7 The continuous random variable \(X\) has the probability density function shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}
- Find the value of the constant \(k\).
- Write down the mean of \(X\), and use integration to find the variance of \(X\).
- Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
- The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance.
\section*{[Question 8 is printed overleaf.]}