Understanding Regression Lines and Coefficients

The question provides two linear equations and asks us to identify which one represents the regression line of y on x and which one represents the regression line of x on y. We are also asked to verify if the provided statements about these lines are correct.

The two given equations are:

1. \(3x - 4y + 8 = 0\)
2. \(4x - 3y - 1 = 0\)

In the context of linear regression, if we have two regression lines, we can determine which is the line of y on x and which is the line of x on y by examining their slopes (regression coefficients). Let the regression coefficient of y on x be \(b_{yx}\) and the regression coefficient of x on y be \(b_{xy}\). A key property is that the product of these coefficients, \(b_{yx} \times b_{xy}\), must be less than or equal to 1 (and also greater than or equal to 0, as they must have the same sign as the correlation coefficient). The square of the correlation coefficient, \(r^2\), is equal to \(b_{yx} \times b_{xy}\), and \(0 \le r^2 \le 1\).

We can test two possibilities:

Possibility 1:

• Assume \(3x - 4y + 8 = 0\) is the regression line of y on x.
• Assume \(4x - 3y - 1 = 0\) is the regression line of x on y.

Let's find the slopes (regression coefficients) under this assumption:

• From \(3x - 4y + 8 = 0\), expressing y in terms of x:

  \(4y = 3x + 8\)

  \(y = \frac{3}{4}x + \frac{8}{4}\)

  \(y = \frac{3}{4}x + 2\)

  The slope, \(b_{yx}\), is the coefficient of x, so \(b_{yx} = \frac{3}{4}\).
• From \(4x - 3y - 1 = 0\), expressing x in terms of y:

  \(4x = 3y + 1\)

  \(x = \frac{3}{4}y + \frac{1}{4}\)

  The slope, \(b_{xy}\), is the coefficient of y, so \(b_{xy} = \frac{3}{4}\).

Now, let's calculate the product of the coefficients:

\(b_{yx} \times b_{xy} = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}\)

Since \( \frac{9}{16} = 0.5625 \le 1 \), this assumption is valid. The coefficients also have the same sign (both positive). Therefore, our initial assumption is correct:

• \(3x - 4y + 8 = 0\) is the regression line of y on x.
• \(4x - 3y - 1 = 0\) is the regression line of x on y.

Possibility 2 (If Possibility 1 failed): We would swap the assumptions and check if the product of coefficients is \(\le 1\). If Possibility 1 was incorrect (product > 1), then Possibility 2 would be the correct assignment. However, since Possibility 1 yielded a valid result, Possibility 2 is not necessary to check for identification, but let's verify the coefficients if we assumed the opposite:

• Assume \(3x - 4y + 8 = 0\) is the regression line of x on y.

  \(3x = 4y - 8\)

  \(x = \frac{4}{3}y - \frac{8}{3}\)

  \(b_{xy} = \frac{4}{3}\).
• Assume \(4x - 3y - 1 = 0\) is the regression line of y on x.

  \(4y = 4x - 1\)

  \(y = x - \frac{1}{4}\)

  \(b_{yx} = 1\).

Product: \(b_{yx} \times b_{xy} = 1 \times \frac{4}{3} = \frac{4}{3}\). Since \( \frac{4}{3} > 1 \), this possibility is invalid. This confirms that Possibility 1 was the correct assignment.

Conclusion from Identification:

• Regression line of y on x is \(y = \frac{3}{4}x + 2\) (from \(3x - 4y + 8 = 0\)).
• Regression line of x on y is \(x = \frac{3}{4}y + \frac{1}{4}\) (from \(4x - 3y - 1 = 0\)).

Now let's examine the given statements:

Statement 1: The regression line of y on x is \( \rm y = \frac{3}{4}x+2 \)

This matches the equation we derived for the regression line of y on x. So, Statement 1 is correct.

Statement 2: The regression line of x on y is \( \rm x = \frac{3}{4}y+\frac{1}{4} \)

This matches the equation we derived for the regression line of x on y. So, Statement 2 is correct.

Both statements are correct based on our identification of the regression lines.

Revision Table: Key Concepts

Additional Information: Properties of Regression Lines

Understanding regression lines is crucial in statistics for modeling the relationship between variables. Here are some additional points:

• Both regression lines pass through the point representing the mean of x and the mean of y \((\bar{x}, \bar{y})\).
• The angle between the two regression lines depends on the correlation coefficient (r).
  • If \(r = 0\) (no correlation), the lines are perpendicular. \(b_{yx} \times b_{xy} = 0\).
  • If \(|r| = 1\) (perfect correlation), the lines coincide. \(b_{yx} \times b_{xy} = 1\).
• The regression line of y on x minimizes the sum of the squared vertical distances (errors in y) from the data points to the line.
• The regression line of x on y minimizes the sum of the squared horizontal distances (errors in x) from the data points to the line.
• The sign of the regression coefficients \(b_{yx}\) and \(b_{xy}\) will always be the same, and it will also be the same as the sign of the correlation coefficient (r).