Question

If, \(U = \frac{X-a}{h}\), \(V = \frac{Y-b}{k}\); \(a, b, h, k>0\), then \(b_{UV}\) is

• A. \( b_{XY} \)
• B. \( khb_{XY} \)
• C. \( \frac{k}{h}b_{XY} \)
• D. \( \frac{(k+a)}{(h+b)}b_{XY} \)

Hint:

Remember how regression coefficients transform: \(b_{YX}\) is affected by the scale factors of both X and Y (\(h\) and \(k\)), while \(b_{XY}\) is also affected. Specifically, \(b_{VU} = \frac{h}{k}b_{YX}\) and \(b_{UV} = \frac{k}{h}b_{XY}\). The coefficient in the numerator corresponds to the independent variable's scale factor.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question deals with the effect of change of origin and scale on the regression coefficient. The variables X and Y are transformed into U and V. We need to find the relationship between the regression coefficient of U on V (\(b_{UV}\)) and the regression coefficient of X on Y (\(b_{XY}\)).

Step 2: Key Formula or Approach:
The regression coefficient of X on Y is given by \(b_{XY} = r_{XY} \frac{\sigma_X}{\sigma_Y}\). Properties to use: 1. Correlation coefficient is independent of change of origin and scale: \(r_{UV} = r_{XY}\). 2. Effect of transformation on standard deviation: \(\sigma_{cX+d} = |c|\sigma_X\).

Step 3: Detailed Explanation:
The transformations are \(U = \frac{X-a}{h}\) and \(V = \frac{Y-b}{k}\). We need to find \(b_{UV}\), the regression coefficient of U on V. The formula for \(b_{UV}\) is: \[ b_{UV} = r_{UV} \frac{\sigma_U}{\sigma_V} \] Let's find the components in terms of X and Y. 1. The correlation coefficient is invariant, so \(r_{UV} = r_{XY}\). 2. The standard deviation of U is: \[ \sigma_U = \sigma_{\left(\frac{X-a}{h}\right)} = \sigma_{\left(\frac{1}{h}X - \frac{a}{h}\right)} = \left|\frac{1}{h}\right|\sigma_X = \frac{1}{h}\sigma_X \quad (\text{since } h>0) \] 3. The standard deviation of V is: \[ \sigma_V = \sigma_{\left(\frac{Y-b}{k}\right)} = \sigma_{\left(\frac{1}{k}Y - \frac{b}{k}\right)} = \left|\frac{1}{k}\right|\sigma_Y = \frac{1}{k}\sigma_Y \quad (\text{since } k>0) \] Now, substitute these into the formula for \(b_{UV}\): \[ b_{UV} = r_{XY} \frac{(1/h)\sigma_X}{(1/k)\sigma_Y} = \frac{k}{h} \left( r_{XY} \frac{\sigma_X}{\sigma_Y} \right) \] Since \(b_{XY} = r_{XY} \frac{\sigma_X}{\sigma_Y}\), we can substitute this into the equation: \[ b_{UV} = \frac{k}{h} b_{XY} \]
Step 4: Final Answer:
The regression coefficient \(b_{UV}\) is \( \frac{k}{h}b_{XY} \).