Which of the following are properties of regression coefficients?

Choose the correct answer from the options given below:

Correct option is C

The correct properties of regression coefficients are:

1. (A) r = √b_yx x  b_xy :​
   This property states that the correlation coefficient rrr is the geometric mean of the two regression coefficients $b$
   and $b$

2. (B) Change in origin are not affected:

   Regression coefficients are independent of changes in origin, meaning adding or subtracting a constant from $x$ or $y$ will not affect the regression coefficients.

3. (D) If both regression coefficients have negative signs, the correlation coefficient will have a negative sign:
   If both byxb_{yx}b_yx and bxyb_{xy}b_xyare negative, their product byx×bxy=r2b_{yx} \times b_{xy} = r^2b_yx×b_xy$=r$², and rrr, being the square root of this product, will also have a negative sign.

4. (E) If one regression coefficient is greater than one, the other must be less than one:
   This is true because byx×bxy=r2b_{yx} \times b_{xy} = r^2b_yx×b_xy$=r$², and r2r^2r²is always less than or equal to 1. Therefore, if one regression coefficient exceeds 1, the other must be less than 1 to satisfy the relationship.

​(C) Change in scale are not affected is incorrect, as regression coefficients do depend on changes in scale. Scaling (multiplying $x$ or $y$ by a constant) will change the magnitude of the regression coefficients.

Information Booster

Key Properties of Regression Coefficients:

1. Product Relationship: byx×bxy=r2b_{yx} \times b_{xy} = r^2b_yx×b_xy$=r$² .
   
2. Effect of Origin: Regression coefficients are independent of a change in origin (adding/subtracting a constant to $x$ or $y$).
3. Effect of Scale: Regression coefficients are affected by a change in scale (multiplication/division of $x$ or $y$ by a constant).
4. Sign of Correlation: If both regression coefficients are negative, the correlation coefficient is also negative.
5. Limits of Coefficients: One regression coefficient greater than 1 implies the other must be less than 1.

Additional Knowledge

1. Effect of Scaling: Multiplying $x$ or $y$ by a constant (x′=mx,y′=ny)(x' = mx, y' = ny)(x'
   =mx,y
   '=ny) changes the slope of the regression line proportionally.