Calculating Regression Coefficient of x on y

The question asks us to find the regression coefficient of x on y, denoted as $b_{xy}$, given the correlation coefficient ($r$) between x and y and the regression coefficient of y on x ($b_{yx}$).

Understanding Correlation and Regression Coefficients

The correlation coefficient ($r$) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1.

Regression coefficients measure the average change in the dependent variable for a unit change in the independent variable. $b_{yx}$ represents the regression coefficient of y on x, meaning the change in y per unit change in x. $b_{xy}$ represents the regression coefficient of x on y, meaning the change in x per unit change in y.

Relationship Between Coefficients

There is a fundamental relationship connecting the correlation coefficient and the two regression coefficients:

$\qquad r^2 = b_{yx} \times b_{xy}$

Also, the sign of the correlation coefficient ($r$) must be the same as the sign of both regression coefficients ($b_{yx}$ and $b_{xy}$). Since $r = 0.5$ (positive) and $b_{yx} = 0.5$ (positive), $b_{xy}$ must also be positive.

Therefore, we can write the relationship as:

$\qquad r = \sqrt{b_{yx} \times b_{xy}}$ (when r, $b_{yx}$, and $b_{xy}$ are positive)

Step-by-Step Calculation

Given:

• Correlation coefficient, $r = 0.5$
• Regression coefficient of y on x, $b_{yx} = 0.5$

We need to find $b_{xy}$.

Using the formula $r = \sqrt{b_{yx} \times b_{xy}}$:

$\qquad 0.5 = \sqrt{0.5 \times b_{xy}}$

To solve for $b_{xy}$, we square both sides of the equation:

$\qquad (0.5)^2 = (\sqrt{0.5 \times b_{xy}})^2$

$\qquad 0.25 = 0.5 \times b_{xy}$

Now, isolate $b_{xy}$ by dividing both sides by 0.5:

$\qquad b_{xy} = \frac{0.25}{0.5}$

$\qquad b_{xy} = \frac{1/4}{1/2}$

$\qquad b_{xy} = \frac{1}{4} \times \frac{2}{1}$

$\qquad b_{xy} = \frac{2}{4}$

$\qquad b_{xy} = 0.5$

Thus, the regression coefficient of x on y is 0.5.

Verification

Let's check if our calculated value of $b_{xy}$ satisfies the original relationship $r^2 = b_{yx} \times b_{xy}$:

Left side: $r^2 = (0.5)^2 = 0.25$

Right side: $b_{yx} \times b_{xy} = 0.5 \times 0.5 = 0.25$

Since Left side = Right side, our calculation is correct.

Summary of Result

Given a correlation coefficient of 0.5 and a regression coefficient of y on x of 0.5, the regression coefficient of x on y is calculated to be 0.5 using the relationship between these three statistical measures.

Revision Table: Correlation and Regression

Additional Information: Properties of Regression Coefficients

• Both regression coefficients ($b_{yx}$ and $b_{xy}$) must have the same sign. This sign is also the sign of the correlation coefficient ($r$).
• The correlation coefficient is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{yx} \times b_{xy}}$. The sign depends on the direction of the relationship.
• If one regression coefficient is greater than 1, the other must be less than 1 (assuming $|r| < 1$). This is because their product equals $r^2$, which is at most 1. However, in this specific case, both are 0.5, and $0.5 \times 0.5 = 0.25 \le 1$.
• Regression coefficients are not symmetric; generally, $b_{yx} \ne b_{xy}$ unless the standard deviations of x and y are equal or $|r|=1$. In this problem, they are equal ($b_{yx} = b_{xy} = 0.5$).