Chapter 10.1, Problem 36E

(a)

To determine

To explain what the value of $r^2$ tells you about how well the least squares line fits the data.

Explanation of Solution

It is given in the question the table in which the values are calculated. Thus, from table we have,

The value of $r^2$ is given in the output as “R-Sq”:

  $r^2=77\%=0.77$

From this, we interpret that $77\%$ of the variation between the variables has been explained by the least squares regression line.

(b)

To determine

To find the mean mileage of the cars in the sample.

Answer to Problem 36E

The mean mileage is $105800$ miles.

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.

The point $(\overline{x},\overline{y})$ lies on the least square regression line thus the mean mileage can be obtained by replacing $x\text{ by }\overline{x}$ in the least square equation and evaluating the expression:

  $\begin{array}{l}\overline{y}=\widehat{y}\\ =-13832+14954\overline{x}\\ =-13832+14954(8)\\ =105800\end{array}$

Thus, the mean mileage is $105800$ miles.

(c)

To determine

To interpret the value of $s$ .

Answer to Problem 36E

The error made when predicting the mileage using the least square regression line is on average $22723$ miles.

Explanation of Solution

In the question it is given the table of calculated values.

Thus, from that it is given the value of $s$ , it is as:

  $s=22723$

Thus, from this we interpret that the error made when predicting the mileage using the least square regression line is on average $22723$ miles.

(d)

To determine

To explain would it be reasonable to use the least square line to predict a car’s mileage from its age for a Council high school teacher.

Explanation of Solution

It is not be reasonable to use the least square line to predict a car’s mileage from its age for a Council high school teacher because the least-squares line was determined using data of students. The data of students is not representative for the teachers and thus we cannot use the least-squares line to predict the mileage of cars of teachers.