## 11.3 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 (and removed one \(13\)-year-old Corolla advertised for sale for $390,000, assuming this was an error). I then produced the scatterplot in Fig. 11.2.

- 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.
- On the scatterplot, locate 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. 11.3 and Fig. 11.4 (jamovi); Fig. 11.5 and Fig. 11.6 (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. 11.4**(jamovi) or**Fig. 11.6**(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. 11.7**). Suppose I made this graph; then:- 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?