3 A random sample of 50 values of the continuous random variable \(X\) was taken. These values are summarised in the following table.
| Interval | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x < 2\) | \(2 \leqslant x < 2.5\) | \(2.5 \leqslant x < 3\) | \(3 \leqslant x < 3.5\) | \(3.5 \leqslant x \leqslant 4\) |
| Observed frequency | 3 | 3 | 8 | 11 | 13 | 12 |
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by
$$f ( x ) = \begin{cases} \frac { 1 } { 24 } \left( \frac { 4 } { x ^ { 2 } } + x ^ { 2 } \right) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
The expected frequencies, correct to 4 decimal places, are given in the following table.
| Interval | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x < 2\) | \(2 \leqslant x < 2.5\) | \(2.5 \leqslant x < 3\) | \(3 \leqslant x < 3.5\) | \(3.5 \leqslant x \leqslant 4\) |
| Expected frequency | 4.4271 | \(a\) | 6.1285 | 8.4549 | \(b\) | 14.9678 |
- Show that \(a = 4.6007\) and find the value of \(b\).
- Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether f is a satisfactory model for the data.