The mean and the standard deviation (s.d.) of | vedclass

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(A) Let the original observations be $x_1, x_2, \dots, x_{10}$. Given mean $\bar{x} = 20$ and s.d. $\sigma = 2$.When each observation is transformed t

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$-20$

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(A) Let the original observations be $x_1, x_2, \dots, x_{10}$. Given mean $\bar{x} = 20$ and s.d. $\sigma = 2$.When each observation is transformed to $y_i = p x_i - q$,the new mean $\bar{y} = p \bar{x} - q$ and the new s.d. $\sigma_y = |p| \sigma$.Given $\bar{y} = \frac{20}{2} = 10$ and $\sigma_y = \frac{2}{2} = 1$.Substituting the values: $10 = 20p - q \dots (i)$ and $1 = |p| 	imes 2 \dots (ii)$.From $(ii)$,$|p| = \frac{1}{2}$,so $p = \frac{1}{2}$ or $p = -\frac{1}{2}$.Case $1$: If $p = \frac{1}{2}$,then $10 = 20(\frac{1}{2}) - q$ $\Rightarrow 10 = 10 - q$ $\Rightarrow q = 0$. But given $q 
eq 0$,this is rejected.Case $2$: If $p = -\frac{1}{2}$,then $10 = 20(-\frac{1}{2}) - q$ $\Rightarrow 10 = -10 - q$ $\Rightarrow q = -20$.

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

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