Answer to Question #187550 in Statistics and Probability for Musenga sichona

Question

11. The following shows the age distribution in years of 40 youths at a youth rally.

Age in years (x)

Frequency (f)

7 − 12. 4

12 − 17

17 − 22. 15

22 − 27

27 − 32. 5

Total. 40

Given that the median age is 19.8 years:

i.

Find the missing frequencies to the nearest whole number.

ii.

Find the mode of this distribution.

iii.

How many youths are older than 17 years?

iv.

Include a column for relative frequencies.

v.

Hence, or otherwise, construct a pie chart for this data.

Answer 1

Given, Median=19.8

(i) Median $M=l+\dfrac{(\frac{N}{2}-cf)}{f}h$

$\Rightarrow 19.8=17+\dfrac{\frac{40}{2}-(4+f_1)}{15}\times 5$

$\Rightarrow 2.8=\dfrac{16-f_1}{3}\Rightarrow f_1=16-8.4=7.6$

Also, $24+f_1+f_2=40$

$f_2=40-24-7.6=8.4$

Missing frequencies are 7.6 and 8.4

(ii) Mode $=l+\dfrac{(f_m-f_{m-1})}{2f_m-f_{m-1}-f_{m+1}}h$

$=17+\dfrac{15-7.6}{2\times 15-7.6-8.4}\times 5$

$=17+2.8=19.8$

(iii) Number of youths older than 17 years are $= 15+f_2+5=20+8.4=28.4$

(iv) Relative frequency table is included in the above chart.

(v) Pie chart for the given distribution is-