Question

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:

(a) What is the equation of the least-squares regression line? Be sure to define any symbols you use.

(b) Interpret the slope of the least-squares line in the context of this problem. (c) One student reported that her $10$-year-old car had$110000$ miles on it. Find the residual for this data value. Show your work

Short answer

(a) The equation of the least-squares regression line is $\hat{y}=-13832+14954x$

(b) The slope of the least-squares line in the context of this problem is $14954$miles per year

(c) The residual for this data value is$25708$miles.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

The least-squares regression line's general equation is:

$\hat{y}=a+bx$

with $\overline{y}$the predicted mileage and x the age

The constant $a$is given in the column "Coef" and in the row "Constant":

$a=-13832$

The slope $b$is given in the row with "Age" and in the column with "Coef"

$b=14954$

The least-square equation then becomes

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

Part (b) Step 1: Given information

Part (b) Step 2: Explanation

From part (a)

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

$\overline{y}=$predicted mileage

$x=$age

The slope is the coefficient of x

Slope $=14954$.

Part (c) Step 1: Given information

Part (c) Step 2: Explanation

From part (a)

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

Replace $x$with $10$

$\overline{y}=-13832+14954\left(10\right)=135708miles$

The discrepancy between the actual and predicted values is the residual:

Residual$=y-\hat{y}=110000-135708=-25708miles$.