Question

The variance of 20 observations is 5. If each observation is multiplied by 2, then the new variance of the resulting observation is:

• A. \( 2^3 \times 5 \)
• B. \( 2^2 \times 5 \)
• C. \( 2 \times 5 \)
• D. \( 2^4 \times 5 \)

Hint:

If data is scaled by \( k \), variance scales by \( k^2 \).

Answer 1

The Correct Option is B

Given the variance: \[ \sigma^2 = 5 \] If each observation is multiplied by a constant \( k = 2 \), the variance changes as: \[ \text{Variance } (\sigma^2) = \frac{1}{n} \sum_{i=1}^{20} (x_i - \overline{x})^2 \] \[ 5 = \frac{1}{20} \sum_{i=1}^{20} (x_i - \overline{x})^2 \] \[ \sum_{i=1}^{20} (x_i - \overline{x})^2 = 100 \quad \text{(i)} \] If each observation is multiplied by 2, the new observations are \( y_i \), such that: \[ y_i = 2x_i \quad \text{or} \quad x_i = \frac{1}{2} y_i \] The mean of the new observations is: \[ \overline{y} = 2 \overline{x} \] Substituting \( x_i \) and \( \overline{x} \) into equation (i), we get: \[ \sum_{i=1}^{20} \left(\frac{1}{2} y_i - \frac{1}{2} \overline{y}\right)^2 = 100 \] \[ \sum_{i=1}^{20} (y_i - \overline{y})^2 = 400 \] Thus, the variance of the new observations is: \[ \frac{1}{20} \times 400 = 20 = 2^2 \times 5 \]

Answer 2

Step 1: Understanding the variance transformation rule
The variance of a set of observations is affected by scaling. If each observation is multiplied by a constant factor, say \( c \), then the variance of the new set of observations becomes \( c^2 \) times the original variance.

Step 2: Given data
We are given that the variance of 20 observations is 5. The constant factor by which each observation is multiplied is 2.

Step 3: Apply the variance transformation rule
When each observation is multiplied by 2, the new variance is given by: \[ \text{New Variance} = 2^2 \times \text{Original Variance} \] Substitute the given variance: \[ \text{New Variance} = 2^2 \times 5 = 4 \times 5 = 20 \] Step 4: Final Answer
The new variance of the resulting observations is:

\( 2^2 \times 5 \)