8 A continuous random variable \(X\) has probability density function given by
$$\mathrm { f } ( x ) = \left\{ \begin{array} { c c }
0.8 \mathrm { e } ^ { - 0.8 x } & x \geq 0
0 & x < 0
\end{array} \right.$$
- Find the mean and variance of \(X\).
The lifetime of a certain organism is thought to have the same distribution as \(X\). The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.
| Range | \(0 \leq x < 1\) | \(1 \leq x < 2\) | \(2 \leq x < 3\) | \(3 \leq x < 4\) | \(x \geq 4\) |
| Observed | 24 | 22 | 10 | 3 | 1 |
| Expected | 33.040 | 14.846 | 6.671 | 2.997 | 2.446 |
- Show how the expected frequency for \(1 \leq x < 2\) is obtained.
- Carry out a goodness of fit test at the \(5 \%\) significance level.