5 The length, \(X \mathrm {~cm}\), of a piece of wooden planking is a random variable with probability density function given by
$$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b
0 & \text { otherwise } \end{cases}$$
where \(b\) is a positive constant.
- Find the mean and variance of \(X\) in terms of \(b\).
The lengths of a random sample of 100 pieces were measured and it was found that \(\Sigma x = 950\).
- Show that the value of \(b\) estimated from this information is 19 .
Using this value of \(b\),
- find the probability that the length of a randomly chosen piece is greater than 11 cm ,
- find the probability that the mean length of a random sample of 336 pieces is less than 9 cm .