A continuous random variable, \(X\), has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8 0 & \text { otherwise } \end{cases}$$
Find \(\mathrm { F } ( x )\), the cumulative distribution function of \(X\).
Sketch \(\mathrm { F } ( x )\).
The median of \(X\) is \(m\). Show that \(m\) satisfies the equation \(m ^ { 3 } - 12 m ^ { 2 } + 148 = 0\). Verify that \(m \approx 4.42\).
The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows.
$$\begin{array} { l l l l l l l l l l l l }
3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78
\end{array}$$
Test at the 5\% level of significance whether a median of 4.42 is consistent with these data.