Question

Let \(x_1, x_2, \dots, x_n\) be observations which have a variance \(\sigma_x^2\). Each \(x_i\) is changed to \(a x_i - h\), where \(a\) and \(h\) are positive constants, then the new series will have variance:

• A. \(\sigma_x^2\)
• B. \(a^2 \sigma_x^2\)
• C. \(\sigma_x^2 - h^2\)
• D. \(a^2 \sigma_x^2 - h^2\)

Hint:

Exam Tip:
Effects of transformations on variance:

• Adding \(c\): Variance unchanged.
• Subtracting \(c\): Variance unchanged.
• Multiplying by \(c\): Variance multiplied by \(c^2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Variance is affected by changes in scale but not by changes in location.
Adding or subtracting a constant does not change the variance.
Multiplying by a constant multiplies the variance by the square of that constant.

Step 2: Key Formula or Approach:
If \(y_i = a x_i - h\), then: \[ \text{Var}(Y) = \text{Var}(aX - h) = a^2 \text{Var}(X) = a^2 \sigma_x^2 \]

Step 3: Detailed Explanation:
The constant \(h\) does not affect the variance because it is a location parameter.
The variance is multiplied by \(a^2\).
So, the new variance is \(a^2 \sigma_x^2\).

Step 4: Final Answer:
Therefore, option (B) is correct.