8 A sample of 216 observations of the continuous random variable \(X\) was obtained and the results are summarised in the following table.
| Interval | \(0 \leqslant x < 1\) | \(1 \leqslant x < 2\) | \(2 \leqslant x < 3\) | \(3 \leqslant x < 4\) | \(4 \leqslant x < 5\) | \(5 \leqslant x < 6\) |
| Observed frequency | 1 | 3 | 15 | 31 | 59 | 107 |
It is suggested that these results are consistent with a distribution having probability density function f given by
$$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x < 6
0 & \text { otherwise } \end{cases}$$
where \(k\) is a positive constant. The relevant expected frequencies are given in the following table.
| Interval | \(0 \leqslant x < 1\) | \(1 \leqslant x < 2\) | \(2 \leqslant x < 3\) | \(3 \leqslant x < 4\) | \(4 \leqslant x < 5\) | \(5 \leqslant x < 6\) |
| Expected frequency | 1 | 7 | \(a\) | \(b\) | \(c\) | 91 |
- Show that \(a = 19\) and find the values of \(b\) and \(c\).
- Carry out a goodness of fit test at the \(10 \%\) significance level.