Chapter 3, Problem R3.3RE

(a)

To determine

To explain is a linear model appropriate for these data.

Answer to Problem R3.3RE

The linear model appropriate for these data.

Explanation of Solution

As the pattern in the scatterplot is roughly linear and does not contain strong curvature. Moreover, the points do not appear to vary much about the regression line and thus the linear model appears appropriate. The pattern in the residual contains no strong curvature contains roughly the same vertical spread everywhere.

(b)

To determine

To find out what is the correlation between the car age and mileage and interpret it.

Answer to Problem R3.3RE

  $r=0.9149$ .

Explanation of Solution

Since the slope given in the computer output is positive thus the correlation will be positive. Thus in the computer output it is given that,

  $\begin{array}{l}{r}^2=83.7\%\\ =0.837\end{array}$

Thus the correlation will be as:

  $\begin{array}{l}r=+\sqrt{r^2}\\ =+\sqrt{0.837}\\ =0.9149\end{array}$

The correlation is positive this means that there is a positive relationship between the age and mileage. And it is close to one then it is strong.

(c)

To determine

To give the equation of the least square regression line for these data.

Answer to Problem R3.3RE

  $\widehat{y}=3704+12188x$ .

Explanation of Solution

It is given in the computer output that:

  $\begin{array}{l}a=3704\\ b=12188\end{array}$

Thus the regression line will be as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=3704+12188x\end{array}$

Where x is age and y is mileage.

(d)

To determine

To find and interpret its residual.

Answer to Problem R3.3RE

Residual is $-11863$ .

Explanation of Solution

The regression line is:

  $\widehat{y}=3704+12188x$

Thus the predicted value is :

  $\begin{array}{l}\widehat{y}=3704+12188x\\ =3704+12188(6)\\ =76832\end{array}$

Thus the residual is as:

  $\begin{array}{l}\text{Residual}=y-\widehat{y}\\ =65000-76832\\ =-11863\end{array}$

This implies that the predicted mileage is $11863$ above the actual mileage when the age is six years.

(e)

To determine

To interpret the values of s and $r^2$ .

Explanation of Solution

It is given in the computer output that:

When using the least-squares regression equation, where x ( x = age of car) is used to predict y ( y = number of miles driven mileage), the prediction will typically be off by $20870$ miles.

  $83.7\%$ of the variation in mileage is accounted for by the linear model relating mileage to age of the car.