5 A motorist is investigating the relationship between tyre pressure and temperature. As the temperature increases during a hot day, she records the pressure (measured in bars) of one of her car tyres at specific temperatures of \(20 ^ { \circ } \mathrm { C } , 22 ^ { \circ } \mathrm { C } , \ldots , 36 ^ { \circ } \mathrm { C }\). The results are shown in Table 5.1.
\begin{table}[h]
| Temperature \(\left( t ^ { \circ } \mathrm { C } \right)\) | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 |
| Tyre pressure \(( P\) bar \()\) | 2.012 | 2.036 | 2.065 | 2.074 | 2.114 | 2.140 | 2.149 | 2.176 | 2.192 |
\captionsetup{labelformat=empty}
\caption{Table 5.1}
\end{table}
- Calculate the equation of the regression line of pressure on temperature. Give your answer in the form \(P = a t + b\), giving the values of \(a\) and \(b\) to \(\mathbf { 4 }\) significant figures.
- Table 5.2 shows the residuals for most of the data values. Complete the copy of the table in the Printed Answer Booklet.
\begin{table}[h]
| Temperature | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 |
| - 0.003 | - 0.002 | 0.004 | - 0.010 | | 0.011 | - 0.003 | 0.001 | |
\captionsetup{labelformat=empty}
\caption{Table 5.2}
- With reference to the values of the residuals, comment on the goodness of fit of the regression line.
- Use your answer to part (a) to calculate an estimate of the pressure in the tyre at each of the following temperatures, giving your answers to \(\mathbf { 3 }\) decimal places.
- \(25 ^ { \circ } \mathrm { C }\)
- \(10 ^ { \circ } \mathrm { C }\)
- Comment on the reliability of each of your estimates.