Until recently, performance tires were fitted mostly on sporty or luxury vehicles. Now they come standard on many everyday sedans. Increased levels of handling and grip have come at the expense of tread wear. The data that follows is abstracted from a report on H-rated performance tires by Consumer Reports8 in which several aspects of performance were evaluated for n = 22 different tires where

y = overall score.............x₁ = dry braking
x₂ = wet braking.............x₃ = handling
x₄ = roll resistance.........x₅ = tread life

The variables x₁ through x₅ are coded using the scale 5 = excellent, 4 = very good, 3 = good, 2 = fair, and 1 = poor.

a. Use a program of your choice to find the correlation matrix for the variables under study including cost. Is cost significantly correlated with any of the study variables? Which variables appear to be highly correlated with y, the total score?

b. Write a model to describe y, total score, as a function of the variables x₁ = dry braking, x₂ = wet braking, x₃ = handling, x₄ = roll resistance, and x₅ = tread life.

c. Use a regression program of your choice to fit the full model using all of the predictors. What proportion of the variation in y is explained by regression? Does this convey the impression that the model adequately explains the inherent variability in y?

d. Which variable or variables appear to be good predictors of y? How might you refine the model in light of these results? Use these variables in refitting the model. What proportion of the variation is explained by this refitted model? Comment on the adequacy of this reduced model in comparison to the full model.