The coefficient of correlation between the values denoted by X and Y is `0.5`. The mean of X is 3 and that of Y is 5. Their standard deviations are 5 and 4 respectively. Find
(i) the two lines of regression.
(ii) the expected value of Y, when X is given 14
(iii) the expected value of when Y is given 9

Answer

To solve the given problem step by step, let's break it down into parts as per the requirements.

### Given Data:
- Coefficient of correlation (r) = 0.5
- Mean of X (x̄) = 3
- Mean of Y (ȳ) = 5
- Standard deviation of X (σx) = 5
- Standard deviation of Y (σy) = 4

### (i) Finding the Two Lines of Regression

1. **Calculate the regression coefficients (bxy and byx)**:
- The formula for bxy (slope of regression line of Y on X) is:
\[ b_{xy} = r \cdot \frac{\sigma_y}{\sigma_x} \]
Substituting the values:
\[ b_{xy} = 0.5 \cdot \frac{4}{5} = 0.4 \]

- The formula for byx (slope of regression line of X on Y) is:
\[ b_{yx} = r \cdot \frac{\sigma_x}{\sigma_y} \]
Substituting the values:
\[ b_{yx} = 0.5 \cdot \frac{5}{4} = 0.625 \]

2. **Equation of the regression line Y on X**:
- The equation is given by:
\[ y - \bar{y} = b_{xy}(x - \bar{x}) \]
- Substituting the known values:
\[ y - 5 = 0.4(x - 3) \]
- Simplifying:
\[ y - 5 = 0.4x - 1.2 \]
\[ y = 0.4x + 3.8 \]

3. **Equation of the regression line X on Y**:
- The equation is given by:
\[ x - \bar{x} = b_{yx}(y - \bar{y}) \]
- Substituting the known values:
\[ x - 3 = 0.625(y - 5) \]
- Simplifying:
\[ x - 3 = 0.625y - 3.125 \]
\[ x = 0.625y - 0.125 \]

### Summary of the Two Lines of Regression:
- **Regression line of Y on X**: \( y = 0.4x + 3.8 \)
- **Regression line of X on Y**: \( x = 0.625y - 0.125 \)

### (ii) Expected Value of Y when X = 14

1. **Use the regression line of Y on X**:
\[ y = 0.4(14) + 3.8 \]
- Calculate:
\[ y = 5.6 + 3.8 = 9.4 \]

### (iii) Expected Value of X when Y = 9

1. **Use the regression line of X on Y**:
\[ x = 0.625(9) - 0.125 \]
- Calculate:
\[ x = 5.625 - 0.125 = 5.5 \]

### Final Answers:
(i) The two lines of regression are:
- \( y = 0.4x + 3.8 \)
- \( x = 0.625y - 0.125 \)

(ii) The expected value of Y when X = 14 is \( 9.4 \).

(iii) The expected value of X when Y = 9 is \( 5.5 \).