Chapter 10.1, Problem 35E

(a)

To determine

To calculate what is the equation of the least squares regression line.

Answer to Problem 35E

The regression line is: $\widehat{y}=-13832+14954x$ .

Explanation of Solution

As we know that the least square regression line is as:

  $\widehat{y}=a+bx$

With $\widehat{y}$ the predicted mileage and $x$ the age.

The constant $a$ is given in the row with “Age” and in the column with “Coef”:

  $a=-13832$

The slope $b$ is given in the row with “Age” and in the column with “Coef”:

  $b=14954$

Then the least square equation then becomes:

  $\widehat{y}=-13832+14954x$

With $\widehat{y}$ the predicted mileage and $x$ the age.

(b)

To determine

To interpret the slope of least square regression line.

Answer to Problem 35E

  $\text{Slope}=14954$ .

The mileage increases on average by about $14954$ miles per year.

Explanation of Solution

We get the result from part (a) that:

  $\widehat{y}=-13832+14954x$

With $\widehat{y}$ the predicted mileage and $x$ the age.

Then the slope is the coefficient of x :

  $\text{Slope}=14954$

This means that the mileage increases on average by about $14954$ miles per year.

(c)

To determine

To find and interpret the residual of the data point.

Answer to Problem 35E

The residual is $-25708\text{miles}$ .

Explanation of Solution

We get the result from part (a) that:

  $\widehat{y}=-13832+14954x$

With $\widehat{y}$ the predicted mileage and $x$ the age.

The data point is given to be $x=10$ . Then replace the value of x in the regression line, then,

  $\begin{array}{l}\widehat{y}=-13832+14954x\\ =-13832+14954(10)\\ =135708\text{miles}\end{array}$

Thus, the residual is the difference between the actual and predicted value then,

  $\begin{array}{l}\text{Residual}=y-\widehat{y}\\ =110000-135708\\ =-25708\text{miles}\end{array}$

This means that the predicts overestimates the mileage by $25708$ miles.