If X is changed to `a+hU and Y` to `b+kV`, then which one of the following is the correct relation between the regression coefficients `b_(XY) and b_(UV)`?

Options

A. `hb_(XY)=kb_(UV)`
B. `kb_(XY)=hb_(UV)`
C. `b_(XY)=b_(UV)`
D. `k^(2)b_(XY)=h^(2)b_(UV)`

Explanation

To solve the problem, we need to establish the relationship between the regression coefficients \( b_{XY} \) and \( b_{UV} \) when the variables \( X \) and \( Y \) are transformed. We are given that \( X \) is changed to \( a + hU \) and \( Y \) is changed to \( b + kV \).

### Step-by-Step Solution:

1. **Understanding the Transformation**:
- We have \( X = a + hU \) and \( Y = b + kV \).
- Here, \( a \) and \( b \) are constants, while \( h \) and \( k \) are scaling factors for the variables \( U \) and \( V \).

2. **Regression Coefficient Definition**:
- The regression coefficient \( b_{XY} \) is defined as the change in \( Y \) for a unit change in \( X \).
- Similarly, \( b_{UV} \) is the change in \( V \) for a unit change in \( U \).

3. **Finding the Relationship**:
- The regression coefficient \( b_{XY} \) can be expressed in terms of the transformed variables:
\[ b_{XY} = \frac{\text{Cov}(Y, X)}{\text{Var}(X)} \]
- Substituting the transformations into this formula, we have:
\[ b_{XY} = \frac{\text{Cov}(kV + b, hU + a)}{\text{Var}(hU + a)} \]
- The covariance can be simplified as:
\[ \text{Cov}(kV, hU) = k \cdot h \cdot \text{Cov}(V, U) \]
- The variance can be simplified as:
\[ \text{Var}(hU) = h^2 \cdot \text{Var}(U) \]
- Therefore, we can rewrite \( b_{XY} \):
\[ b_{XY} = \frac{k \cdot h \cdot \text{Cov}(V, U)}{h^2 \cdot \text{Var}(U)} = \frac{k}{h} \cdot \frac{\text{Cov}(V, U)}{\text{Var}(U)} \]

4. **Finding \( b_{UV} \)**:
- Similarly, we can express \( b_{UV} \):
\[ b_{UV} = \frac{\text{Cov}(U, V)}{\text{Var}(U)} \]

5. **Establishing the Final Relation**:
- We can relate \( b_{XY} \) and \( b_{UV} \):
\[ b_{XY} = \frac{k}{h} \cdot b_{UV} \]

### Conclusion:
Thus, the relationship between the regression coefficients is:
\[ b_{XY} = \frac{k}{h} \cdot b_{UV} \]