If the regrassion coefficient of Y on X is 1.6 that of X on Y is 0.4 , and ` theta ` be the angle between two regression lines , find the value of ` tan theta `

Answer

To solve the problem, we need to find the value of \( \tan \theta \) where \( \theta \) is the angle between the two regression lines. We are given the regression coefficients of \( Y \) on \( X \) and \( X \) on \( Y \).

### Step-by-Step Solution:

1. **Identify the given values**:
- The regression coefficient of \( Y \) on \( X \) is \( b_{YX} = 1.6 \).
- The regression coefficient of \( X \) on \( Y \) is \( b_{XY} = 0.4 \).

2. **Use the relationship between the regression coefficients and the angle**:
- The tangent of the angle \( \theta \) between the two regression lines is given by the formula:
\[ \tan \theta = \frac{b_{YX} + b_{XY}}{1 - b_{YX} \cdot b_{XY}} \]

3. **Substitute the values into the formula**:
- Substitute \( b_{YX} = 1.6 \) and \( b_{XY} = 0.4 \) into the formula:
\[ \tan \theta = \frac{1.6 + 0.4}{1 - (1.6 \cdot 0.4)} \]

4. **Calculate the numerator**:
- The numerator is:
\[ 1.6 + 0.4 = 2.0 \]

5. **Calculate the denominator**:
- First, calculate \( 1.6 \cdot 0.4 \):
\[ 1.6 \cdot 0.4 = 0.64 \]
- Now, calculate the denominator:
\[ 1 - 0.64 = 0.36 \]

6. **Combine the results**:
- Now, substitute back into the equation for \( \tan \theta \):
\[ \tan \theta = \frac{2.0}{0.36} \]

7. **Perform the division**:
- Calculate \( \frac{2.0}{0.36} \):
\[ \tan \theta \approx 5.5556 \]

8. **Final Result**:
- Therefore, the value of \( \tan \theta \) is approximately \( 5.56 \).