Question

Let $x_1,x_2,\dots ,x_n$ be $n$ observations. Let $w_i=l_{x_i}+k$ for $i=1,2,\dots ,n$, where $l$ and $k$ are constants. If the mean of $x_i$ 's is 48 and their standard deviation is 12, the mean of $w_i$ 's is 55 and standard deviation of $w_i$ 's is 15 . The values of $l$ and $k$ should be

• A. $I=1.25,k=-5$
• B. $I=-1.25,k=5$
• C. $I=2.5,k=-5$
• D. $I=2.5,k=5$

Answer 1

Answer: (A)

Given, $w_i=l{x}_i+k$
$M\left(x_j\right)=\bar{x}=48,\sigma \left(x_i\right)=12$
$M(\bar{w})=55$ and $\sigma (w)=15$
$M\left(w_i\right)=IM\left(x_i\right)+M(k)$
$55=I\times 48+k....$(i)
and $\sigma \left(w_i\right)=\mathrm{lo}\left(x_i\right)+\sigma (k)$
$\Rightarrow 15=I(12)+0$
$\Rightarrow I=\frac{15}{12}$
$=1.25$
From Eq. (i),
$55=1.25\times 48+k$
$\Rightarrow k=55-60$
$\therefore k=-5$