The table shows the frequency distribution of... | WAEC SSCE - Further Mathematics - 2021 (Essay)

Question 1 Report

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
Frequency	27	58	130	105	50	25	5

(a) (i) Construct a cumulative frequency table. (ii) Use the table to draw a cumulative frequency curve.

(b) Using the curve, estimate the: (i)median height; (ii) inter quartile range (iii) percentage of students whose heights are most 130cm.

Answer

(a)(i) Cumulative frequency table (cf against upper boundaries). \(N = 400\).

Height	Freq	Upper boundary	Cumulative freq
100 - 109	27	109.5	27
110 - 119	58	119.5	85
120 - 129	130	129.5	215
130 - 139	105	139.5	320
140 - 149	50	149.5	370
150 - 159	25	159.5	395
160 - 169	5	169.5	400

(a)(ii) Plot (109.5, 27), (119.5, 85), ... , (169.5, 400) and join with a smooth ogive.

(b)(i) Median at position \( \tfrac{400}{2} = 200\), in class 120 - 129 (\(L=119.5, \text{cf}=85, f=130\)):

\[ \text{Median} = 119.5 + \frac{200 - 85}{130}\times 10 = 119.5 + 8.85 \approx \mathbf{128.3 \text{ cm}} \]

(b)(ii) Interquartile range. \(Q_1\) at 100th (class 120 - 129):

\[ Q_1 = 119.5 + \frac{100 - 85}{130}\times 10 = 119.5 + 1.15 = 120.65 \]

\(Q_3\) at 300th (class 130 - 139, \(L=129.5, \text{cf}=215, f=105\)):

\[ Q_3 = 129.5 + \frac{300 - 215}{105}\times 10 = 129.5 + 8.10 = 137.60 \] \[ \text{IQR} = Q_3 - Q_1 = 137.60 - 120.65 \approx \mathbf{17.0 \text{ cm}} \]

(b)(iii) Percentage with height at most 130 cm. Read cf at 130 (class 130 - 139):

\[ \text{cf}(130) = 215 + \frac{130 - 129.5}{10}\times 105 = 215 + 5.25 = 220.25 \] \[ \text{Percentage} = \frac{220.25}{400}\times 100 \approx \mathbf{55\%} \]

Read lesson note on Statistics (WAEC)

Answer Details

(a)(i) Cumulative frequency table (cf against upper boundaries). \(N = 400\).

Height	Freq	Upper boundary	Cumulative freq
100 - 109	27	109.5	27
110 - 119	58	119.5	85
120 - 129	130	129.5	215
130 - 139	105	139.5	320
140 - 149	50	149.5	370
150 - 159	25	159.5	395
160 - 169	5	169.5	400