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 61 48 50

  2 54 51 47

3 46 44 51

4 50 41    51

(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.

H0A: α1 = α2 = α3 = α4 = 0 HaA: no αi = 0

H0A: α1 = α2 = α3 = α4 = 0 HaA: at least one αi ≠ 0

H0A: α1 ≠ α2 ≠ α3 ≠ α4 HaA: at least one αi = 0

H0A: α1 ≠ α2 ≠ α3 ≠ α4 HaA: all αi's = 0

State the appropriate hypotheses for the soil type factor. H0B: β1 = β2 = β3 = 0 HaB: at least one βj ≠ 0 H0B: β1 = β2 = β3 = 0 HaB: no βj = 0 H0B: β1 ≠ β2 ≠ β3 HaB: at least one βj = 0 H0B: β1 ≠ β2 ≠ β3 HaB: all βj'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 H0A? Reject H0A. The data suggests that there is a coating effect. Fail to reject H0A. The data suggests that there is a coating effect. Reject H0A. The data does not suggest that there is a coating effect. Fail to reject H0A. The data does not suggest that there is a coating effect.

What can you conclude for H0B? Fail to reject H0B. The data suggests that there is a soil type effect. Fail to reject H0B. The data does not suggest that there is a soil type effect. Reject H0B. The data does not suggest that there is a soil type effect. Reject H0B. The data suggests that there is a soil type effect.

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

mu hat =

alpha hat1 =

alpha hat2 =

alpha hat3 =

alpha hat4 =

beta hat1 =

beta hat2 =

beta hat3 =

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

a)

H0A: α1 = α2 = α3 = α4 = 0

HaA: at least one αi ≠ 0

H0B: β1 = β2 = β3 = 0

HaB: at least one βj ≠ 0

Source of Variation	df	SS	MS	F-stat		F-critical
Rows	3	73.67	24.56	1.25		4.76
Columns	2	91.50	45.75	2.33		5.14
Error	6	117.83	19.64
Total	11	283.00

Fail to reject H0A. The data does not suggest that there is a coating effect.

Fail to reject H0B. The data does not suggest that there is a soil type effect.

b)

mu hat =49.50

alpha hat1 = 3.50

alpha hat2 = 1.17

alpha hat3 = -2.50

alpha hat4 = -2.17

beta hat1 = 3.25

beta hat2 = -3.50

beta hat3 = 0.25