Question

If the mean and standard deviations of $100$ observations $x_1,x_2,...,x_{100}$ are $5$ and $4$ respectively, then the mean of $(x_1+1{)}^2,(x_1+1{)}^2,...,(x_{100}+1{)}^2$ is

• A. $51$
• B. $52$
• C. $151$
• D. $152$

Answer 1

The correct option is B $52$

Mean of $(x_1+1{)}^2,(x_1+1{)}^2,...,(x_{100}+1{)}^2$ is

$\overline{X}=\frac{1}{100}\sum _{i=1}^{100}(x_i+1{)}^2$
$=\frac{1}{100}\sum _{i=1}^{100}{x}_i^2+\frac{2}{100}\sum _{i=1}^{100}{x}_i+1$
$={\sigma }^2+(\overline{x}{)}^2+2\times 5+1$
$=4^2+5^2+10+1$
$=52$