Question

Let $X_1,X_2,X_3,\cdots X_n$ be $n$ observations and let $\overline{X}$ be the arithmetic mean and ${\sigma }^2$ be the variance.

Statement $1:$ Variance of
$2X_1,2X_2,....,2X_n$ is $4{\sigma }^2$

Statement $2:$ Arithmetic mean of $2X_1,2X_2,2X_3,......,2X_n$ is $4\overline{X}$

• A. Statement $1$ is false, statement $2$ is true.
• B. Statement $1$ is true, statement $2$ is true; statement $2$ is a correct explanation of statement $1.$
• C. Statement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation of statement $1.$
• D. Statement $1$ is true, statement $2$ is false.

Answer 1

The correct option is D Statement $1$ is true, statement $2$ is false.

Arithmetic mean of $2X_1,2X_2,.....,2X_n$ $=\frac{2X_1+2X_2+......+2X_n}{n}$
$=2\times \frac{X_1+X_2+.....+X_n}{n}=2\overline{X}$

So, statement $2$ is false.

Variance of $\left(2X_i\right)$
$=2^2\times \text{Variance}\left(X_i\right)=4{\sigma }^2$

So, statement $1$ is true.