Question

In two Towns A and B, daily pocket money and the standard deviation are given below:

Town	Average Daily Pocket	Standard Deviation	No. of Teenagers
A	34.5	5.0	476
B	28.5	4.5	524

(i) Which town, A or B, pays out the larger amount of daily pocket money?
(ii) What is the average daily pocket money of all teenagers taken together?
(iii) Calculate coefficient of variation of each town. Which town is more variable in terms of pocket money?

Answer 1

(i) Town A
Daily pocket money = Average Daily Pocket × No. of teenagers
= 34.5 × 476
= Rs 16422

Town B
Daily Pocket Money = 28.5 × 524
= Rs 14934

Town 'A' pays larger amount of daily pocket money.

(ii) Solution:

In order to find out average daily pocket money of all teenagers, we need find out the combined mean of their pocket money.

Thus,

$$\begin{gathered} {\overline{X}}_A=34.5,N_{\mathrm{A}}=476 \\ {\overline{X}}_B=28.5,N_{\mathrm{B}}=524 \end{gathered}$$

$$\begin{gathered} {\overline{X}}_{A,B}=\frac{{\overline{X}}_A{N}_A+{\overline{X}}_B{N}_B}{N_A+N_B} \\ or,{\overline{X}}_{A,B}=\frac{34.5\times 476+28.5\times 524}{476+524} \\ or,{\overline{X}}_{A,B}=\frac{16422+14934}{1000} \\ or,{\overline{X}}_{A,B}=\frac{31356}{1000} \\ \Rightarrow {\overline{X}}_{A,B}=Rs31.356=Rs31.36approx \end{gathered}$$

(iii)
$$\begin{gathered} \mathrm{C}.\mathrm{V}.\mathrm{of}\mathrm{A}=\frac{{\sigma }_A}{{\overline{X}}_A}\times 100=\frac{5}{34.5}\times 100=14.49\% \\ \mathrm{C}.\mathrm{V}.\mathrm{of}\mathrm{B}=\frac{{\sigma }_B}{{\overline{X}}_B}\times 100=\frac{4.5}{28.5}\times 100=15.79\% \end{gathered}$$

Town B is more variable in terms of pocket money as its C.V. is higher than that of C.V. of Town A.