Mark (x) 1 2 3 4 5 Total Frequency (f) m + 2 m - 1 2m - 3 m + 5 3m - 4 8m - 1 fx m + 2 2m - 2 6m - 9 4m + 20 15m - 20 28m - 9 (a) Mean \(\bar{x} = \frac{\sum fx}{\sum f}\) \(\frac{75}{23} = \frac{28m - 9}{8m - 1}\) \(75(8m - 1) = 23(28m - 9) \implies 600m - 75 = 644m - 207\) \(-75 + 207 = 644m - 600m\) \(132 = 44m \implies m = 3\) (b)(i) Interquartile range = Third quartile - First quartile Frequency = 8(3) - 1 = 24 - 1 = 23 \(Q_{3} = \frac{3}{4} \times 23 = 17.25th\) position = 4 \(Q_{1} = \frac{1}{4} \times 23 = 5.75th\) position = 2 Interquartile range : 4 - 2 = 2 (ii) P(at least 4 marks) = \(\frac{(m + 5 + 3m - 4)}{23} = \frac{4m + 1}{23}\) = \(\frac{4(3) + 1}{23} \) = \(\frac{13}{23}\)