Understanding Regression Lines and Key Statistics

Regression lines are used to model the relationship between two variables. The intersection point of the two regression lines, one for y on x and one for x on y, represents the mean values of x and y, denoted as \(\bar{x}\) and \(\bar{y}\) respectively.

Finding the Means (x̅, y̅)

We are given the equations of the two regression lines:

1. \(6x + y = 30\)
2. \(3x + 2y = 25\)

To find the point of intersection, we need to solve this system of linear equations. We can use the elimination method.

Multiply the first equation by 2:

\(2 \times (6x + y) = 2 \times 30\)

\(12x + 2y = 60\) (Equation 3)

Now, subtract the second equation from the third equation:

\((12x + 2y) - (3x + 2y) = 60 - 25\)

\(12x + 2y - 3x - 2y = 35\)

\(9x = 35\)

\(x = \frac{35}{9}\)

Now substitute the value of x into the first equation \(6x + y = 30\):

\(6 \times \frac{35}{9} + y = 30\)

\(\frac{210}{9} + y = 30\)

\(\frac{70}{3} + y = 30\)

\(y = 30 - \frac{70}{3}\)

\(y = \frac{90}{3} - \frac{70}{3}\)

\(y = \frac{20}{3}\)

So, the mean values are \(\bar{x} = \frac{35}{9}\) and \(\bar{y} = \frac{20}{3}\).

Determining Regression Coefficients and Correlation Coefficient (r)

The slopes of the regression lines are the regression coefficients. Let \(b_{yx}\) be the regression coefficient of y on x and \(b_{xy}\) be the regression coefficient of x on y.

The regression equation of y on x is typically written as \(y - \bar{y} = b_{yx}(x - \bar{x})\) or \(y = b_{yx}x + a_y\). The regression equation of x on y is \(x - \bar{x} = b_{xy}(y - \bar{y})\) or \(x = b_{xy}y + a_x\).

From the given equations:

Equation 1: \(6x + y = 30\)

• If this is y on x: \(y = -6x + 30\). So, \(b_{yx} = -6\).
• If this is x on y: \(6x = -y + 30 \implies x = -\frac{1}{6}y + 5\). So, \(b_{xy} = -\frac{1}{6}\).

Equation 2: \(3x + 2y = 25\)

• If this is y on x: \(2y = -3x + 25 \implies y = -\frac{3}{2}x + \frac{25}{2}\). So, \(b_{yx} = -\frac{3}{2}\).
• If this is x on y: \(3x = -2y + 25 \implies x = -\frac{2}{3}y + \frac{25}{3}\). So, \(b_{xy} = -\frac{2}{3}\).

A key property of regression coefficients is that the product of their absolute values must be less than or equal to 1: \(|b_{yx} \times b_{xy}| \le 1\).

Let's test the possibilities:

Possibility 1: Equation 1 is y on x, Equation 2 is x on y.

• \(b_{yx} = -6\) (from Eq 1)
• \(b_{xy} = -\frac{2}{3}\) (from Eq 2)

Product: \(b_{yx} \times b_{xy} = (-6) \times (-\frac{2}{3}) = \frac{12}{3} = 4\). Since \(|4| = 4 > 1\), this assignment is invalid.

Possibility 2: Equation 1 is x on y, Equation 2 is y on x.

• \(b_{xy} = -\frac{1}{6}\) (from Eq 1)
• \(b_{yx} = -\frac{3}{2}\) (from Eq 2)

Product: \(b_{yx} \times b_{xy} = (-\frac{3}{2}) \times (-\frac{1}{6}) = \frac{3}{12} = \frac{1}{4}\). Since \(|\frac{1}{4}| = \frac{1}{4} \le 1\), this assignment is valid.

Therefore, the correct regression coefficients are \(b_{yx} = -\frac{3}{2}\) and \(b_{xy} = -\frac{1}{6}\).

The correlation coefficient \(r\) is the geometric mean of the regression coefficients, and its sign is the same as the sign of the regression coefficients:

\(r = \pm \sqrt{b_{yx} \times b_{xy}}\)

Since both \(b_{yx}\) and \(b_{xy}\) are negative, \(r\) must be negative.

\(r = - \sqrt{(-\frac{3}{2}) \times (-\frac{1}{6})}\)

\(r = - \sqrt{\frac{3}{12}}\)

\(r = - \sqrt{\frac{1}{4}}\)

\(r = - \frac{1}{2}\)

\(r = -0.5\)

Summary of Results

Based on the calculations:

• Mean of x, \(\bar{x} = \frac{35}{9}\)
• Mean of y, \(\bar{y} = \frac{20}{3}\)
• Correlation coefficient, \(r = -0.5\)

Revision Table: Key Statistics Concepts

Additional Information: Properties of Regression Lines

Here are some important properties related to regression lines and coefficients:

• Both regression lines pass through the point \((\bar{x}, \bar{y})\).
• The correlation coefficient \(r\) has the same sign as the regression coefficients \(b_{yx}\) and \(b_{xy}\).
• The value of \(r^2\) (the coefficient of determination) represents the proportion of the variance in the dependent variable that is predictable from the independent variable. \(r^2 = b_{yx} \times b_{xy}\).
• The angle between the two regression lines is related to the correlation coefficient. If \(r = \pm 1\), the lines coincide. If \(r = 0\), the lines are perpendicular (if the variables are measured in their own standard deviations).
• The condition \(|b_{yx} \times b_{xy}| \le 1\) helps identify which equation corresponds to which regression line if it's not explicitly stated.