Question 1. A tire manufacturing company was interested in the effect of 4 alter

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Question 1.

A tire manufacturing company was interested in the effect of 4 alternate rubber compounds on the life of automobile tires. They randomly selected 4 tires from each of the 4 alternate rubber compounds. The company also purchased 4 cars of identical make for use in the experiment. For each different rubber compound, randomly assign a different tire position (front left, front right, back left, back right) then randomly assign each tire to an automobile and put it in the already assigned position on that car, because they felt that there could be variation in how the tires wear at the different positions, though they were not scientifically interested in this variation. Then each automobile was driven for 40,000 miles and the amount of wear on each of the four tires was recorded on a 30 point scale. See the results below.

a). Which factor is the treatment factor? Which factor is the block factor? Explain why.

b). Compute the ANOVA table.

c). Test the overall model at the .05 level of significance. Clearly state all pieces needed to conduct this test.

d). Test for block effects at the .05 level of significance. Clearly state all pieces needed to conduct this test.

e). Test at the .05 level of significance whether or not the type of compound significantly effects the life of a tire. Clearly state all pieces needed to conduct this test.

Front Left Front Right Back Left Back Right Totals Compound 1 22 20 15 18 75 Compound 2 21 18 19 22 80 Compound 3 25 22 21 23 91 Compound 4 29 20 23 27 99 Totals 97 80 78 90 345

Explanation / Answer

a) Here treatment factor is rubber compound and block factor is tire position. Here, the main interest is to check the effect of four different rubber compound of tires. So, clearly this is the treatment factor that we want to test. But, tire positions are assigned randomly to each of the tires of each of the compounds, to reduce the unexplainable variation, unknowingly, as they felt that the position may lead to some kind of variation, as mentioned in the question. So, clearly, the tire position is introduced to arrange experimental units in several 'homogeneous' groups in order to reduce (residual) variability that can not be explained by the model, which is the main aim behind blocking in design of experiments. Hence, tire compound is the treatment effect and position is the blocking effect.

b) Clearly this is a problem of randomized block design. The anova table for such model and for the given data comes as,

c) The total Sum of squares explained by Position and Tire compound is (87.69+59.19)=146.88 with (3+3)=6 degrees of freedom. That gives MS(Tire+Position)=24.48 and after dividing it by MS(error) we get F-value for the overall model as= 6.283 on 6 and 9 DF. Which gives,

P-value, i.e. Pr(F with DF 6 and 9> observed F-value) at level 0.05.

P-value for the model as 0.0077 which is less than 0.05, i.e. the level of significance. Hence, we can say that this model is significant in explaining variablity at 5% level.

d) With the similar arguement we get

P-value for block effect as

P(F with DF 3 and 9> 5.064)= 0.025 , which is less than the level of significance 0.05. Hence, block effect or the tire position has significant effect at 5% level.

e) For the treatment effect i.e. the compound of the tire we get, p-value as

P( F with DF 3 and 9 > 7.503) = .008, which is less than the given level 0.05 and infact less than 0.01 also. So, it can be inferred that the compound of the tire has a high significance in the life of the tire.

Source DF SS MS F Tire 3 87.69 29.229 7.503 Position 3 59.19 19.729 5.064 Error 9 35.06 3.896 X Total 15 181.94 X X

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