Question

If a constant 35 is added to each of the value of X and Y, the regression coefficient is :

• A. increased by 35
• B. $\frac{1}{35}^{\text{th}}$ of the original regression coefficient
• C. reduced by 35
• D. same as the original regression coefficient

Hint:

Remember:
- Correlation and regression coefficients are independent of a change of origin (addition/subtraction of a constant).
- Regression coefficients are not independent of a change of scale (multiplication/division), whereas correlation coefficients are independent of both origin and scale.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the properties of the regression coefficient ($b_{yx}$ or $b_{xy}$) under linear transformations of the variables, specifically changes of origin and scale.
Key Formula or Approach:
Let $U = X + c$ and $V = Y + d$ represent changes of origin, where $c$ and $d$ are constants.
The regression coefficient of $Y$ on $X$ is defined as: \[ b_{yx} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} \] Since covariance and variance are completely unaffected by shifts in origin: \[ \text{Cov}(U, V) = \text{Cov}(X, Y) \] \[ \text{Var}(U) = \text{Var}(X) \] Therefore, the regression coefficient remains unchanged: \[ b_{vu} = b_{yx} \]

Step 2: Detailed Explanation:
When we add a constant (such as 35) to all values of $X$ and $Y$:
- The mean of $X$ and $Y$ shifts by 35, representing a shift in the origin.
- The spread, variability, and covariation between the data points remain exactly the same.
- Because regression coefficients are independent of any change of origin, adding a constant does not alter the slope of the regression line.
- Thus, the regression coefficient remains exactly the same as the original coefficient.

Step 3: Final Answer:
The regression coefficient remains the same as the original, corresponding to Option (D).