AP EAMCET · Maths · Statistics
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.
- A Both Statements are true and II is a correct explanation of \(I\).
- B Both Statements are true but II is not a correct explanation of I.
- C Statements-I is true and Statement-II is false.
- D Statements-I is false and Statement-II is true.
Answer & Solution
Correct Answer
(A) Both Statements are true and II is a correct explanation of \(I\).
Step-by-step Solution
Detailed explanation
\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…
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