4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, \(x _ { i }\), and the time taken, \(y _ { i }\) hours. His results are shown in the table.
| Roof \(( \boldsymbol { i } )\) | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) | \(\mathbf { 9 }\) | \(\mathbf { 1 0 }\) | \(\mathbf { 1 1 }\) |
| \(\boldsymbol { x } _ { \boldsymbol { i } }\) | 8 | 11 | 14 | 14 | 16 | 20 | 22 | 22 | 25 | 27 | 30 |
| \(\boldsymbol { y } _ { \boldsymbol { i } }\) | 5.0 | 5.2 | 6.3 | 7.2 | 8.0 | 8.8 | 10.6 | 11.0 | 11.8 | 12.1 | 13.0 |
- The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page.
Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
- Calculate the equation of the least squares regression line of \(y _ { i }\) on \(x _ { i }\), and draw your line on the scatter diagram.
- Interpret your values for the gradient and for the intercept of this regression line.
- Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
- Given that \(r _ { i }\) denotes the residual for the point representing roof \(i\) :
- calculate the value of \(r _ { 6 }\);
- state why the value of \(\sum _ { i = 1 } ^ { 11 } r _ { i }\) gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
[0pt]
[1 mark]
\section*{Answer space for question 4}
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