A continuous random variable \(X\) has probability density function
$$\mathrm { f } ( x ) = \lambda x ^ { c } , \quad 0 \leqslant x \leqslant 1 ,$$
where \(c\) is a constant and the parameter \(\lambda\) is greater than 1 .
Find \(c\) in terms of \(\lambda\).
Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
Show that \(\operatorname { Var } ( X ) = \frac { \lambda } { ( \lambda + 2 ) ( \lambda + 1 ) ^ { 2 } }\).
Every day, Godfrey does a puzzle from the newspaper and records the time taken in minutes. Last year, his median time was 32 minutes. His times for a random sample of 12 puzzles this year are as follows.
$$\begin{array} { l l l l l l l l l l l l }
40 & 20 & 18 & 11 & 47 & 36 & 38 & 35 & 22 & 14 & 12 & 21
\end{array}$$
Use an appropriate test, with a 5\% significance level, to examine whether Godfrey's times this year have decreased on the whole.