Based on the following statements, choose the correct option Statement-I: The variance of the first $n$ even…
Based on the following statements, choose the correct option
Statement-I: The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement-II: The difference between the variance of the first 20 even natural numbers and their arithmetic mean is 112.
Both Statements are true and II is a correct explanation of $I$.
Both Statements are true but II is not a correct explanation of I.
Statements-I is true and Statement-II is false.
Statements-I is false and Statement-II is true.
Solution
$\begin{aligned}
& \mathrm{S}=2+4+\ldots .+2 n=2 \times \frac{n(n+1)}{2}=n(n+1) \\
& \text { Mean of first } n \text { natural numbers }=\frac{n(n+1)}{n}=(n+1) \\
& \text { Variance }=\frac{1}{n} \sum_{i=1}^n\left(x_i\right)^2-(\bar{x})^2 \\
& =\frac{1}{n}\left(2^2+4^2+\ldots .+4 n^2\right)-(\bar{x})^2 \\
& =\frac{4}{n} \times \frac{n(n+1)(2 n+1)}{6}-(n+1)^2=\frac{n^2-1}{3}
\end{aligned}$ So, statement I is false
Now, mean of first 20 even natural numbers
$=20+1=21$ Variance of first 20 even natural numbers
$=\frac{(20)^2-1}{3}=133$ Difference $=133-21=112$
So, statement II is true.