5 A statistician suggested that the weekly sales \(X\) thousand litres at a petrol station could be modelled by the following probability density function. $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( 2 x + 3 ) & 0 \leqslant x < 5
0 & \text { otherwise } \end{cases}$$

1. Show that, using this model, \(\mathrm { P } ( a \leqslant X < a + 1 ) = \frac { a + 2 } { 20 }\) for \(0 \leqslant a \leqslant 4\). Sales in 100 randomly chosen weeks gave the following grouped frequency table.

   \(x\)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Frequency	16	12	18	30	24

2. Carry out a goodness of fit test at the \(10 \%\) significance level of whether \(\mathrm { f } ( x )\) fits the data.