Question
The mean of a set of observation is $$\overline x $$. If each observation is divided by $$\alpha ,\,\alpha \ne 0$$ and then is increased by $$10,$$ then the mean of the new set is :
A.
$$\frac{{\overline x }}{\alpha }$$
B.
$$\frac{{\overline x + 10}}{\alpha }$$
C.
$$\frac{{\overline x + 10\alpha }}{\alpha }$$
D.
$$\alpha \overline x + 10$$
Answer :
$$\frac{{\overline x + 10\alpha }}{\alpha }$$
Solution :
$$\eqalign{
& {\text{Let}}\,{x_1}{\text{,}}\,{x_2}{\text{,}}......{\text{,}}{x_n}\,{\text{be }}n{\text{ observations}}{\text{.}} \cr
& {\text{Then, }}\overline x = \frac{1}{n}\sum {{x_i}} \,;\,{\text{Let }}{y_i} = \frac{{{x_i}}}{\alpha } + 10 \cr
& {\text{then, }}\frac{1}{n}\sum\limits_{i = 1}^n {{y_i}} = \frac{1}{\alpha }\left( {\frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} } \right) + \frac{1}{n}\left( {10n} \right) \cr
& \Rightarrow \overline y = \frac{1}{\alpha }\overline x + 10 = \frac{{\overline x + 10\alpha }}{\alpha } \cr} $$