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

Question 1 Report

The table shows the distribution of monthly income (in thousands of naira) of workers in a factory

Monthly Income (N'1000)	135-139	140-149	150-154	155-164	165-169
Number of workers	20	42	28	38	22

(a) Draw a histogram for the distribution.

(b) Use your graph to estimate the mode of the distribution.

Answer

(a) Histogram

Since the class intervals have unequal widths, the heights of the rectangles are the frequency densities:

\[\text{Frequency density}=\frac{\text{frequency}}{\text{class width}}.\]

Monthly income (₦'000)	Class boundaries (₦'000)	Frequency	Class width	Frequency density
135–139	134.5–139.5	20	5	4.0
140–149	139.5–149.5	42	10	4.2
150–154	149.5–154.5	28	5	5.6
155–164	154.5–164.5	38	10	3.8
165–169	164.5–169.5	22	5	4.4

Plot the class boundaries on the horizontal axis and frequency density on the vertical axis. The resulting histogram is:

graph — Histogram with contiguous rectangles whose areas represent the numbers of workers. Heights are frequency densities.

(b) Estimated mode

The modal class is \(149.5\text{–}154.5\), since it has the greatest frequency density, \(5.6\).

Using the standard intersecting-diagonals construction on the modal rectangle gives

\[\begin{aligned} \text{Mode} &=149.5+\frac{5.6-4.2}{(5.6-4.2)+(5.6-3.8)}\times 5\\ &=149.5+\frac{1.4}{3.2}\times5\\ &=151.6875\approx151.7. \end{aligned}\]

Therefore, the estimated modal monthly income is \(151.7\) thousand naira, that is, approximately ₦151,700.

Read lesson note on Statistics (WAEC)

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