Question

Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated with one of the four coatings and buried in one of the three types of soil for a fixed time, after which the amount of corrosion (depth of maximum pits, in 0.0001 in.) is determined. The data appears in the table.

		Soil Type (B)
		1	2	3
Coating (A)	1	62	47	51
	2	53	51	45
	3	48	45	50
	4	50	41	50

(a) Assuming the validity of the additive model, carry out the ANOVA analysis using an ANOVA table to see whether the amount of corrosion depends on either the type of coating used or the type of soil.
State the appropriate hypotheses for the coating effect.

H_0A: α₁ ≠ α₂ ≠ α₃ ≠ α₄
H_aA: at least one α_i = 0H_0A: α₁ ≠ α₂ ≠ α₃ ≠ α₄
H_aA: all α_i's = 0    H_0A: α₁ = α₂ = α₃ = α₄ = 0
H_aA: at least one α_i ≠ 0H_0A: α₁ = α₂ = α₃ = α₄ = 0
H_aA: no α_i = 0

State the appropriate hypotheses for the soil type factor.

H_0B: β₁ ≠ β₂ ≠ β₃
H_aB: at least one β_j = 0H_0B: β₁ = β₂ = β₃ = 0
H_aB: at least one β_j ≠ 0    H_0B: β₁ = β₂ = β₃ = 0
H_aB: no β_j = 0H_0B: β₁ ≠ β₂ ≠ β₃
H_aB: all β_j's are equal

Complete the ANOVA table. Use α = 0.05. (Round your answers to two decimal places.)

Source	df	SS	MS	f	F0.05
A
B
Error
Total

What can you conclude for H_0A?

Fail to reject H_0A. The data suggests that there is a coating effect.Reject H_0A. The data suggests that there is a coating effect.    Fail to reject H_0A. The data does not suggest that there is a coating effect.Reject H_0A. The data does not suggest that there is a coating effect.

What can you conclude for H_0B?

Reject H_0B. The data suggests that there is a soil type effect.Reject H_0B. The data does not suggest that there is a soil type effect.    Fail to reject H_0B. The data suggests that there is a soil type effect.Fail to reject H_0B. The data does not suggest that there is a soil type effect.

(b) Compute the model parameters. (Round your answers to two decimal places.)

	=
1	=
2	=
3	=
4	=
1	=
2	=
3	=

Answer 1

a)

H_0A: α₁ = α₂ = α₃ = α₄ = 0
H_aA: at least one α_i ≠ 0

H_0B: β₁ = β₂ = β₃ = 0
H_aB: at least one β_j ≠ 0

Applying two way ANOVA: (use excel: data: data analysis: two way ANOVA: without replication select Array):

Source of Variation	df	SS	MS	F	F crit
A	3	72.92	24.31	1.26	4.76
B	2	106.17	53.08	2.75	5.14
Error	6	115.83	19.31
Total	11	294.92

  Fail to reject H_0A. The data does not suggest that there is a coating effect.

Fail to reject H_0B. The data does not suggest that there is a soil type effect.

b)

µ =	49.42
α1 =	3.92
α2 =	0.25
α3 =	-1.75
α4 =	-2.42
ß1 =	3.83
ß2 =	-3.42
ß3 =	-0.42