The coefficient of correlation between the values denoted by x and y is 0.5. The standard deviation of x is 5 and that of y is 4. Find the angle between the lnes of regression.

Answer

To find the angle between the lines of regression given the coefficient of correlation, standard deviations of x and y, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Coefficient of correlation (r) = 0.5
- Standard deviation of x (σ_x) = 5
- Standard deviation of y (σ_y) = 4

2. **Calculate the Regression Coefficient of x on y (b_xy):**
The formula for the regression coefficient of x on y is given by:
\[ b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y} \]
Substituting the values:
\[ b_{xy} = 0.5 \cdot \frac{5}{4} = 0.5 \cdot 1.25 = 0.625 \]

3. **Calculate the Regression Coefficient of y on x (b_yx):**
The formula for the regression coefficient of y on x is given by:
\[ b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} \]
Substituting the values:
\[ b_{yx} = 0.5 \cdot \frac{4}{5} = 0.5 \cdot 0.8 = 0.4 \]

4. **Find the Tangent of the Angle (θ) Between the Lines of Regression:**
The tangent of the angle between the two lines of regression is given by:
\[ \tan(\theta) = \frac{b_{xy} - b_{yx}}{1 + b_{xy} \cdot b_{yx}} \]
Substituting the values:
\[ \tan(\theta) = \frac{0.625 - 0.4}{1 + (0.625 \cdot 0.4)} \]
Calculate the numerator:
\[ 0.625 - 0.4 = 0.225 \]
Calculate the denominator:
\[ 1 + (0.625 \cdot 0.4) = 1 + 0.25 = 1.25 \]
Therefore:
\[ \tan(\theta) = \frac{0.225}{1.25} = 0.18 \]

5. **Calculate the Angle (θ):**
To find θ, we take the arctangent:
\[ \theta = \tan^{-1}(0.18) \]
Using a calculator, we find:
\[ \theta \approx 10.3^\circ \]

### Final Answer:
The angle between the lines of regression is approximately \( 10.3^\circ \).