Question

Stats teachers’ cars A random sample of AP® Statistics teachers were asked to

report the age (in years) and mileage of their primary vehicles. Here are a scatterplot, a residual plot, and other computer output:

a. Is a linear model appropriate for these data? Explain how you know this.

b. What’s the correlation between car age and mileage? Interpret this value in context.

c. Give the equation of the least-squares regression line for these data. Identify any variables you use.

d. One teacher reported that her 6-year-old car had 65,000 miles on it. Find and interpret its residual.

e. Interpret the values of s and r2.

Short answer

Part (a) The linear model appropriate for these data.

Part (b)$r=0.9149$

Part (c)$\stackrel{⏜}{y}=3704+12188x$

Part (d) Residual is$-11863$

Part (e) The linear model connecting mileage to vehicle age accounts for $83.7\%$percent of the variation in mileage.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

Because the scatterplot's pattern is essentially linear and lacks substantial curvature. Furthermore, the points appear to vary little around the regression line, implying that the linear model is adequate. The residual pattern has no noticeable curvature and has nearly the same vertical spread throughout.

Part (b) Step 1: Calculation

The correlation will be positive because the slope in the computer output is positive. As a result, it is stated in the computer output that,

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

Thus the correlation will be as:

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

There is a positive association. This indicates that age and mileage have a favorable association. And if it's near to one, it's powerful.

Part (c) Step 1: Explanation

It is given in the computer output that:

$$\begin{gathered} a=3704 \\ b=12188 \end{gathered}$$

Thus the regression line will be as:

$$\begin{gathered} \stackrel{⏜}{y}=a+bx \\ \Rightarrow \stackrel{⏜}{y}=3704+12188x \end{gathered}$$

Where $x$ is age and $y$ is mileage.

Part (d) Step 1: Explanation

The regression line is:

$\stackrel{⏜}{y}=3704+12188x$

Thus the predicted value is :

$$\begin{gathered} \stackrel{⏜}{y}=3704+12188x \\ =3704+12188\left(6\right) \\ =76832 \end{gathered}$$

Thus the residual is as:

$$\begin{gathered} Residual=y-\stackrel{⏜}{y} \\ =65000-76832 \\ =-11863 \end{gathered}$$

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

Part (e) Step 1: Explanation

It is given in the computer output that:

The least-squares regression equation, which uses $x$ (vehicle age) to estimate $y$ (number of miles driven mileage), is normally wrong by $20870$ miles. The linear model connecting mileage to vehicle age accounts for $83.7$ percent of the variation in mileage.