Step 1: Concept Explanation:
This problem explores how changing the origin and scale affects the regression coefficient. Variables X and Y are transformed to U and V. Our goal is to relate the regression coefficient of U on V (\(b_{UV}\)) to the regression coefficient of X on Y (\(b_{XY}\)).
Step 2: Core Formula and Principles:
The regression coefficient of X on Y is defined as \(b_{XY} = r_{XY} \frac{\sigma_X}{\sigma_Y}\).
Key properties:
1. The correlation coefficient remains unchanged by origin and scale shifts: \(r_{UV} = r_{XY}\).
2. The standard deviation transforms as: \(\sigma_{cX+d} = |c|\sigma_X\).
Step 3: Detailed Solution:
Given transformations: \(U = \frac{X-a}{h}\) and \(V = \frac{Y-b}{k}\).
We aim to determine \(b_{UV}\), the regression coefficient of U on V, using the formula:
\[ b_{UV} = r_{UV} \frac{\sigma_U}{\sigma_V} \]Expressing components in terms of X and Y:
1. Due to invariance, \(r_{UV} = r_{XY}\).2. The standard deviation of U is derived as: \[ \sigma_U = \sigma_{\left(\frac{X-a}{h}\right)} = \sigma_{\left(\frac{1}{h}X - \frac{a}{h}\right)} = \left|\frac{1}{h}\right|\sigma_X = \frac{1}{h}\sigma_X \quad (\text{since } h>0) \]3. Similarly, the standard deviation of V is: \[ \sigma_V = \sigma_{\left(\frac{Y-b}{k}\right)} = \sigma_{\left(\frac{1}{k}Y - \frac{b}{k}\right)} = \left|\frac{1}{k}\right|\sigma_Y = \frac{1}{k}\sigma_Y \quad (\text{since } k>0) \]Substituting these into the \(b_{UV}\) formula:\[ b_{UV} = r_{XY} \frac{(1/h)\sigma_X}{(1/k)\sigma_Y} = \frac{k}{h} \left( r_{XY} \frac{\sigma_X}{\sigma_Y} \right) \]Replacing \(r_{XY} \frac{\sigma_X}{\sigma_Y}\) with \(b_{XY}\):\[ b_{UV} = \frac{k}{h} b_{XY} \]
Step 4: Final Result:
The regression coefficient \(b_{UV}\) equals \( \frac{k}{h}b_{XY} \).