9 A random sample of 200 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
| Interval | \(1 \leqslant x < 2\) | \(2 \leqslant x < 3\) | \(3 \leqslant x < 4\) | \(4 \leqslant x < 5\) | \(5 \leqslant x < 6\) | \(6 \leqslant x < 7\) | \(7 \leqslant x < 8\) |
| Observed frequency | 63 | 45 | 32 | 25 | 22 | 7 | 6 |
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by
$$f ( x ) = \begin{cases} \frac { 1 } { x \ln 8 } & 1 \leqslant x < 8
0 & \text { otherwise } \end{cases}$$
The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
| Interval | \(1 \leqslant x < 2\) | \(2 \leqslant x < 3\) | \(3 \leqslant x < 4\) | \(4 \leqslant x < 5\) | \(5 \leqslant x < 6\) | \(6 \leqslant x < 7\) | \(7 \leqslant x < 8\) |
| Expected frequency | 66.67 | \(p\) | 27.67 | \(q\) | 17.54 | 14.83 | 12.84 |
Show that \(p = 39.00\), correct to 2 decimal places, and find the value of \(q\).
Carry out a goodness of fit test at the 5\% significance level.