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Nov 2014 p62 q6
2471
On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.
Height (cm)
4 – 10
11 – 15
16 – 20
21 – 25
26 – 30
Frequency
22
32
78
40
28
Draw a cumulative frequency graph to illustrate the data.
28% of these daffodils are of height h cm or more. Estimate h.
You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils.
Solution
(i) To draw the cumulative frequency graph, calculate the cumulative frequencies:
Height (cm)
Cumulative Frequency
\(< 10.5\)
22
\(< 15.5\)
54
\(< 20.5\)
132
\(< 25.5\)
172
\(< 30.5\)
200
Plot these points and join them with a smooth curve.
(ii) 28% of the daffodils are of height \(h\) cm or more, which means 72% are less than \(h\). 72% of 200 is 144, so \(h\) corresponds to the cumulative frequency of 144. From the graph, \(h \approx 22.5 \text{ cm}\).
(iii) To calculate the standard deviation:
Use the midpoints of the intervals: \(7, 13, 18, 23, 28\).