9 A random sample of 50 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
| Interval | \(0 \leqslant x < 0.8\) | \(0.8 \leqslant x < 1.6\) | \(1.6 \leqslant x < 2.4\) | \(2.4 \leqslant x < 3.2\) | \(3.2 \leqslant x < 4\) |
| Observed frequency | 18 | 16 | 8 | 6 | 2 |
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by
$$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 4 - x ) ^ { \frac { 1 } { 2 } } & 0 \leqslant x < 4
0 & \text { otherwise. } \end{cases}$$
The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
| Interval | \(0 \leqslant x < 0.8\) | \(0.8 \leqslant x < 1.6\) | \(1.6 \leqslant x < 2.4\) | \(2.4 \leqslant x < 3.2\) | \(3.2 \leqslant x < 4\) |
| Expected frequency | 14.22 | 12.54 | 10.59 | 8.18 | 4.47 |
- Show how the expected frequency for \(1.6 \leqslant x < 2.4\) is obtained.
- Carry out a goodness of fit test at the \(5 \%\) significance level.