Question

sharon was interested in purchasing a specific model of a used car. she researched used cars and found 15 models of this car that were being sold in her local area. she recorded the number of miles driven (in thousands of miles) and the advertised price for each of the 15 cars. the output shown in the table is from a least - squares regression to predict price from the number of miles driven (in thousands of miles).
term coef se coef
constant 26,411.96 1,139.82
mileage (1000s of miles) - 117.84 16.06
sharon finds another model of this car that has an advertised price of $18,998 and was driven for 49,500 miles. based on the residual, does the regression model overestimate or underestimate the price of the car?
a underestimate, because the residual is positive.
b underestimate, because the residual is negative.
c overestimate, because the residual is positive.
d overestimate, because the residual is negative.
e neither, because the residual is 0.

Explanation:

Step1: Write the regression equation

The regression equation is of the form $y = a+bx$, where $a = 26411.96$ (constant coefficient) and $b=- 117.84$ (mileage coefficient). So the equation is $y=26411.96 - 117.84x$, where $x$ is the number of miles driven in thousands of miles.

Step2: Convert the mileage of the new - car to thousands of miles

The new car has been driven 49.5 thousand miles ($x = 49.5$) since $49500\div1000 = 49.5$.

Step3: Calculate the predicted price

Substitute $x = 49.5$ into the regression equation: $y=26411.96-117.84\times49.5=26411.96 - 5833.08=20578.88$.

Step4: Calculate the residual

The residual $e$ is given by $e = \text{observed}-\text{predicted}$. The observed price is $18998$. So $e=18998 - 20578.88=- 1580.88$. A negative residual means the regression model over - estimates the price.

Answer:

D. Overestimate, because the residual is negative