1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where \(k\) is a constant.

1. Show that \(k = \frac { 5 } { 8 }\).
2. Find the modal time.
3. Find \(\mathrm { E } ( T )\) and show that \(\operatorname { Var } ( T ) = \frac { 8 } { 63 }\).
4. A large random sample of \(n\) fireworks of this type is tested. Write down in terms of \(n\) the approximate distribution of \(\bar { T }\), the sample mean time.
5. For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.