Question

Number of regression equations with 2 variables will be

• A. 8
• B. 4
• C. 2
• D. 6

Hint:

Bivariate Regression: For 2 variables ($X, Y$), there are ALWAYS 2 regression equations ($Y$ on $X$ and $X$ on $Y$), intersecting at $(\bar{X}, \bar{Y})$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Bivariate linear regression: two interdependent continuous random variables yield exactly two distinct least-squares regression lines ($Y$ on $X$ and $X$ on $Y$).
Key Formula or Approach:
\[ \text{1. Regression of } Y \text{ on } X: \quad (Y - \bar{Y}) = b_{yx}(X - \bar{X}) \]
\[ \text{2. Regression of } X \text{ on } Y: \quad (X - \bar{X}) = b_{xy}(Y - \bar{Y}) \]

Step 2: Detailed Explanation:
For any two variables $X$ and $Y$ in bivariate statistical analysis:
- There are two distinct regression lines and equations:
1. Regression of $Y$ on $X$: Minimizes the sum of squared vertical residuals (errors in $Y$), predicting dependent variable $Y$ from independent variable $X$ ($Y = a_1 + b_{yx} X$).
2. Regression of $X$ on $Y$: Minimizes the sum of squared horizontal residuals (errors in $X$), predicting $X$ from $Y$ ($X = a_2 + b_{xy} Y$).
- Both lines intersect at the centroid point $(\bar{X}, \bar{Y})$.

Step 3: Final Answer:
Thus, the number of regression equations with 2 variables will be 2, corresponding to option (C).