Correlation Calculation Using Regression Coefficients

The coefficient of correlation ($r$) quantifies the linear relationship between two variables. When the two regression coefficients ($b_x$ and $b_y$) are known, the correlation coefficient can be directly determined.

Relationship Between Regression and Correlation Coefficients

The regression coefficients, $b_x$ (regression of X on Y) and $b_y$ (regression of Y on X), are related to the coefficient of correlation ($r$) by the formula:

$ r = \pm \sqrt{b_x \times b_y} $

The sign of the correlation coefficient '$r$' must match the signs of the individual regression coefficients.

Calculation Steps

1. Identify Given Coefficients: The two regression coefficients are provided as $0.8$ and $0.2$. Let $b_x = 0.8$ and $b_y = 0.2$.
2. Apply the Formula: Substitute these values into the formula for '$r$':

   $ r = \pm \sqrt{0.8 \times 0.2} $

3. Compute the Value: Calculate the product and the square root:

   $ r = \pm \sqrt{0.16} $

   $ r = \pm 0.4 $

4. Determine the Sign: Both given regression coefficients ($0.8$ and $0.2$) are positive. Therefore, the coefficient of correlation '$r$' must also be positive.
5. State the Result: The coefficient of correlation is $+0.4$.

Conclusion

The value of the coefficient of correlation is +0.40.