Calculating Correlation Coefficient from Regression Coefficients

The correlation coefficient ($r$) of a bivariate distribution is related to the regression coefficients ($b_{yx}$ and $b_{xy}$). The relationship is given by the formula:

$$\strong{r = \pm \sqrt{b_{yx} \times b_{xy}}}$$

The sign of the correlation coefficient ($r$) must be the same as the sign of both regression coefficients ($b_{yx}$ and $b_{xy}$).

Given Regression Coefficients

• Regression coefficient of Y on X, $b_{yx} = -0.64$
• Regression coefficient of X on Y, $b_{xy} = -0.36$

Determining the Sign of Correlation Coefficient

Since both given regression coefficients ($b_{yx} = -0.64$ and $b_{xy} = -0.36$) are negative, the correlation coefficient ($r$) must also be negative.

Calculating the Correlation Coefficient

Now, we can calculate the value of the correlation coefficient using the formula and the determined negative sign:

$$r = - \sqrt{b_{yx} \times b_{xy}}$$

Substitute the given values:

$$r = - \sqrt{(-0.64) \times (-0.36)}$$

Multiply the values under the square root. A negative number multiplied by a negative number gives a positive number:

$$r = - \sqrt{0.64 \times 0.36}$$

We can calculate the square root of each term separately:

$$r = - (\sqrt{0.64} \times \sqrt{0.36})$$

Calculate the square roots:

$$\sqrt{0.64} = 0.8$$

$$\sqrt{0.36} = 0.6$$

Now, substitute these values back into the equation for $r$:

$$r = - (0.8 \times 0.6)$$

Perform the multiplication:

$$r = - 0.48$$

Conclusion

The correlation coefficient of the bivariate distribution with regression coefficients -0.64 and -0.36 is -0.48.