The table below shows the distribution of the... | WAEC SSCE - General Mathematics - 1991 (Essay)

Question 1 Report

The table below shows the distribution of the waiting times for some customers in a certain petrol station.

Waiting time (in mins)	No of customers
1.5 - 1.9	3
2.0 - 2.4	10
2.5 - 2.9	18
3.0 - 3.4	10
3.5 - 3.9	7
4.0 - 4.4	2

(a) Write down the class boundaries of the distribution.

(b) Construct a cumulative frequency curve for the data;

(c) Using your graph, estimate: (i) the interquartile range of the distribution ; (ii) the proportion of customers who could have waited for more than 3 minutes.

Answer

(a) Class boundaries. Subtract 0.05 from each lower limit and add 0.05 to each upper limit.

Class	Class boundary	Frequency	Cumulative frequency
1.5-1.9	1.45-1.95	3	3
2.0-2.4	1.95-2.45	10	13
2.5-2.9	2.45-2.95	18	31
3.0-3.4	2.95-3.45	10	41
3.5-3.9	3.45-3.95	7	48
4.0-4.4	3.95-4.45	2	50

(b) Plot cumulative frequency against the upper boundaries: \((1.95,3),(2.45,13),(2.95,31),(3.45,41),(3.95,48),(4.45,50)\) and join with a smooth curve. \(N=50\).

(c)(i) Interquartile range. \(Q_1\) at \(\frac{N}{4}=12.5\) (in class 1.95-2.45):

\[ Q_1=1.95+\frac{12.5-3}{10}\times 0.5 = 1.95+0.475 = 2.43 \]

\(Q_3\) at \(\frac{3N}{4}=37.5\) (in class 2.95-3.45):

\[ Q_3=2.95+\frac{37.5-31}{10}\times 0.5 = 2.95+0.325 = 3.28 \] \[ \text{IQR}=Q_3-Q_1=3.28-2.43 \approx 0.85 \text{ min} \]

(ii) Proportion waiting more than 3 minutes. Read the c.f. at \(3.0\): between \(2.95\,(31)\) and \(3.45\,(41)\),

\[ \text{c.f.}(3.0)=31+\frac{3.0-2.95}{0.5}\times 10 = 32 \]

Number waiting more than 3 min \(=50-32=18\), so the proportion is

\[ \frac{18}{50}=0.36 \;\;(36\%) \]

Read lesson note on Statistics (WAEC)

Read lesson note on Measures Of Dispersion (JAMB)

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