## 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.

Compute an approximate 95% confidence interval for the population slope (use the software output in

**Fig. 10.4**(jamovi) or**Fig. 10.6**(SPSS)).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?