Question

Drive my car (3.2, 4.3)

(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 $\overline{x}=8$years. Find the mean mileage of the cars in the sample. Show your work.

(c) Interpret the value of $s$in the context of this setting.

(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

Short answer

1. The least-squares regression line explained $77.0\%$of the variation between the variables.
2. When a car is $8$years old, the average mileage of the cars in the sample is $105,800$miles.
3. There is an average difference of 22723 miles between actual and expected distance.
4. No, the least-squares line was calculated using student data. Because student data is not representative of instructor data, we cannot use the least-squares line to forecast teacher car mileage.

Step-by-step solution

Part (a) Step 1: Given Information 

The value of $r^2$and the data fit of the least square line. For these data, the dot plot and Minitab output for least squares regression analysis.

Part (a) Step 2: explanation 

$R-sq$has a value of $77\%$ according to Minitab's report. This means that the linear relationship between automobile age (in years) and mileage explains $77\%$ of the variation in mileage.

Part (b) Step 1: Explanation 

The mean age of the students’ cars in the sample was $\overline{x}=8$. We have to find the mean mileage of the cars in the sample

Part (b) Step 2: Explanation 

Let's use the regression line equation:$\hat{y}=a+bx$

The constant $a$is stated in the column "Coef" and in the row "Constant": $a=-13832$

In the row with "Age" and the column with "Coef," the slope $b$is given: $b=14954$

The least-squares equation is then:

localid="1652895688419" $$\begin{gathered} \hat{y}=-13832+14954x \\ \overline{y}=-13,832+14954\overline{x} \\ \overline{y}=-13832+14954\times 8=105800 \end{gathered}$$

Part (c) Step 1: Given Information 

We have to interpret the value of $s$in the context of this setting.

Part (c) Step 2: Explanation 

Given in the question that, $22723$is the standard deviation from the minitab

To predict the mileage of cars based on their age, the least square regression line is utilised. The approximation of prediction error is given by the standard deviation of the residuals (s).

Part (d) Step 1: Given Information

Is it appropriate for a Council high school teacher to use the least squares line to predict a car's mileage based on its age?

Part (d) Step 2: Explanation 

The data and Minitab output are used to anticipate the mileage of student-owned cars based on their age. It is not fair to use the least square line to forecast a car's mileage from its age for a teacher because the least square line is based on a sample of automobiles owned by students, not teachers.