- A metallurgist measured the length, \(l \mathrm {~mm}\), of a copper rod at various temperatures, \(t ^ { \circ } \mathrm { C }\), and recorded the following results.
| \(t\) | \(l\) |
| 20.4 | 2461.12 |
| 27.3 | 2461.41 |
| 32.1 | 2461.73 |
| 39.0 | 2461.88 |
| 42.9 | 2462.03 |
| 49.7 | 2462.37 |
| 58.3 | 2462.69 |
| 67.4 | 2463.05 |
The results were then coded such that \(x = t\) and \(y = l - 2460.00\).
- Calculate \(S _ { x y }\) and \(S _ { x x }\).
(You may use \(\Sigma x ^ { 2 } = 15965.01\) and \(\Sigma x y = 757.467\) ) - Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
- Estimate the length of the rod at \(40 ^ { \circ } \mathrm { C }\).
- Find the equation of the regression line of \(l\) on \(t\).
- Estimate the length of the rod at \(90 ^ { \circ } \mathrm { C }\).
- Comment on the reliability of your estimate in part (e).