The table shows the age distributions of the... | WAEC SSCE - General Mathematics - 2005 (Essay)

Question 1 Report

The table shows the age distributions of the members of a club.

Age (years)	10-14	15-19	20-24	25-29	30-34	35-39
Frequency	7	18	25	17	9	4

(a) Calculate, correct to one decimal place, the mean age.

(b) (i) Draw a histogram to illustrate the information.

(ii) Use the histogram to estimate the modal age .

Answer

(a) Mean age

Take the class mid-value (class-mark) \(x\) of each interval and form \(fx\).

Age (years)	Mid-value \(x\)	Frequency \(f\)	\(fx\)
10 - 14	12	7	84
15 - 19	17	18	306
20 - 24	22	25	550
25 - 29	27	17	459
30 - 34	32	9	288
35 - 39	37	4	148
Total		80	1835

\[ \bar{x}=\frac{\sum fx}{\sum f}=\frac{1835}{80}=22.9375 \]

Mean age \( \approx \mathbf{22.9\ years}\) (to 1 d.p.).

(b)(i) Histogram

First convert each class to its continuous class boundaries, then draw bars whose heights equal the frequencies (the bars touch, since the boundaries are continuous).

Age (years)	Class boundaries	Frequency \(f\)
10 - 14	9.5 - 14.5	7
15 - 19	14.5 - 19.5	18
20 - 24	19.5 - 24.5	25
25 - 29	24.5 - 29.5	17
30 - 34	29.5 - 34.5	9
35 - 39	34.5 - 39.5	4

Histogram of the age distribution (bars drawn on continuous class boundaries). The two red construction lines across the modal bar cross above the mode; the dashed vertical line reads the modal age as about 21.8 years.

(b)(ii) Modal age from the histogram

The tallest bar is the modal class \(20\text{-}24\) (boundaries \(19.5\text{-}24.5\)). To read the mode, join the top-left corner of the modal bar to the top-left corner of the bar after it, and the top-right corner of the modal bar to the top-right corner of the bar before it. From the point where these two lines cross, drop a vertical line to the age axis.

The vertical line meets the axis at approximately \(\mathbf{21.8\ years}\).

This agrees with the calculation \[ \text{Mode}=L+\frac{\Delta_1}{\Delta_1+\Delta_2}\times c =19.5+\frac{25-18}{(25-18)+(25-17)}\times 5 =19.5+\frac{7}{15}\times 5 =19.5+2.3=21.8\ \text{years}. \]

(c) Probability of being in the modal class

The modal class \(20\text{-}24\) contains \(25\) members out of a total of \(80\):

\[ P(\text{modal class})=\frac{25}{80}=\frac{5}{16}=0.3125 \]

Read lesson note on Probability (WAEC)

Read lesson note on Measures Of Location (JAMB)

Read lesson note on Statistics (WAEC)

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