The table shows the frequency distribution of heights (in cm) of pupils in a certain school. Heights 100-109 110-119 120-129 130-139 140-149 150-159 160-169...

(a) (i) Cumulative frequency table

The total number of pupils is \(N=400\). Include the starting point \((99.5,0)\).

(a) (ii) Less-than cumulative frequency curve (ogive)

Plot the upper class boundaries against their cumulative frequencies and join successive points with a smooth increasing curve.

(b) Estimates from the ogive

(i) Median height

\[\frac{N}{2}=\frac{400}{2}=200\]

At cumulative frequency 200, the corresponding height is approximately \(128.3\text{ cm}\).

Median height \(\approx 128.3\text{ cm}\).

(ii) Interquartile range

\[Q_1=\frac{N}{4}=100,\qquad Q_3=\frac{3N}{4}=300\]

From the curve, \(Q_1\approx120.7\text{ cm}\) and \(Q_3\approx137.6\text{ cm}\).

\[\text{Interquartile range}=Q_3-Q_1=137.6-120.7\approx16.9\text{ cm}.\]

(iii) Percentage of pupils whose heights are at most \(130\text{ cm}\)

At \(130\text{ cm}\), the cumulative frequency is approximately \(220\).

\[\text{Percentage}=\frac{220}{400}\times100\%\approx55\%.\]

Therefore, approximately \(55\%\) of the pupils have heights at most \(130\text{ cm}\).