**Question:**

The mean and the standard deviation (s.d.) 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 s.d. become half of their original values, then $q$ is equal to:

Correct Option: , 3

**Solution:**

Let $\bar{x}$ and $\sigma$ be the mean and standard deviations of given observations.

If each observation is multiplied with $p$ and then $q$ is subtracted.

New mean $\left(\bar{x}_{1}\right)=p \bar{x}-q$

$\Rightarrow \quad 10=p(20)-q$ $\ldots$ (i)

and new standard deviations $\sigma_{1}=|p| \sigma$

$\Rightarrow 1=|p|(2) \Rightarrow|p|=\frac{1}{2} \Rightarrow p=\pm \frac{1}{2}$

If $p=\frac{1}{2}$, then $q=0$ (from equation (i))

If $p=-\frac{1}{2}$, then $q=-20$

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