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:

Remember: Adding a constant to all observations does NOT change the variance, but multiplying by a constant \(k\) changes it by \(k^2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Variance is a measure of dispersion. If every data point in a set is scaled by a constant factor \( k \), the spread of the data increases quadratically.

Step 2: Key Formula or Approach:
If \( \text{Var}(X) = \sigma^2 \), then: \[ \text{Var}(kX) = k^2 \text{Var}(X) \]

Step 3: Detailed Explanation:
Given the initial variance \(\sigma^2 = 5\). The constant multiplier is \( k = 2 \). Using the property: \[ \text{New Variance} = k^2 \times \text{Old Variance} \] \[ \text{New Variance} = 2^2 \times 5 = 4 \times 5 = 20 \] In the format of the options, this is \( 2^2 \times 5 \).

Step 4: Final Answer
The new variance is \( 2^2 \times 5 \).