7 A continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1
a x ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$
where \(a\) is a constant.
- Show that \(a = \frac { 12 } { 31 }\).
- Find \(\mathrm { E } ( X )\).
It is thought that the time taken by a student to complete a task can be well modelled by \(X\). The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
| Time | \(0 \leqslant x < 0.5\) | \(0.5 \leqslant x < 1\) | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x \leqslant 2\) |
| Observed frequency | 8 | 92 | 279 | 613 |
| Expected frequency | 6 | 90 | | |
- Find the other expected frequencies and test, at the \(5 \%\) level of significance, whether the data can be well modelled by \(X\).