The regression coefficients remain unchanged due to a shift of origin but change due to a shift of scale. This property states that if the original pair of variables is (x, y) and if they are changed to the pair (u, v) where

The correct answer is option 'A', which states that the regression coefficients remain unchanged due to a shift of origin. In this explanation, we will discuss why the other options are incorrect and provide a detailed understanding of why the regression coefficients remain unaffected by a shift of origin.

Explanation:
1. Shift of Origin:
- A shift of origin refers to a change in the position of the coordinate system used to represent the data.
- When a shift of origin occurs, the entire dataset is moved by a fixed amount in both the x-axis and y-axis directions.
- However, the regression line represents the relationship between the variables, and the coefficients of the regression equation capture this relationship.
- The regression coefficients are calculated based on the relative positions and relationships of the data points, rather than their absolute positions.
- Therefore, a shift of origin does not affect the slope or intercept of the regression line, and the regression coefficients remain unchanged.

2. Shift of Scale:
- A shift of scale refers to a change in the units or measurement scale of the variables.
- For example, converting the variable from inches to centimeters or from meters to kilometers.
- When a shift of scale occurs, the regression coefficients will be affected as they are based on the units of measurement.
- The slope of the regression line represents the change in the dependent variable for a one-unit change in the independent variable.
- If the scale of the independent variable changes, the slope will also change proportionally, and hence, the regression coefficients will be affected.

3. Both (a) and (b):
- This option suggests that both a shift of origin and a shift of scale can affect the regression coefficients.
- However, as explained above, a shift of origin does not affect the regression coefficients, only a shift of scale does.
- Therefore, this option is incorrect.

4. (a) or (b):
- This option suggests that either a shift of origin or a shift of scale can affect the regression coefficients.
- However, as discussed earlier, a shift of origin does not affect the regression coefficients.
- Therefore, this option is also incorrect.

In conclusion, the correct answer is option 'A' because the regression coefficients remain unchanged due to a shift of origin.