Sketch the graph of \(y = f(x)\).

Find the cumulative distribution function for \(X\).

Find the interquartile range for \(X\).

         A1

Note:     Ignore open / closed endpoints and vertical lines.

Note:     Award A1 for a correct graph with scales on both axes and a clear indication of the relevant values.

[1 mark]

\(F(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{x}{2}}&{0 \le x < 1} \\ {\frac{x}{4} + \frac{1}{4}}&{1 \le x < 3} \\ 1&{x \ge 3} \end{array}} \right.\)

considering the areas in their sketch or using integration     (M1)

\(F(x) = 0,{\text{ }}x < 0,{\text{ }}F(x) = 1,{\text{ }}x \ge 3\)     A1

\(F(x) = \frac{x}{2},{\text{ }}0 \le x < 1\)     A1

\(F(x) = \frac{x}{4} + \frac{1}{4},{\text{ }}1 \le x < 3\)     A1A1

Note:     Accept \( < \) for \( \le \) in all places and also \( > \) for \( \ge \) first A1.

[5 marks]

\({Q_3} = 2,{\text{ }}{Q_1} = 0.5\)     A1A1

\({\text{IQR is }}2 - 0.5 = 1.5\)     A1

[3 marks]

Total [9 marks]

Part (a) was correctly answered by most candidates. Some graphs were difficult to mark because candidates drew their lines on top of the ruled lines in the answer book. Candidates should be advised not to do this. Candidates should also be aware that the command term ‘sketch’ requires relevant values to be indicated.

In (b), most candidates realised that the cumulative distribution function had to be found by integration but the limits were sometimes incorrect.

In (c), candidates who found the upper and lower quartiles correctly sometimes gave the interquartile range as \([0.5,{\text{ }}2]\). It is important for candidates to realise that that the word range has a different meaning in statistics compared with other branches of mathematics.