Question

In an experiment to see whether the amount of coverage of light-blue interior latex paint depends...

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.

H0A: α1 = α2 = α3 = α4 = 0
HaA: no αi = 0H0A: α1 = α2 = α3 = α4 = 0
HaA: at least one αi ≠ 0    H0A: α1α2α3α4
HaA: all αi's = 0H0A: α1α2α3α4
HaA: at least one αi = 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.

H0B: β1β2β3
HaB: at least one βj = 0H0B: β1 = β2 = β3 = 0
HaB: no βj = 0    H0B: β1 = β2 = β3 = 0
HaB: at least one βj ≠ 0H0B: β1β2β3
HaB: all βj'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.)

x1. and x2.x1. and x3.x1. and x4.x2. and x3.x2. and x4.x3. and x4.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

Homework Answers

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)

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

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

c)

H0B: β1 = β2 = β3 = 0
HaB: at least one βj ≠ 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

x1. and x3.

.x1. and x4.

Brand 1 is clearly preferable to the others

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