Question

The coefficient of correlation ${r_{xy}}$ and the two regression coefficients ${b_{xy}};{b_{yx}}$ are related as
A. ${b_{xy}}/{b_{yx}}$
B. ${r_{xy}} = {b_{xy}}.{b_{yx}}$
C. ${r_{xy}} = {b_{xy}} + {b_{yx}}$
D. ${r_{xy}} = (\operatorname{sgn} {\text{ }}{b_{yx}})\sqrt {\left| {{b_{xy}}} \right|\left| {{b_{yx}}} \right|}$

Answer 1

In this statistical problem, we have given the coefficient of correlation and the two regression coefficients. And here we are asked to put how they are related with each other. That is we need to say the multiplication or the addition of two regression coefficients equals the coefficient correlation or it is related by the sign of the regression coefficient.

Complete step by step solution:
Given that, the coefficient correlation is ${r_{xy}}$ and the two regression coefficients ${b_{xy}};{b_{yx}}$.
If ${b_{yx}}$ is the regression coefficient of $y$ on $x$ and ${b_{xy}}$ is the regression coefficient of $x$ on $y$, then ${b_{yx}} \times {b_{xy}} = {r^2}_{xy}$ where ${r_{xy}}$ is the correlation coefficient of $x$ and $y$.
Therefore, ${r^2} =$coefficient of regression $y$on $x$ $\times$ coefficient regression $x$on$y$
$\Rightarrow {r^2} = {b_{yx}} \times {b_{xy}}$
$\Rightarrow r = \pm \sqrt {{b_{yx}}.{b_{xy}}}$
$r = (\operatorname{sgn} {\text{of }}{b_{xy}})\sqrt {\left| {{b_{yx}}} \right|.\left| {{b_{xy}}} \right|}$
This implies that, ${b_{xy}};{b_{yx}}$ and $r(x,y)$ have the same sign.
$\therefore$ The coefficient of correlation ${r_{xy}}$ and the two regression coefficients ${b_{xy}};{b_{yx}}$ are related as ${r_{xy}} = (\operatorname{sgn} {\text{ }}{b_{yx}})\sqrt {\left| {{b_{xy}}} \right|\left| {{b_{yx}}} \right|}$.

Hence, the answer is option (D)

Additional Information: A correlation coefficient is a numerical measure of some types of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.

Note: We can observe that, the relation between correlation coefficient and regression coefficient: Correlation coefficient is defined as the covariance of x and y divided by the product of the standard deviations of x and y. The regression coefficient is defined as the covariance of x and y divided by the variance of the independent variables, x or y. Also the regression coefficient is always positive.