Given the following information about the production and demand of a commodity.
Obtain the two regression lines :
`{:(,"PRODUCTION(X)", "DEMAND(Y)"),("MEAN",85,90),("VARIANCE", 25, 36):}`
Coefficient of correlation between X and Y is 0.6. Also estimate the demand when the production is 100 units.

Answer

To solve the problem, we need to find the two regression lines based on the given data and then estimate the demand when the production is 100 units.

### Step-by-Step Solution

1. **Identify the Given Data:**
- Mean of Production (X̄) = 85
- Mean of Demand (Ȳ) = 90
- Variance of Production (σ²_X) = 25 → Standard Deviation (σ_X) = √25 = 5
- Variance of Demand (σ²_Y) = 36 → Standard Deviation (σ_Y) = √36 = 6
- Coefficient of Correlation (r) = 0.6

**Hint:** Write down all the given values clearly to avoid confusion later.

2. **Calculate the Regression Coefficient of Y on X (B_YX):**
\[ B_{YX} = r \cdot \frac{\sigma_Y}{\sigma_X} = 0.6 \cdot \frac{6}{5} = \frac{36}{50} = 0.72 \]

**Hint:** Use the formula for regression coefficients carefully, ensuring you substitute the correct values.

3. **Calculate the Regression Coefficient of X on Y (B_XY):**
\[ B_{XY} = r \cdot \frac{\sigma_X}{\sigma_Y} = 0.6 \cdot \frac{5}{6} = \frac{30}{36} = \frac{5}{6} \]

**Hint:** Remember that the regression coefficients are derived from the correlation coefficient and the standard deviations.

4. **Formulate the Regression Equation of Y on X:**
\[ Y - Ȳ = B_{YX} \cdot (X - X̄) \]
Substituting the known values:
\[ Y - 90 = 0.72 \cdot (X - 85) \]
Rearranging gives:
\[ Y = 0.72X - 0.72 \cdot 85 + 90 \]
\[ Y = 0.72X - 61.2 + 90 \]
\[ Y = 0.72X + 28.8 \]

**Hint:** Ensure to simplify the equation correctly after substituting the values.

5. **Estimate Demand (Y) when Production (X) is 100:**
Substitute X = 100 into the regression equation:
\[ Y = 0.72(100) + 28.8 = 72 + 28.8 = 100.8 \]

**Hint:** Always double-check your arithmetic when substituting values into equations.

6. **Formulate the Regression Equation of X on Y:**
\[ X - X̄ = B_{XY} \cdot (Y - Ȳ) \]
Substituting the known values:
\[ X - 85 = \frac{5}{6} \cdot (Y - 90) \]
Rearranging gives:
\[ X = \frac{5}{6}(Y - 90) + 85 \]
\[ X = \frac{5}{6}Y - 75 + 85 \]
\[ X = \frac{5}{6}Y + 10 \]

**Hint:** Follow the same steps as before to ensure the equation is correctly derived.

### Final Results
- The regression equation of Y on X is:
\[ Y = 0.72X + 28.8 \]
- The regression equation of X on Y is:
\[ X = \frac{5}{6}Y + 10 \]
- Estimated Demand when Production is 100 units:
\[ Y = 100.8 \]