The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10…
The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by $q$, where $p \neq 0$ and $q \neq 0$. If the new mean and new standard deviation (s.d.) become half of the original values, then q is equal to
$\quad-20$
-5
10
-10
Solution
Mean $=20$ and $\mathrm{SD}=2, \mathrm{n}=10$
If each observation is multiplied by p and then reduced by q.
New Mean $=\bar{x}_1=\mathrm{p} \bar{x}-\mathrm{q}$
$\begin{aligned}
& 10=p(20)-q \\
& \Rightarrow 20 p-q=10
...(i)\end{aligned}$ New $\mathrm{SD}=\sigma_1=|\mathrm{p}| \sigma$
Squaring on both sides,
$\begin{aligned}
& \Rightarrow 1=\mathrm{p}^2 \times 4 \\
& \Rightarrow \mathrm{p}^2=\frac{1}{4} \\
& \Rightarrow \mathrm{p}= \pm \frac{1}{2}
\end{aligned}$ From (i)
When, $\mathrm{p}=\frac{-1}{2}, \mathrm{q}=-20$
When, $\mathrm{p}=\frac{1}{2}, \mathrm{q}=0$