11 The length of time in minutes for which a particular geyser erupts is modelled by the continuous random variable \(T\) with cumulative distribution function given by
\(\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 2 ,
k \left( 8 t ^ { 2 } - t ^ { 3 } - 24 \right) & 2 < t < 4 ,
1 & t \geqslant 4 , \end{cases}\)
where \(k\) is a positive constant.
- Show that \(k = \frac { 1 } { 40 }\).
- Find the probability that a randomly selected eruption time lies between 2.5 and 3.5 minutes.
- Show that the median \(m\) of the distribution satisfies the equation \(m ^ { 3 } - 8 m ^ { 2 } + 44 = 0\).
- Verify that the median eruption time is 2.95 minutes, correct to 2 decimal places.
The mean and standard deviation of \(T\) are denoted by \(\mu\) and \(\sigma\) respectively.
- Find \(\mathrm { P } ( \mu - \sigma < T < \mu + \sigma )\).
- Sketch the graph of the probability density function of \(T\).
- A Normally distributed random variable \(X\) has the same mean and standard deviation as \(T\).
By considering the shape of the Normal distribution, and without doing any calculations, explain whether \(\mathrm { P } ( \mu - \sigma < X < \mu + \sigma )\) will be greater than, equal to or less than the probability that you calculated in part (e).