Understanding the Angle of the Regression Line of Y on X

The question asks for the angle made by the line of regression of Y on X with the positive x-axis. This angle, denoted by $\theta$, is related to the slope of the regression line. The line of regression of Y on X is used to predict the value of Y based on a given value of X.

The general form of the regression line of Y on X is given by:

$$Y - \bar{Y} = b_{YX}(X - \bar{X})$$

where:

• $Y$ is the dependent variable
• $X$ is the independent variable
• $\bar{Y}$ is the mean of Y
• $\bar{X}$ is the mean of X
• $b_{YX}$ is the regression coefficient of Y on X, which represents the slope of the regression line of Y on X.

The angle $\theta$ that this line makes with the positive direction of the X-axis is such that its tangent is equal to the slope of the line. Therefore:

$$\tan \theta = b_{YX}$$

Calculating the Regression Coefficient \(b_{YX}\)

The regression coefficient of Y on X, $b_{YX}$, can be calculated using the correlation coefficient between X and Y ($r$), the standard deviation of Y ($\sigma_Y$), and the standard deviation of X ($\sigma_X$). The formula is:

$$b_{YX} = r \frac{\sigma_Y}{\sigma_X}$$

Using the Given Information

We are given the following information:

• The relationship between the standard deviations: $\sigma_Y = 2\sigma_X$
• The correlation coefficient between X and Y: $r = 0.3$

Now, let's substitute these values into the formula for $b_{YX}$:

$$b_{YX} = r \frac{\sigma_Y}{\sigma_X} = (0.3) \frac{2\sigma_X}{\sigma_X}$$

We can cancel out $\sigma_X$ from the numerator and the denominator (assuming $\sigma_X \neq 0$, which is typically the case for meaningful regression):

$$b_{YX} = 0.3 \times 2$$

$$b_{YX} = 0.6$$

Finding the Angle \(\theta\)

We know that $\tan \theta = b_{YX}$. Substituting the calculated value of $b_{YX}$:

$$\tan \theta = 0.6$$

To find the angle $\theta$, we take the inverse tangent (arctan) of 0.6:

$$\theta = \tan^{-1}(0.6)$$

This is the angle made by the regression line of Y on X with the positive x-axis.

Comparing with Options

Let's compare our result with the given options:

1. $\tan^{-1}\ 0.3$
2. $\cot^{-1}\ 0.3$
3. $\tan^{-1}\ 0.6$
4. $\cot^{-1}\ 0.6$

Our calculated value, $\theta = \tan^{-1}(0.6)$, matches option 3.

Conclusion

Based on the given standard deviations and correlation coefficient, the regression coefficient of Y on X is 0.6. The angle made by the regression line of Y on X with the x-axis is the angle whose tangent is this coefficient. Therefore, the value of $\theta$ is $\tan^{-1}(0.6)$.

Revision Table: Regression Line Concepts

Additional Information on Regression and Correlation

Correlation Coefficient (r): This value measures the strength and direction of a linear relationship between two variables, X and Y. It ranges from -1 to +1. A value of 0.3 indicates a weak positive linear relationship.

Regression Coefficients ($b_{YX}$ and $b_{XY}$): These are measures of the average functional relationship between variables. $b_{YX}$ tells us how much Y changes for a unit change in X, and $b_{XY}$ tells us how much X changes for a unit change in Y. They are generally not equal unless $\sigma_X = \sigma_Y$.

Relationship between Regression Coefficients and Correlation: The correlation coefficient $r$ is the geometric mean of the two regression coefficients, provided they have the same sign:

$$r^2 = b_{YX} \times b_{XY}$$

The sign of $r$, $b_{YX}$, and $b_{XY}$ is always the same, indicating the direction of the linear relationship (positive or negative).

Angle between Regression Lines: The two regression lines (Y on X and X on Y) intersect at the point $(\bar{X}, \bar{Y})$. The angle between these two lines is given by:

$$\tan \phi = \frac{1 - r^2}{r} \frac{\sigma_X \sigma_Y}{\sigma_X^2 + \sigma_Y^2} \quad \text{or} \quad \tan \phi = \frac{b_{XY} - b_{YX}}{1 + b_{YX} b_{XY}}$$

If the correlation is perfect ($|r|=1$), the two regression lines coincide, and the angle between them is 0. If the variables are uncorrelated ($r=0$), the regression lines are perpendicular, and the angle between them is 90 degrees ($\tan \phi$ would be undefined). In this question, $r=0.3$, which is between 0 and 1, so the lines will not be perpendicular or coincident.