Stine and Foster Problem 20-37: Used Accords

Used Accords Cars depreciate over time. These data show the prices of Honda Accords listed for sale by individuals in The Philadelphia Inquirer. One column gives the asking price (in thousands of dollars) and a second column gives the age (in years).

a)Do you expect the resale value of a car to drop by a fixed amount each year?
No, cars usually depreciate the most during the early years of the car. The percentage depreciation might be constant, but the absolute amount of depreciation should decrease over time.

b)Fit a linear equation with price as the response and age as the explanatory variable. What do the slope and intercept tell you, if you accept this equation’s description of the pattern in the data?

Term / Estimate / Std Error / t Ratio / Prob>|t| / Lower 95% / Upper 95%
Intercept / 15.462647 / 0.825117 / 18.74 / <.0001* / 13.789232 / 17.136062
Age (years) / -0.946414 / 0.079924 / -11.84 / <.0001* / -1.108508 / -0.78432

The intercept suggests a new “used” car (just driven off the deal’s lot) is about $15,463. The slope indicates these cars drop in resale value by about $946 per year.

c)Plot the residuals from the linear equation on age. Do the residuals suggest a problem with the linear equation?


The linear regression misses the nonlinear pattern. The tell-tale smile in the residuals indicators a nonlinear pattern.

d)Fit the equation . Do the residuals from this fit “fix” the problem found in part (c)?

Summary of Fit

RSquare / 0.928067
RSquareAdj / 0.926069
Root Mean Square Error / 1.300732
Mean of Response / 6.646053
Observations (or Sum Wgts) / 38
Term / Estimate / Std Error / t Ratio / Prob>|t| / Lower 95% / Upper 95%
Intercept / 22.992566 / 0.787292 / 29.20 / <.0001* / 21.395864 / 24.589268
Log Age / -7.819326 / 0.362822 / -21.55 / <.0001* / -8.555162 / -7.083489

These residuals appear more random when compared to the pattern that is evident in the residuals from the linear equation.


e)Compare the fitted values from this equationwith those from the linear model. Show both in the same scatterplot. In particular, compare what this graph has to say about the effects of increasing age on resale value.

The natural logarithmic transformation for sales captures the nonlinear effect of a constant percentage decrease in the value of the Accords.

f)Compare the values of R-Squared and se between these two equations. Give units where appropriate. Does this comparison agree with your impression of the better model? Should these summary statistics be compared?

The R-Squared increases from 79% to 92% by using the logarithmic transformation. The standard error of the regression decreases from $2.19 to $1.30. The R-Squared and standard errors of regression are comparable in this case because we have the same response variable which is measured in dollars.

g)Interpret the intercept and slope in this equation.
The intercept gives the asking price for a car that is on the dealer’s lot is $22,993. If we increase the age of the car by 1% (.01) gives a change in price of 0.01(-7.8) = .0078 thousand dollars or $7.8.

h)Compare the change in asking price for cars that are 1 and 2 years old to that for cars that are 11 and 12 years old. Use the equation with the log of age as the explanatory variable. Is the difference the same or different?

Intercept / 22.992
Slope / -7.819
Age / Ln of Age / Equation / Profiler / Difference
1 / 0 / 23.0 / 23.0
2 / 0.693147 / 17.6 / 17.6 / 5.43
11 / 2.397895 / 4.2 / 4.2
12 / 2.484907 / 3.6 / 3.6 / 0.68

The car depreciates by $5,430 between years 1 and 2. It only depreciates $680 between years 11 and 12. This matches the answer that we anticipated in part (a) of this question.