Question

Suppose a population $A$ has $100$ observations $101,102,\ldots ,200$ and another population $B$ has $100$ observations $151,152,\ldots \ldots ,250.$ If $V_{A}$ and $V_{B}$ represents the variances of the two populations respectively, then the value of $\frac{V_{A}}{V_{B}}$ is

Answer 1

Let $x_{1},x_{2}\ldots \ldots $ be $n$ values of $x$
then $\sigma =\frac{1}{n}\displaystyle \sum _{i = 1}^{n}\left(x_{i} - \bar{x}\right)^{2}$
the variable $ax+b$ takes values $ax_{1}+b,ax_{2}+b,\ldots \ldots ,ax_{n}+b$
with mean $\bar{x}+b$
$\therefore SD$ of $\left(a x + b\right)$
$=\sqrt{\frac{1}{n} \displaystyle \sum _{i = 1}^{n} \left(\left\{\left(a x_{i} - b\right) - \left(\right. a \bar{x} - b \left.\right)\right\}\right)^{2}}$
$=\sqrt{a^{2} \cdot \frac{1}{n} \displaystyle \sum _{i = 1}^{n} \left(x_{i} - \bar{x}\right)^{2}}=\left|a\right|\sigma $