5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts.
The number of attempts made by the children in the group are summarised in the table below.
| Number of attempts | 1 | 2 | 3 | 4 |
| Number of children | 43 | 26 | 13 | 3 |
- Calculate the mean and standard deviation of the number of attempts made by each child.
It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function
$$P ( X = x ) = \left\{ \begin{array} { c c }
k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4
0 , & \text { otherwise }
\end{array} \right.$$ - Show that \(k = \frac { 1 } { 50 }\).
- Find \(\mathrm { E } ( X )\).
- Comment on the suitability of this model.