Question

If the regression coefficient of X on Y is \(b_{xy}\), and the variables are transformed as \(U = \frac{X - a}{h}\), \(V = \frac{Y - b}{k}\), then the regression coefficient of U on V is:

• A. \(b_{xy}\)
• B. \(\frac{h}{k} b_{xy}\)
• C. \(\frac{h}{k} (b_{xy} - a - b)\)
• D. \(\frac{k}{h} b_{xy}\)

Hint:

Exam Tip:
Effect of linear transformations on regression coefficients:
If \(U = \frac{X - a}{h}\) and \(V = \frac{Y - b}{k}\), then \(b_{uv} = \frac{k}{h} b_{xy}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find how the regression coefficient changes under linear transformations of variables.

Step 2: Key Formula or Approach:
The regression coefficient of X on Y is: \[ b_{xy} = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} \] Given: \[ U = \frac{X - a}{h}, \quad V = \frac{Y - b}{k} \] So, \(X = a + hU\), \(Y = b + kV\).
Then, \[ \text{Cov}(U, V) = \text{Cov}\left(\frac{X - a}{h}, \frac{Y - b}{k}\right) = \frac{1}{hk} \text{Cov}(X, Y) \] \[ \text{Var}(V) = \text{Var}\left(\frac{Y - b}{k}\right) = \frac{1}{k^2} \text{Var}(Y) \] The regression coefficient of U on V is: \[ b_{uv} = \frac{\text{Cov}(U, V)}{\text{Var}(V)} = \frac{\frac{1}{hk} \text{Cov}(X, Y)}{\frac{1}{k^2} \text{Var}(Y)} = \frac{k}{h} \cdot \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} = \frac{k}{h} b_{xy} \]

Step 3: Final Answer:
Therefore, option (D) is correct.