10.2 Interpreting regressions
I was wondering about how the age of second-hand cars impact their price. On 25 June 2014, I searched Gum Tree for ‘Toyota Corolla’ in the ‘Cars, Vans & Utes’ category.
The age and the price of each (second-hand) car was recorded from the first two pages of results that were returned.
I then restricted the data to cars 15 years old or younger. (I also removed one 13-year-old Corolla advertised for sale for $390,000, assuming this was an error.)
I then produced the scatterplot in Fig. 10.2.
- Describe the relationship displayed in the graph in words.
- What else could influence the price of a second-hand Corollas?
- From the scatterplot, draw (if you can) or estimate by eye an approximation of the regression line.
- Using a cross symbol, place a mark corresponding to a seven-year-old Corolla selling for $15,000. Would this be cheap or expensive?
- As stated, I removed one observation: a 13-year-old Corolla for sale at $390,000. What do you think the price was meant to be listed as, by looking at the scatterplot? Explain.
- Estimate the value of \(b_0\) (the intercept) from the line you drew. What does this mean? Do you think this value is meaningful?
- Estimate the value of \(b_1\) (the slope) from the line you drew. What does this mean? Do you think this value is meaningful?
- From the line you drew above, write down an estimate of the regression equation.
- Use the software output (Fig. 10.3 and Fig. 10.4 (jamovi); Fig. 10.5 and Fig. 10.6 (SPSS)) relating the price (in thousands of dollars) to age to write down the regression equation.
- Using the software output, write down the value of \(r\). Using this value of \(r\), compute the value of \(R^2\). What does this mean?
Use the regression equation from the software output to estimate the sale price of a Corolla that is 20 years old, and explain your answer.
Would a Corolla 6-years-old advertised for sale at $15,000 appear to be good value? Estimate the sale price and explain your answer.
Using the software output, perform a suitable hypothesis test to determine if there is evidence that lower prices are associated with older Corollas.
I could have drawn a scatterplot with Price on the vertical (up-and-down) axis and Year of manufacture on the horizontal (left-to-right) axis (Fig. 10.7). From this graph:
- What is the value of the correlation coefficient?
- How would the value of \(R^2\) change?
- How would the value of the slope change?
- How would the value of the intercept change?