Question

The correlation coefficient between two variables X and Y is 0.60 and it is given that \(\sigma_X = 2, \sigma_Y = 4\). Then, the angle between two lines of regression, is

• A. \( \tan^{-1}(0.2462) \)
• B. \( \tan^{-1}(0.4267) \)
• C. \( \tan^{-1}(0.6052) \)
• D. \( \tan^{-1}(0.90) \)

Hint:

Note the extreme cases for the angle formula. If \(r=0\), \(\tan \theta = \infty\), so \(\theta = 90^\circ\). The lines are perpendicular. If \(r = \pm 1\), \(\tan \theta = 0\), so \(\theta = 0^\circ\). The lines are coincident. This can help you check if your answer is reasonable.

Answer 1

The Correct Option is B

Step 1: Concept Overview:
The regression lines (Y on X and X on Y) intersect at \((\bar{X}, \bar{Y})\). The angle (\(\theta\)) between them is determined by the correlation coefficient and the standard deviations.

Step 2: Key Formula:
The acute angle \(\theta\) between the regression lines is calculated as follows: \[ \tan \theta = \left| \frac{1-r^2}{r} \frac{\sigma_X \sigma_Y}{\sigma_X^2 + \sigma_Y^2} \right| \] where \(r\) is the correlation coefficient, and \(\sigma_X\) and \(\sigma_Y\) are the standard deviations.

Step 3: Detailed Calculation:
Given: - Correlation coefficient: \(r = 0.60\) - Standard deviation of X: \(\sigma_X = 2\) - Standard deviation of Y: \(\sigma_Y = 4\) Calculate the components: - \(r^2 = (0.60)^2 = 0.36\) - \(1 - r^2 = 1 - 0.36 = 0.64\) - \(\sigma_X \sigma_Y = 2 \times 4 = 8\) - \(\sigma_X^2 + \sigma_Y^2 = 2^2 + 4^2 = 4 + 16 = 20\) Substitute into the formula: \[ \tan \theta = \frac{0.64}{0.60} \frac{8}{20} \] \[ \tan \theta = \frac{64}{60} \times \frac{8}{20} = \frac{16}{15} \times \frac{2}{5} = \frac{32}{75} \] Convert to decimal: \[ \tan \theta = \frac{32}{75} \approx 0.42666... \] Therefore: \[ \theta = \tan^{-1}(0.4267) \]
Step 4: Final Result:
The angle between the regression lines is \( \tan^{-1}(0.4267) \).