Thirty randomly selected seniors at Council High School were asked to report the age (in years) and mileage of their main vehicles. Here is a scatterplot of the data:

We used Minitab to perform a least-squares regression analysis for these data. Part of the computer output from this regression is shown here.

a. Explain what the value of $r^2$tells you about how well the least-squares line fits the data.

b. The mean age of the students’ cars in the sample was $\overline{x}=5$years. Find the mean mileage of the cars in the sample.

c. Interpret the value of $s$.

d. Would it be reasonable to use the least-squares line to predict a car’s mileage from its age for a Council High School teacher? Justify your answer.

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.

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

$\hat{y}=a+bx$

With the predicted mileage and 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:

$\hat{y}=-13832+14954x$

With $\hat{y}$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 $x$by $\overline{x}$in the least square equation and evaluating the expression:

$$\begin{gathered} \overline{y}=\hat{y} \\ =-13832+14954\overline{x} \\ =-13832+14954\left(8\right) \\ =105800 \end{gathered}$$

Thus, the mean mileage islocalid="1664182337396" $105800$.

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.

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.