Question

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. Statement-I is true and Statement-II is false.
• D. Statement-I is false and Statement-II is true.

Hint:

For even natural numbers, use the correct variance formula: \[ \sigma^2 = \frac{n^2 - 1}{3}. \] Always verify formulas before applying them in problems.

Answer 1

The Correct Option is D

Step 1: Checking the Variance Formula for First \( n \) Even Natural Numbers
The first \( n \) even natural numbers are: \[ 2, 4, 6, \dots, 2n. \] Their mean is given by: \[ \text{Mean} = \frac{\sum 2k}{n} = \frac{2(1+2+\dots+n)}{n} = \frac{2 \times \frac{n(n+1)}{2}}{n} = n+1. \] Variance is given by: \[ \sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2. \] Using standard results, the correct formula is: \[ \sigma^2 = \frac{n^2 - 1}{3}. \] This does not match the given formula \( \frac{n^2 - 1}{4} \), so Statement-I is incorrect.
Step 2: Checking Statement-II For \( n = 20 \), \[ \sigma^2 = \frac{20^2 - 1}{3} = \frac{399}{3} = 133. \] The arithmetic mean is \( 21 \), and \[ 133 - 21 = 112. \] Since this matches the given statement, Statement-II is true.
Thus, the correct answer is \( \mathbf{Statement-I} \) is false and \( \mathbf{Statement-II} \) is true.