5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, \(\pounds x\), and the length of commute, \(y\) minutes, for employees at the firm. She collected this information from 9 randomly selected employees.
The salary of each employee was then coded using \(w = \frac { x - 20000 } { 1000 }\)
The table shows the values of \(w\) and \(y\) for the 9 employees.
| \(w\) | 6 | 8 | 8 | - 1 | 25 | 15 | 3 | - 2 | 19 |
| \(y\) | 45 | 50 | 35 | 65 | 25 | 40 | 50 | 75 | 20 |
(You may use \(\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500\) )
- Calculate the salary of the employee with \(w = - 2\)
- Show that, to 3 significant figures, the value of the product moment correlation coefficient between \(w\) and \(y\) is - 0.899
- State, giving a reason, the value of the product moment correlation coefficient between \(x\) and \(y\)
The least squares regression line of \(y\) on \(w\) is \(y = 60.75 - 1.75 w\)
- Find the equation of the least squares regression line of \(y\) on \(x\) giving your answer in the form \(y = a + b x\)
- Estimate the length of commute for an employee with a salary of \(\pounds 21000\)
Franca uses the regression line to estimate the length of commute for employees with salaries between \(\pounds 25000\) and \(\pounds 40000\)
- State, giving a reason, whether or not these estimates are reliable.