6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of \(x\), the test rig speed in miles per hour (mph), and the temperature, \(y ^ { \circ } \mathrm { C }\), generated in the shoulder of the tyre for a particular tyre material.
| \(x ( \mathrm { mph } )\) | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| \(y \left( { } ^ { \circ } \mathrm { C } \right)\) | 53 | 55 | 63 | 65 | 78 | 83 | 91 | 101 |
- Draw a scatter diagram to represent these data.
- Give a reason to support the fitting of a regression line of the form \(y = a + b x\) through these points.
- Find the values of \(a\) and \(b\).
(You may use \(\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615\) ) - Give an interpretation for each of \(a\) and \(b\).
- Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table.
A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
- Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.