Understanding Regression Coefficients and Correlation

This question asks us to find the regression coefficient of X on Y, given the regression coefficient of Y on X and the correlation coefficient between X and Y. Let's define the terms involved:

• Regression coefficient of Y on X (\(b_{YX}\)): Measures the average change in Y associated with a one-unit change in X. It tells us how Y is expected to change when X changes.
• Regression coefficient of X on Y (\(b_{XY}\)): Measures the average change in X associated with a one-unit change in Y. It tells us how X is expected to change when Y changes.
• Correlation coefficient (\(r\)): Measures the strength and direction of the linear relationship between two variables, X and Y. It ranges from -1 to +1.

Relationship Between Regression Coefficients and Correlation Coefficient

There is a fundamental relationship connecting the two regression coefficients (\(b_{YX}\) and \(b_{XY}\)) and the correlation coefficient (\(r\)). The square of the correlation coefficient is equal to the product of the two regression coefficients:

\(r^2 = b_{YX} \times b_{XY}\)

Also, it's important to remember that the sign of \(b_{YX}\), \(b_{XY}\), and \(r\) must always be the same. If \(r\) is positive, both regression coefficients must be positive. If \(r\) is negative, both regression coefficients must be negative.

Step-by-Step Calculation

We are given the following information:

• Regression coefficient of Y on X, \(b_{YX} = -6\)
• Correlation coefficient between X and Y, \(r = -\frac{1}{2}\)

We need to find the regression coefficient of X on Y, \(b_{XY}\).

Using the formula \(r^2 = b_{YX} \times b_{XY}\), we can substitute the given values:

\(\left(-\frac{1}{2}\right)^2 = (-6) \times b_{XY}\)

Let's calculate the square of the correlation coefficient:

\(\left(-\frac{1}{2}\right)^2 = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) = \frac{(-1) \times (-1)}{2 \times 2} = \frac{1}{4}\)

Now substitute this back into the equation:

\(\frac{1}{4} = -6 \times b_{XY}\)

To isolate \(b_{XY}\), we need to divide both sides of the equation by -6:

\(b_{XY} = \frac{\frac{1}{4}}{-6}\)

\(b_{XY} = \frac{1}{4} \times \frac{1}{-6}\)

\(b_{XY} = \frac{1 \times 1}{4 \times (-6)}\)

\(b_{XY} = \frac{1}{-24}\)

\(b_{XY} = -\frac{1}{24}\)

Verification of Sign Consistency

We found \(b_{XY} = -\frac{1}{24}\). Let's check if the signs are consistent with the given values:

• \(r = -\frac{1}{2}\) (Negative)
• \(b_{YX} = -6\) (Negative)
• \(b_{XY} = -\frac{1}{24}\) (Negative)

All three values are negative, which is consistent with the property that \(r\), \(b_{YX}\), and \(b_{XY}\) must have the same sign.

Thus, the regression coefficient of X on Y is \(-\frac{1}{24}\).

Revision Table: Key Concepts in Regression Analysis

Additional Information on Regression and Correlation

Regression analysis is a powerful statistical method used to examine the relationship between two or more variables. Simple linear regression involves two variables, one independent (X) and one dependent (Y).

• The regression line of Y on X is given by \(\hat{Y} = a + b_{YX}X\), where \(\hat{Y}\) is the predicted value of Y.
• The regression line of X on Y is given by \(\hat{X} = c + b_{XY}Y\), where \(\hat{X}\) is the predicted value of X.
• The constants \(a\) and \(c\) are the intercepts.
• The correlation coefficient \(r\) is a standardized measure, meaning its value doesn't depend on the units of measurement of X and Y.
• The regression coefficients \(b_{YX}\) and \(b_{XY}\), however, are not standardized and depend on the units of measurement.
• When the two regression lines are perpendicular, the correlation coefficient \(r\) is 0. This indicates no linear relationship between the variables.
• If the correlation is perfect (\(r = +1\) or \(r = -1\)), the two regression lines coincide.

Understanding the relationship \(r^2 = b_{YX} \times b_{XY}\) is crucial for solving problems involving both correlation and regression coefficients.