Question

Suppose a population $A$ has $100$ observations $101,102,\dots ,200$ and another population $B$ has $100$ observations $151,152,\dots \dots ,250.$ 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

Answer: 1

Let $x_1,x_2\dots \dots$ be $n$ values of $x$
then $\sigma =\frac{1}{n}\sum _{i=1}^n{\left(x_i-\bar{x}\right)}^2$
the variable $ax+b$ takes values $a{x}_1+b,a{x}_2+b,\dots \dots ,a{x}_n+b$
with mean $\bar{x}+b$
$\therefore SD$ of $\left(ax+b\right)$
$=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(\left\{\left(a{x}_i-b\right)-(a\bar{x}-b)\right\}\right)}^2}$
$=\sqrt{a^2\cdot \frac{1}{n}\sum _{i=1}^n{\left(x_i-\bar{x}\right)}^2}=\left|a\right|\sigma$