor​ A, Level 3 significantly​ different?

Answer: A.Yes ( p value 0.001 which is < 0.01 level)

B. No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ A, Level 2 and Factor​ A, Level 3 significantly​ different?

A.Yes

Answer: B.No( p value 0.047 which is > 0.01 level)

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor A means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 1 and Factor​ B, Level 2 significantly​ different?

Answer A.Yes ( p value 0.005 which is < 0.01 level)

B.No

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 1 and Factor​ B, Level 3 significantly​ different?

A.No

Answer B.Yes( p value 0.000 which is < 0.01 level)

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.

Are the means for Factor​ B, Level 2 and Factor​ B, Level 3 significantly​ different?

Answer:A.No( p value 0.060 which is > 0.01 level)

B.Yes

C.The comparison is unwarranted because there is insufficient evidence to conclude that not all Factor B means are equal.

D.The comparison is unwarranted because Factor A and Factor B interact.