Question

The mean of a set of observations 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 1

Answer: (C)

$\frac{\overline{x}+10\alpha }{\alpha }$

Let $x_1,x_2,\dots ,x_n$ be $n$ observations.
Then, $\overline{x}=\frac{1}{n}\sum x_i$
let $y_i=\frac{x_i}{\alpha }+10$
Then, $\frac{1}{n}\sum _{i=1}^n{y}_i=\frac{1}{\alpha }\left(\frac{1}{n}\sum x_i\right)+\frac{1}{n}\left(10n\right)$
$\Rightarrow \overline{y}=\frac{1}{\alpha }\overline{x}+10$
$\Rightarrow \overline{y}=\frac{\overline{x}+10\alpha }{\alpha }$