In a study of the effects of octane rating of gasoline and speed (mph) at which an automobile is driven on gasoline mileage (mpg), 20 test runs of cars of the same make were driven with different fuels and at various speeds. Data are available in a file named octane.sav.

1. Plot mpg against each of the predictors, and plot octane against speed, and comment on the apparent relationships.
   
2. Compute all bivariate correlations. Comment on direction and strength of each.
3. Find the multiple regression equation relating mpg to speed and octane rating.
   
4. Explain, in terms pertinent to this problem, what the coefficient on octane in the multiple regression tells you.
   
5. How do you explain the fact that the correlation between speed and mpg was positive, but the coefficient on speed in the multiple regression is negative?
   
6. The intercept of this equation is not interpretable, since it makes no sense to talk about cars run at speed = 0 with octane = 0. To obtain an interpretable intercept, shift the origin to the lowest practical levels for the independent variables. The lowest octane level was 87, and the slowest speed was 46, so define two new variables,
   

X ₁ '= X ₁ – 87

X ₂ '= X ₂ – 46

and re-run the regression, using these variables as predictors.

What does the intercept of this equation tell you?

Did the shifting of the origin affect the slopes relating mpg to octane and speed?

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