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Speed: 25 35 45 55 65

Mileage: 20 24 26 24 20

The lease-squares line for predicting mileage from speed is

mileage= 22.8 +(0*speed)

Question: The correlation between mileage and speed is r=0. What does this say about the usefulness of the regression line in predicting mileage?

I just make a scatterplot of the data and draw this line on the plot and noticed that it shows a strong non-linear relationship. However, I don't know how to answer this question. Is there anybody could help? Thanks!

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You are totally correct and a linear regression leads to this "stupid" result. But, I suppose that you observed that a scatterplot of the data reveals something looking like a parabola and effectively a quadratic regression $$M=a+b S+ c S^2$$ gives something which is much better.

Brute force would lead to $$M=-0.0142857 S^2+1.28571 S-3.27143$$ which is quite good but with the problem that the mileage would be negative for a zero speed ! So, let us make a more reasonable model $$M=a S+ b S^2$$ for which one could obtain $$M=1.13252 S-0.0126465 S^2$$ Using this last correlation for the different speeds $S$, we obtain as estimates of mileage $M$ the following values : $20.41$, $24.15$, $25.35$, $24.03$ and $20.18$.

But, if you want to stay with a linear model, you could instead consider $$\frac MS=a + b S$$ and get $$\frac MS=1.11679 -0.0123397 S$$ which is quite good.

Using this last correlation for the different speeds $S$, we obtain as estimates of mileage $M$ the following values : $20.20$, $23.97$, $25.27$, $24.10$ and $20.46$.

If you want to play with really nonlinear model, you could use $$M=a S + b S^c$$ and get $$M=1.00113 S-0.00308981 S^{2.2973}$$ Using this last correlation for the different speeds $S$, we obtain as estimates of mileage $M$ the following values : $20.00$, $24.15$, $25.65$, $24.30$ and $19.91$.

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