Question

In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, one gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of square feet covered).

		Roller Brand
		1	2	3
Paint Brand	1	454	446	451
	2	446	442	446
	3	439	442	444
	4	444	437	442

(a)

Construct the ANOVA table. [Hint: The computations can be expedited by subtracting 400 (or any other convenient number) from each observation. This does not affect the final results.] (Round your answers to two decimal places.)

Source	df	SS	MS	f	F0.05
Paint Brand
Roller Brand
Error
Total

(b)

Test hypotheses appropriate for deciding whether paint brand has any effect on coverage. Use

α = 0.05.

State the appropriate hypotheses.

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

What can you conclude?

Fail to reject H₀. The data suggests that paint brand has an effect.Reject H₀. The data suggests that paint brand has an effect.    Reject H₀. The data does not suggest that paint brand has an effect.Fail to reject H₀. The data does not suggest that paint brand has an effect.

(c)

Repeat part (b) for brand of roller.

State the appropriate hypotheses.

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

What can you conclude?

Fail to reject H₀. The data suggests that roller brand has an effect.Reject H₀. The data does not suggest that roller brand has an effect.    Reject H₀. The data suggests that roller brand has an effect.Fail to reject H₀. The data does not suggest that roller brand has an effect.

(d)

Use Tukey's method to identify significant differences among paint brands. (Round your answer to two decimal places.)

w =

Which means differ significantly? (Select all that apply.)

x_1. and x_2.x_1. and x_3.x_1. and x_4.x_2. and x_3.x_2. and x_4.x_3. and x_4.There are no significant differences.

Is there one brand that seems clearly preferable to the others?

Paint Brand 1Paint Brand 2    Paint Brand 3Paint Brand 4None of the brands is clearly preferable to the others

Answer 1

a_)

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
Rows	3	162.92	54.31	8.28	4.76
Columns	2	42.67	21.33	3.25	5.14
Error	6	39.33	6.56
Total	11	244.92

b)

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

.Reject H₀. The data suggests that paint brand has an effect.

c)

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

Fail to reject H₀. The data does not suggest that roller brand has an effect.

d)

MSE=			6.56
df(error)=			6
number of treatments =			4
pooled standard deviation=Sp =√MSE=			2.560

				critical q with 0.05 level and k=4, N-k=6 df=			4.90
				Tukey's (HSD)=w =(q/√2)*(sp*√(1/ni+1/nj) =			7.24

x_1. and x_3.

_.x_1. and x_4.

Brand 1 is clearly preferable to the others