Understanding Correlation Coefficient and Regression Lines

This question asks about the relationship between the correlation coefficient (\(r_{xy}\)) and the two lines of regression (regression of Y on X and regression of X on Y). The correlation coefficient measures the strength and direction of the linear relationship between two variables, X and Y. The lines of regression represent the best linear fit to the data points, allowing us to predict one variable based on the other.

Let's analyze each statement carefully based on what we know about regression lines and the correlation coefficient.

Evaluating Statement 1: Correlation Coefficient \({r_{xy}} = 0\)

Statement 1 says that if the correlation coefficient \(r_{xy} = 0\), the two lines of regression are parallel to each other.

When the correlation coefficient \(r_{xy} = 0\), it means there is no linear relationship between the variables X and Y. In this case, the best prediction for Y for any given X is simply the mean of Y, and the best prediction for X for any given Y is simply the mean of X.

Let's look at the equations of the regression lines:

• The regression line of Y on X is given by: \(Y - \bar{Y} = b_{YX}(X - \bar{X})\)
• The regression line of X on Y is given by: \(X - \bar{X} = b_{XY}(Y - \bar{Y})\)

The slopes of these lines are related to the correlation coefficient and the standard deviations (\(\sigma_X\) and \(\sigma_Y\)):

• Slope of regression of Y on X (\(b_{YX}\)): \(b_{YX} = r_{xy} \frac{\sigma_Y}{\sigma_X}\)
• Slope of regression of X on Y (\(b_{XY}\)): \(b_{XY} = r_{xy} \frac{\sigma_X}{\sigma_Y}\)

If \(r_{xy} = 0\), then:

• \(b_{YX} = 0 \times \frac{\sigma_Y}{\sigma_X} = 0\)
• \(b_{XY} = 0 \times \frac{\sigma_X}{\sigma_Y} = 0\)

Substituting these slopes back into the regression equations:

• Regression of Y on X: \(Y - \bar{Y} = 0(X - \bar{X}) \implies Y = \bar{Y}\)
• Regression of X on Y: \(X - \bar{X} = 0(Y - \bar{Y}) \implies X = \bar{X}\)

The equation \(Y = \bar{Y}\) represents a horizontal line (parallel to the X-axis) at the mean of Y. The equation \(X = \bar{X}\) represents a vertical line (parallel to the Y-axis) at the mean of X.

A horizontal line and a vertical line are perpendicular to each other, not parallel (unless they are the same line, which is not the case here as they intersect at \((\bar{X}, \bar{Y})\)). Therefore, statement 1 is incorrect.

Evaluating Statement 2: Correlation Coefficient \({r_{xy}} = +1\)

Statement 2 says that if the correlation coefficient \(r_{xy} = +1\), the two lines of regression are perpendicular to each other.

When the correlation coefficient \(r_{xy} = +1\), it signifies perfect positive linear correlation. This means all the data points (X, Y) lie exactly on a single straight line with a positive slope. Since the regression lines represent the best linear fit, and all points lie on a single line, both regression lines (Y on X and X on Y) will coincide and be the same line.

Let's consider the slopes again when \(r_{xy} = +1\):

• Slope of regression of Y on X (\(b_{YX}\)): \(b_{YX} = +1 \times \frac{\sigma_Y}{\sigma_X} = \frac{\sigma_Y}{\sigma_X}\)
• Slope of regression of X on Y (\(b_{XY}\)): \(b_{XY} = +1 \times \frac{\sigma_X}{\sigma_Y} = \frac{\sigma_X}{\sigma_Y}\)

The slope of the regression line of X on Y is \(b_{XY}\). The slope of the line when Y is expressed in terms of X from the second regression equation (\(X - \bar{X} = b_{XY}(Y - \bar{Y})\)) is \(Y - \bar{Y} = \frac{1}{b_{XY}}(X - \bar{X})\), so the slope is \(\frac{1}{b_{XY}}\).

If the two lines are the same, their slopes must be equal. The slope of the Y on X line is \(\frac{\sigma_Y}{\sigma_X}\). The slope of the X on Y line, when rearranged to predict Y, is \(\frac{1}{b_{XY}} = \frac{1}{r_{xy} \frac{\sigma_X}{\sigma_Y}} = \frac{1}{+1 \times \frac{\sigma_X}{\sigma_Y}} = \frac{\sigma_Y}{\sigma_X}\).

Since \(\frac{\sigma_Y}{\sigma_X} = \frac{\sigma_Y}{\sigma_X}\), the slopes are indeed equal. The lines are the same line.

Alternatively, the angle \(\theta\) between the two lines of regression can be calculated using the formula:

\(\tan \theta = \frac{1 - r^2}{|r|} \frac{\sigma_X \sigma_Y}{\sigma_X^2 + \sigma_Y^2}\)

When \(r_{xy} = +1\), \(r^2 = (+1)^2 = 1\).

\(\tan \theta = \frac{1 - 1}{+1} \frac{\sigma_X \sigma_Y}{\sigma_X^2 + \sigma_Y^2} = \frac{0}{1} \frac{\sigma_X \sigma_Y}{\sigma_X^2 + \sigma_Y^2} = 0\)

If \(\tan \theta = 0\), then \(\theta = 0\) degrees. An angle of 0 degrees between two lines means they coincide.

Perpendicular lines have an angle of 90 degrees between them. Since the lines coincide when \(r_{xy} = +1\), they are not perpendicular. Therefore, statement 2 is incorrect.

Conclusion

Based on our analysis, both statement 1 and statement 2 are incorrect.

• When \(r_{xy} = 0\), the lines of regression are perpendicular (one horizontal, one vertical).
• When \(r_{xy} = +1\) (or \(r_{xy} = -1\)), the two lines of regression coincide (they are the same line).

Thus, neither statement is correct.

Revision Table: Correlation and Regression Lines

Additional Information: Properties of Regression Lines

Here are some key properties of the lines of regression related to the correlation coefficient and other statistical measures:

• Both regression lines always pass through the point of means \((\bar{X}, \bar{Y})\).
• The correlation coefficient \(r_{xy}\) is the geometric mean of the two regression coefficients (\(b_{YX}\) and \(b_{XY}\)) with the same sign: \(r_{xy}^2 = b_{YX} \times b_{XY}\). The sign of \(r_{xy}\) is the same as the sign of the regression coefficients.
• The angle between the two regression lines is smaller (closer to 0) when the correlation is strong (absolute value of \(r_{xy}\) is close to 1) and larger (closer to 90 degrees) when the correlation is weak (absolute value of \(r_{xy}\) is close to 0).
• When \(r_{xy} = 0\), the regression line of Y on X is \(Y = \bar{Y}\) and the regression line of X on Y is \(X = \bar{X}\). These lines are indeed perpendicular and intersect at \((\bar{X}, \bar{Y})\).
• When \(r_{xy} = \pm 1\), the two regression lines are identical. The equation of this single line represents the perfect linear relationship between X and Y.