Question

Let the mean and the variance of 20 observations $x _1, x _2, \ldots x _{20}$ be 15 and 9 , respectively. For $\alpha \in R$, if the mean of $\left( x _1+\alpha\right)^2,\left( x _2+\alpha\right)^2, \ldots,\left( x _{20}+\alpha\right)^2$ is 178 , then the square of the maximum value of $\alpha$ is equal to

Answer 1

$\sum x_1=15 \times 20=300$...(i)
$ \frac{\sum x_1^2}{20}-(15)^2=9$...(ii)
$ \sum x_1^2=234 \times 20=4680 $
$ \frac{\sum\left(x_1+\alpha\right)^2}{20}=178 \Rightarrow \sum\left(x_1+\alpha\right)^2=3560$
$ \Rightarrow \sum x_1^2+2 \alpha \sum x_1+\sum \alpha^2=3560$
$ 4680+600 \alpha+20 \alpha^2=3560 $
$ \Rightarrow \alpha^2+30 \alpha+56=0 $
$ \Rightarrow(\alpha+28)(\alpha+2)=0 $
$\alpha=-2,-28$
Square of maximum value of $\alpha$ is 4