The answer
(a)(i) \(180 < h \leqslant 200\)
(ii) \(\approx 179\) cm
(b)(i) \(\approx 23\)
(ii) \(\approx 50\) cm
(c) the second set's heights are less spread out (more consistent)
O-Level E-Math 2019 Paper 1 Question 24 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 24 of the O-Level E-Math 2019 Paper 1. It tests median class, in the Averages & spread (grouped data) area. It is worth 5 marks: (a) 1 + 1, (b) 1 + 1, (c) 1. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
The histogram gives frequencies 4, 5, 8, 11, 14, 12, 6 for the intervals 100 to 120, 120 to 140, 140 to 160, 160 to 180, 180 to 200, 200 to 220, 220 to 240 (total 60).
(a)(i) Median = average of the 30th and 31st values. Cumulative frequencies: 4, 9, 17, 28, 42, 54, 60. The 30th and 31st lie in \(180 < h \leqslant 200\).
(ii) Using midpoints 110, 130, 150, 170, 190, 210, 230: mean \(= \dfrac{110(4)+130(5)+150(8)+170(11)+190(14)+210(12)+230(6)}{60} = \dfrac{10720}{60} = 178.7\) cm (3 s.f.).
(b)(i) \(1.92 \text{ m} = 192\) cm. From the curve the cumulative frequency at 192 cm is about 37, so the number with height at least 192 cm is about \(60 - 37 = 23\). (Read from graph; accept roughly 22 to 25.)
(ii) From the curve, \(Q_1\) (15th value) \(\approx 155\) cm and \(Q_3\) (45th value) \(\approx 205\) cm, so IQR \(\approx 205 - 155 = 50\) cm. (Read from graph.)
(c) An interquartile range of 36 cm is smaller than about 50 cm, so the middle half of the second set's heights is less spread out: the second set of trees has more consistent (less varied) heights.
Answer: (a)(i) \(180 < h \leqslant 200\)
(ii) \(\approx 179\) cm
(b)(i) \(\approx 23\)
(ii) \(\approx 50\) cm
(c) the second set's heights are less spread out (more consistent)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a median class question from Averages & spread (grouped data), worth 5 marks: (a) 1 + 1, (b) 1 + 1, (c) 1.
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