Question

Let $x_1,x_2,\cdots \cdot x_n$ be $n$ observations, and let $\overline{x}$ be their arithmetic mean and ${\sigma }^2$ be their variance.

Statement 1: Variance of $2x_1,2x_2,\dots ,2x_n$ is $4{\sigma }^2$.

Statement 2: Arithmetic mean of $2x_1,2x_2,\dots ,2x_n$ is $4\overline{x}$.

• A. Statement 1 is false, Statement 2 is true,
• B. Statement 1 is true, Statement 2 is true, Statament 2 is a correct explanation for Statement 1
• C. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
• D. Statement 1 is true, Statement 2 is false.

Answer 1

Answer: (D)

Statement 1 is true, Statement 2 is false.

${\sigma }^2=\sum \frac{x_1^2}{n}-{(\sum \frac{x_i}{n})}^2$
Variance of $2x_1.....2x_n$
$=\sum \frac{(2x_i{)}^2}{n}-{(\sum \frac{2x_i}{n})}^2=4{\sigma }^2$.
Statement 1 is true.
A.M. of $2x_i......2x_n$
$=\frac{2x_1+2x_2....2x_n}{n}=2\overline{x}$
Statement 2 is false.
Hence, option 'D' is correct.