5 A random variable \(X\) is believed to have (cumulative) distribution function given by
$$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
1 - \mathrm { e } ^ { - x ^ { 2 } } & x \geqslant 0 . \end{cases}$$
In order to test this, a random sample of 150 observations of \(X\) were taken, and their values are summarised in the following grouped frequency table.
| Values | \(0 \leqslant x < 0.5\) | \(0.5 \leqslant x < 1\) | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x < 2\) | \(x \geqslant 2\) |
| Frequency | 41 | 50 | 32 | 23 | 4 |
The expected frequencies, correct to 1 decimal place, corresponding to the above distribution, are 33.2, 61.6 and 39.4 respectively for the first 3 cells.
- Find the expected frequencies for the last 2 cells.
- Carry out a goodness of fit test at the \(2 \frac { 1 } { 2 } \%\) significance level.