Given below are two statements:

Statement I: The product of two regression coefficients, bxy and byx, equals r² and cannot exceed 1.
Statement II: The average of two regression coefficients ((bxy + byx)/2) is always greater than the correlation coefficient, rxy.

In the light of the above statements, choose the correct answer from the options given below:

Correct option is A

Statement I is correct because it is a well-known identity in regression analysis:

b_xy × b_yx = r²

​where rrr is the correlation coefficient between variables X and Y. Since r²>1, the product of the regression coefficients cannot exceed 1.

Statement II is also considered correct in the context where the regression coefficients are positive or when the variables are positively correlated. The average of the two regression coefficients

is always greater than or equal to the correlation coefficient $r$. This is because each regression coefficient can be written as:

and by the AM–GM inequality (Arithmetic Mean - Geometric Mean inequality), the average of b_xy and b_yx is always greater than or equal to r (except when s_x = s_y, where it equals r).

Thus, both statements hold true.

Information Booster:

• The regression coefficients show how one variable changes in response to another.

• The product of both regression coefficients always gives the square of the correlation, which is always less than or equal to 1.

• The correlation coefficient measures the strength and direction of a linear relationship between two variables.

• The average of the two regression coefficients is usually greater than or equal to the correlation when both coefficients are positive.

• The regression coefficients are affected by how spread out the variables are (i.e., their variability).

• These relationships are useful to understand how closely two variables move together and how one can be predicted using the other.