8 A random sample of 100 students were given a task and the time taken by each student to complete the task was recorded. The maximum time allowed to complete the task was one minute and all students completed the task within the maximum time. The times, \(T\) minutes, for the random sample of students are summarised as follows.
\(n = 100 \quad \sum t = 61.88\)
A researcher proposes that \(T\) can be modelled by the continuous random variable with probability density function
\(f ( t ) = \begin{cases} \alpha t ^ { \alpha - 1 } & 0 \leqslant t \leqslant 1 ,
0 & \text { otherwise, } \end{cases}\)
where \(\alpha\) is a positive constant.
\section*{(a) In this question you must show detailed reasoning.}
By finding \(\mathbf { E } ( T )\) according to the researcher's model, determine an approximation for the value of \(\alpha\). Give your answer correct to \(\mathbf { 3 }\) significant figures.
Further information about the times taken for the sample of 100 students to complete the task is given in the table.
| Time \(t\) | \(0 \leqslant t < \frac { 1 } { 3 }\) | \(\frac { 1 } { 3 } \leqslant t < \frac { 2 } { 3 }\) | \(\frac { 2 } { 3 } \leqslant t \leqslant 1\) |
| Frequency | 18 | 37 | 45 |
(b) Using the value of \(\alpha\) found in part (a), determine the extent to which the proposed model is a good model. (Do not carry out a goodness of fit test.)