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, corr...
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 .
(c) If a member is selected at random, what is the probability that he/she is in the modal class?
Answer Details
(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 |
(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 \(80\):
\[ P(\text{modal class})=\frac{25}{80}=\frac{5}{16}=0.3125 \]
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