6. The chief executive of Rex cars wants to investigate the relationship between the number of new car sales and the amount of money spent on advertising. She collects data from company records on the number of new car sales, \(c\), and the cost of advertising each year, \(p\) (£000). The data are shown in the table below.
| Year | Number of new car sale, \(c\) | Cost of advertising (£000), \(p\) |
| 1990 | 4240 | 120 |
| 1991 | 4380 | 126 |
| 1992 | 4420 | 132 |
| 1993 | 4440 | 134 |
| 1994 | 4430 | 137 |
| 1995 | 4520 | 144 |
| 1996 | 4590 | 148 |
| 1997 | 4660 | 150 |
| 1998 | 4700 | 153 |
| 1999 | 4790 | 158 |
- Using the coding \(x = ( p - 100 )\) and \(y = \frac { 1 } { 10 } ( c - 4000 )\), draw a scatter diagram to represent these data. Explain why \(x\) is the explanatory variable.
- Find the equation of the least squares regression line of \(y\) on \(x\).
$$\text { [Use } \left. \Sigma x = 402 , \Sigma y = 517 , \Sigma x ^ { 2 } = 17538 \text { and } \Sigma x y = 22611 . \right]$$
- Deduce the equation of the least squares regression line of \(c\) on \(p\) in the form \(c = a + b p\).
- Interpret the value of \(a\).
- Predict the number of extra new cars sales for an increase of \(\pounds 2000\) in advertising budget. Comment on the validity of your answer.
(2)