State True or False: If `u=x-a` and `v= y-b` then `b_(xy)= b_(uv)`.

Answer

To determine whether the statement "If \( u = x - a \) and \( v = y - b \), then \( b_{xy} = b_{uv} \)" is true or false, we will analyze the relationship between the regression coefficients \( b_{xy} \) and \( b_{uv} \).

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( u = x - a \) and \( v = y - b \).
- This implies \( x = u + a \) and \( y = v + b \).

2. **Understand the Regression Coefficient**:
- The regression coefficient \( b_{xy} \) (the regression of \( y \) on \( x \)) is given by:
\[ b_{xy} = r_{xy} \frac{\sigma_x}{\sigma_y} \]
- Similarly, the regression coefficient \( b_{uv} \) (the regression of \( v \) on \( u \)) is given by:
\[ b_{uv} = r_{uv} \frac{\sigma_u}{\sigma_v} \]

3. **Correlation Coefficient Independence**:
- The correlation coefficient \( r \) is independent of the origin and scale. Therefore, we have:
\[ r_{xy} = r_{uv} \]

4. **Standard Deviations**:
- The standard deviations \( \sigma_x \) and \( \sigma_y \) are related to \( \sigma_u \) and \( \sigma_v \) as follows:
- Since \( u = x - a \), the standard deviation \( \sigma_u = \sigma_x \).
- Since \( v = y - b \), the standard deviation \( \sigma_v = \sigma_y \).

5. **Substituting into the Regression Coefficients**:
- Substitute the values into the regression coefficients:
\[ b_{xy} = r_{xy} \frac{\sigma_x}{\sigma_y} = r_{uv} \frac{\sigma_u}{\sigma_v} \]
- Since \( r_{xy} = r_{uv} \) and \( \sigma_u = \sigma_x \), \( \sigma_v = \sigma_y \), we can conclude:
\[ b_{xy} = b_{uv} \]

6. **Conclusion**:
- Therefore, the statement \( b_{xy} = b_{uv} \) is true.

### Final Answer:
The statement is **True**.