If the two coefficients of regression are - 0.6 and – 1.4, find the acute angle between the regression lines.

Answer

To find the acute angle between the regression lines given the coefficients of regression \( b_{xy} = -0.6 \) and \( b_{yx} = -1.4 \), we can follow these steps:

### Step 1: Identify the coefficients of regression
Let:
- \( b_{xy} = -0.6 \)
- \( b_{yx} = -1.4 \)

### Step 2: Use the formula for the tangent of the angle between the regression lines
The formula to find the tangent of the angle \( \theta \) between the two regression lines is given by:

\[ \tan \theta = \frac{b_{xy} \cdot b_{yx} - 1}{b_{xy} + b_{yx}} \]

### Step 3: Substitute the values into the formula
Substituting the values of \( b_{xy} \) and \( b_{yx} \):

\[ \tan \theta = \frac{(-0.6) \cdot (-1.4) - 1}{(-0.6) + (-1.4)} \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:

\[ (-0.6) \cdot (-1.4) = 0.84 \]
So, the numerator becomes:

\[ 0.84 - 1 = -0.16 \]

Now, calculating the denominator:

\[ -0.6 - 1.4 = -2.0 \]

### Step 5: Substitute back to find \( \tan \theta \)
Now substituting back into the formula gives:

\[ \tan \theta = \frac{-0.16}{-2.0} = \frac{0.16}{2.0} = \frac{16}{200} = \frac{4}{25} \]

### Step 6: Find the angle \( \theta \)
To find \( \theta \), we take the inverse tangent:

\[ \theta = \tan^{-1}\left(\frac{4}{25}\right) \]

### Final Result
Thus, the acute angle \( \theta \) between the regression lines is:

\[ \theta = \tan^{-1}\left(\frac{4}{25}\right) \]