3 A random sample of 200 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
| Interval | \(0 \leqslant x < 0.5\) | \(0.5 \leqslant x < 1\) | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x < 2\) | \(2 \leqslant x < 2.5\) | \(2.5 \leqslant x < 3\) |
| Observed frequency | 5 | 23 | 40 | 41 | 46 | 45 |
It is required to test the goodness of fit of the distribution with probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 9 } x ( 4 - x ) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
Most of the relevant expected frequencies, correct to 2 decimal places, are given in the following table.
| Interval | \(0 \leqslant x < 0.5\) | \(0.5 \leqslant x < 1\) | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x < 2\) | \(2 \leqslant x < 2.5\) | \(2.5 \leqslant x < 3\) |
| Expected frequency | \(p\) | \(q\) | 37.96 | 43.52 | 43.52 | 37.96 |
- Show that \(p = 10.19\) and find the value of \(q\).
- Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether f is a satisfactory model for the data.