Stats teachers’ cars A random sample of AP Statistics teachers were asked to report the age (in years) and mileage of their primary vehicles. A scatterplot of the data, a least-squares regression printout, and a residual plot are provided below.

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

(b) One teacher reported that her $6$-year-old car had$65,000$ miles on it. Find its residual.

(c) Interpret the slope of the line in context.

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

(e) How well does the regression line fit the data? Justify your answer using the residual plot and s.

A scatter plot with least square regression and residual plot are given.

Linear regression is commonly used for predictive analysis and modeling.

The following is the regression equation to be determined:

The variables used in the study are as follows, based on the information provided:

The dependent variable is $Y$ which stands for car mileage.

Let's call the independent variable $X$ the car's age.

The following is the regression equation:$\stackrel{⏜}{y}=7288.84+11630.6x$

The car is $6$ years old and has $65,000$ kilometers on it.

Substitution method is used.

The residual value is obtained as below,

Given the data point values of $x=6,y=65,000$

The value of $x$ is clearly $6$ based on the data point provided.

$$\begin{gathered} \stackrel{⏜}{y}=7288.84+11630.6x \\ \stackrel{⏜}{y}=7288.84+11630.6\left(6\right) \\ \stackrel{⏜}{y}=7288.84+69783.6 \\ \stackrel{⏜}{y}=77072.44 \end{gathered}$$

The residual value is denoted by the letter $e$

As a result, the residual is as follows:

$$\begin{gathered} e=y-\stackrel{⏜}{y} \\ =65,000-77072.44 \\ e=-12,072.44 \end{gathered}$$

Thus, the residual value is $-12,072.44$

It follows $x$ and $y$ variables.

The slope of the regression line should be interpreted.

The following is how the slope is interpreted:

The slope value is $11630.6$based on the information provided.

Interpret: That example, the dependent variable mileage of the car$\left(y\right)$is projected to rise by $11630.6$miles for every one year increase in the age of the car, the independent variable $\left(x\right)$

The independent variable $x$is equal to a one-year rise in the car's age, while the dependent variable $y$ is equal to an increase of $11630.6$miles.

The correlation between car age and mile age is to be interpreted.

Formula used$r=\sqrt{r^2}$

From the output, the correlation is, $r=\sqrt{r^2}$

$$\begin{gathered} r=\sqrt{0.82} \\ r=0.91 \end{gathered}$$

The correlation is close to $1$ in this case. As a result, the age of an automobile and its mileage have a perfect positive relationship.

The regression line correctly suited the data.

The residual plot's attributes are as follows:

• The horizontal axis of a residual plot generated against the predictor variables must be slightly centered and symmetric.
• A residual plot generated against the response variable's expected values must be cantered and symmetric around the horizontal axis.
• The residuals' normal probability plot must be linear or nearly linear.

It is clear from the graph of the residual vs the fitted values that only one horizontal axis about the symmetry is displayed. It means that there is more variety among the residuals. There is a little discrepancy between the symmetry values. The residuals are not distributed evenly on either side of the horizontal axis. As a result, the assumption of the conditional standard deviation's constancy is likely to be violated. The assumption of constancy of the conditional standard deviation is likely to be violated.