4 The lengths of time, in seconds, between vehicles passing a fixed observation point on a road were recorded at a time when traffic was flowing freely. The frequency distribution in Table 1 is a summary of the data from 100 observations.
\begin{table}[h]
| Time interval \(( x\) seconds \()\) | \(0 < x \leqslant 5\) | \(5 < x \leqslant 10\) | \(10 < x \leqslant 20\) | \(20 < x \leqslant 40\) | \(40 < x\) |
| Observed frequency | 49 | 22 | 20 | 7 | 2 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
It is thought that the distribution of times might be modelled by the continuous random variable \(X\) with probability density function given by
$$f ( x ) = \begin{cases} 0.1 e ^ { - 0.1 x } & x > 0 \\ 0 & \text { otherwise } \end{cases}$$
Using this model, the expected frequencies (correct to 2 decimal places) for the given time intervals are shown in Table 2.
\begin{table}[h]
| Time interval \(( x\) seconds \()\) | \(0 < x \leqslant 5\) | \(5 < x \leqslant 10\) | \(10 < x \leqslant 20\) | \(20 < x \leqslant 40\) | \(40 < x\) |
| Expected frequency | 39.35 | 23.87 | 23.25 | 11.70 | 1.83 |
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{table}
- Show how the expected frequency of 23.87, corresponding to the interval \(5 < x \leqslant 10\), is obtained.
- Test, at the 10\% significance level, the goodness of fit of the model to the data.