9 A sample of 100 observations of the continuous random variable \(T\) was obtained and the values are summarised in the following table.
| Interval | \(1 \leqslant t < 1.5\) | \(1.5 \leqslant t < 2\) | \(2 \leqslant t < 2.5\) | \(2.5 \leqslant t < 3\) |
| Frequency | 64 | 17 | 16 | 3 |
It is required to test the goodness of fit of the distribution with probability density function given by
$$f ( t ) = \begin{cases} \frac { 9 } { 4 t ^ { 3 } } & 1 \leqslant t < 3
0 & \text { otherwise } \end{cases}$$
The relevant expected values are as follows.
| Interval | \(1 \leqslant t < 1.5\) | \(1.5 \leqslant t < 2\) | \(2 \leqslant t < 2.5\) | \(2.5 \leqslant t < 3\) |
| Expected frequency | 62.5 | 21.875 | 10.125 | 5.5 |
Show how the expected value 10.125 is obtained.
Carry out the test, at the \(10 \%\) significance level.