3
MILEAGE AND PRICE OF USED CARS
The relationship between mileage and the price of used cars
Emily Parkins
Vesalius College
Author Note
Assignment 1 for STA101 Introduction to Statistics (Professor Luc Hens)
28 August 2014
Abstract
The selling price at auction of a random sample of 100 used Ford Tauruses tends to decrease with the mileage. The coefficient of correlation (-0.81) indicates a strong negative linear relationship. The line of best fit shows that an additional 1 000 miles on the odometer lowers the predicted selling price by about US$ 62. This effect is rather small and statistically significant (the p-value is less than 1%). The line of best fit explains 65% of the variation in the selling price.
EconLit subject codes: C210, L620
The relationship between mileage and the price of used cars
A car dealer wants to find the relationship between the mileage and the selling price of used cars. Economic theory suggests that the price of a consumer durable (like a car) is equal to the net present value of the expected future flow of services the good provides to the owner. As a used car with a higher mileage tends to last shorter, we expect a lower price. To investigate the relationship, I used information for a random sample of 100 used Ford Tauruses, provided by Keller & Warrack (2003, p. 610). The variables are the price at auction (with a mean of US$ 14 822 and a standard deviation of US$ 510) and the odometer reading in miles (with a mean of 36 009 miles and a standard deviation of 660 miles). The histograms (not shown) reveal that the distributions of both variables are roughly bell-shaped. A scatter plot and the line of best fit show the relationship between price and mileage (figure 1).
The coefficient of correlation is -0.81, indicating a strong and negative linear relationship between the mileage and the selling price of used cars. The equation of the line of best fit is:
predicted selling price = US$ 17 067 - (US$ 0.062/mile) ´ odometer reading
(standard error) (169.03) ( 0.0045)
R2 = 0.65
As expected, the higher the mileage of a second-hand Ford Taurus, the lower the predicted selling price. The estimated gradient (-0.062, with a 95%-confidence interval of [0.072;-0.053]) means that an additional mile lowers the predicted selling price by about US$ 0.062, or, put differently, that an additional 1 000 miles lowers the predicted selling price by about US$ 62; this effect is rather small. The gradient is statistically significant: using the conventional levels of significance (α = 5% and α = 1%), we can reject the null hypothesis that mileage has no effect of price, as the p-value (the observed probability of making a type I error) is very small and well below α. The y-intercept implies that a second-hand Ford Taurus which has never been driven is predicted to cost $17 067. However, as a mileage of 0 miles is very far removed from the sample mean of 36 009 miles (see figure 1), little weight should be given to this interpretation of the y-intercept. The line of best fit explains 65% of the variation in the selling price (R2 = 0.65).
The line of best fit predicts the following price for a used Ford Taurus with a mileage of 40000:
(predicted selling price, US$ | odometer reading = 40 000 miles)
= US$ 17 067 - (US$ 0.0623/mile) ´ 40.000 miles = US$ 14 575
That is, a used Ford Taurus with 40 000 miles is predicted to sell for US$ 14 575 at auction.
References
Keller, G., & Warrack, B. (2003). Statistics for Management and Economics (6th edition). Pacific Grove: Thomson / Brooks Cole.
R Development Core Team (2006). R: A Language and Environment for Statistical Computing [Computer software]. Vienna: R Foundation for Statistical Computing. Retrieved from http://www.R-project.org.
Figure 1. The relationship between mileage and selling price at auction of used Ford Tauruses (sample size = 100). Note. Data from Keller & Warrack (2003, p. 610). Scatter plot and line of best fit were obtained using the R statistical environment (R Development Core Team, 2006).