Question

Consider the first 10 natural numbers. If we multiply each number by \( -1 \) and add 1 to each number, the variance of the numbers so obtained is

• A. 6.5
• B. 8.25
• C. 2.87
• D. 3.87

Hint:

Variance is a measure of how spread out the numbers in a set are. To calculate it, find the mean and then calculate the squared differences from the mean.

Answer 1

The Correct Option is B

Let the first 10 natural numbers be \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \). If we multiply each of these numbers by \( -1 \) and add 1, the new set of numbers will be: \[ \{-1+1, -2+1, -3+1, -4+1, -5+1, -6+1, -7+1, -8+1, -9+1, -10+1\} \] This simplifies to: \[ \{0, -1, -2, -3, -4, -5, -6, -7, -8, -9\} \] Now, we calculate the variance of this new set of numbers. The variance is given by the formula: \[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \] where \( \mu \) is the mean of the numbers and \( x_i \) are the individual numbers in the set. Step 1: Find the mean of the new set The mean \( \mu \) is the sum of the numbers divided by the number of elements: \[ \mu = \frac{0 + (-1) + (-2) + (-3) + (-4) + (-5) + (-6) + (-7) + (-8) + (-9)}{10} = \frac{-45}{10} = -4.5 \] Step 2: Calculate the variance The variance is: \[ \text{Variance} = \frac{1}{10} \left[ (0 - (-4.5))^2 + (-1 - (-4.5))^2 + \dots + (-9 - (-4.5))^2 \right] \] This simplifies to: \[ \text{Variance} = \frac{1}{10} \left[ 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 \right] \] \[ \text{Variance} = \frac{1}{10} \times 80.5 = 8.25 \] Thus, the variance of the numbers is 8.25.