Question

Let $x_1,x_2,x_3....x_k$ be $k$ observations and $w_i=a{x}_i+b$ for $i=1,2,\mathrm{3.....}k$ , where $a$ and $b$ are constants. If mean of $x_i$ is $52$ and their standard deviation is $12$ and mean of $w_i$ is $60$ and their standard deviation is $15$ , then the value of $a$ and $b$ should be

• A. $a=1.25,b=-5$
• B. $a=-1.25,b=5$
• C. $a=2.5,b=-5$
• D. $a=2.5,b=5$

Answer 1

Answer: (A)

$w_i=a{x}_i+b$
$\Rightarrow \bar{w}=a\bar{x}+b$
$\Rightarrow 60=a\left(52\right)+b$ .... (i)
$S.Dofw=\left|a\right|\left(S.Dof{x}_i\right)$
$15=\left|a\right|12$
$\Rightarrow a=\pm 1.25$ .... (ii)
From (i) and (ii), we get,
If $a=1.25\Rightarrow 60=65+b\Rightarrow b=-5$
If $a=-1.25\Rightarrow 60=-65+b\Rightarrow b=125$