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June 2011 p63 q3
2474
The following cumulative frequency table shows the examination marks for 300 candidates in country A and 300 candidates in country B.
Mark
\(< 10\)
\(< 20\)
\(< 35\)
\(< 50\)
\(< 70\)
\(< 100\)
Cumulative frequency, A
25
68
159
234
260
300
Cumulative frequency, B
10
46
72
144
198
300
Without drawing a graph, show that the median for country B is higher than the median for country A.
Find the number of candidates in country A who scored between 20 and 34 marks inclusive.
Calculate an estimate of the mean mark for candidates in country A.
Solution
(i) For country A, the median lies between 35 and 50 marks because
\(159 < 150\) (half of \(300\)), so the median is less than \(35\).
For country B, since \(144 < 150\) and the next cumulative frequency is at \(198 > 150\),
the median lies between \(50\) and \(70\).
Therefore, the median for country B is higher than that for country A.
(ii) Number of candidates in country A who scored between 20 and 34 marks:
\[
159 - 68 = 91.
\]
(iii) To estimate the mean mark for country A, use class midpoints:
\(4.5,\ 14.5,\ 27,\ 42.5,\ 60,\ 85\).
\[
\text{Mean} \approx
\frac{(4.5 \times 25) + (14.5 \times 43) + (27 \times 91) + (42.5 \times 75) + (60 \times 26) + (85 \times 40)}{300}
= 37.6.
\]
So, the estimated mean mark is approximately \(37.6\).