1. **Problem Statement:** We want to fit a multiple linear regression model to forecast Quantity Demand ($Q$) based on Price ($P$) and Income ($I$). 2. **Model Form:** The multiple linear regression equation is: $$Q = \beta_0 + \beta_1 P + \beta_2 I + \epsilon$$ where $\beta_0$ is the intercept, $\beta_1$ and $\beta_2$ are coefficients for Price and Income respectively, and $\epsilon$ is the error term. 3. **Data:** | Quantity Demand (Q) | Price (P) | Income (I) | |---------------------|-----------|------------| | 10 | 2 | 8 | | 8 | 4 | 6 | | 10 | 3 | 7 | | 9 | 4 | 6 | | 7 | 5 | 5 | | 6 | 5 | 4 | | 4 | 6 | 2 | | 8 | 3 | 6 | | 10 | 3 | 9 | | 12 | 2 | 10 | 4. **Calculate means:** $$\bar{Q} = \frac{10+8+10+9+7+6+4+8+10+12}{10} = 8.4$$ $$\bar{P} = \frac{2+4+3+4+5+5+6+3+3+2}{10} = 3.7$$ $$\bar{I} = \frac{8+6+7+6+5+4+2+6+9+10}{10} = 6.3$$ 5. **Calculate sums of squares and cross products:** $$SS_{PP} = \sum (P_i - \bar{P})^2 = 17.1$$ $$SS_{II} = \sum (I_i - \bar{I})^2 = 38.1$$ $$SS_{PI} = \sum (P_i - \bar{P})(I_i - \bar{I}) = -15.1$$ $$SS_{PQ} = \sum (P_i - \bar{P})(Q_i - \bar{Q}) = -19.3$$ $$SS_{IQ} = \sum (I_i - \bar{I})(Q_i - \bar{Q}) = 33.7$$ 6. **Calculate coefficients $\beta_1$ and $\beta_2$ using matrix form:** $$\begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix} = \begin{bmatrix} SS_{PP} & SS_{PI} \\ SS_{PI} & SS_{II} \end{bmatrix}^{-1} \begin{bmatrix} SS_{PQ} \\ SS_{IQ} \end{bmatrix}$$ Calculate determinant: $$D = SS_{PP} \times SS_{II} - (SS_{PI})^2 = 17.1 \times 38.1 - (-15.1)^2 = 651.51 - 228.01 = 423.5$$ Inverse matrix: $$\frac{1}{D} \begin{bmatrix} SS_{II} & -SS_{PI} \\ -SS_{PI} & SS_{PP} \end{bmatrix} = \frac{1}{423.5} \begin{bmatrix} 38.1 & 15.1 \\ 15.1 & 17.1 \end{bmatrix}$$ Multiply inverse by vector: $$\begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix} = \frac{1}{423.5} \begin{bmatrix} 38.1 & 15.1 \\ 15.1 & 17.1 \end{bmatrix} \begin{bmatrix} -19.3 \\ 33.7 \end{bmatrix} = \frac{1}{423.5} \begin{bmatrix} 38.1 \times (-19.3) + 15.1 \times 33.7 \\ 15.1 \times (-19.3) + 17.1 \times 33.7 \end{bmatrix}$$ Calculate numerator: $$\beta_1: 38.1 \times (-19.3) + 15.1 \times 33.7 = -735.33 + 509.87 = -225.46$$ $$\beta_2: 15.1 \times (-19.3) + 17.1 \times 33.7 = -291.43 + 576.27 = 284.84$$ Divide by determinant: $$\beta_1 = \frac{-225.46}{423.5} = -0.532$$ $$\beta_2 = \frac{284.84}{423.5} = 0.673$$ 7. **Calculate intercept $\beta_0$:** $$\beta_0 = \bar{Q} - \beta_1 \bar{P} - \beta_2 \bar{I} = 8.4 - (-0.532)(3.7) - 0.673(6.3) = 8.4 + 1.968 - 4.239 = 6.129$$ 8. **Final regression equation:** $$Q = 6.129 - 0.532 P + 0.673 I$$ 9. **Forecast demand when $P=8$ and $I=15$:** $$Q = 6.129 - 0.532 \times 8 + 0.673 \times 15 = 6.129 - 4.256 + 10.095 = 9.968$$ **Answer:** The forecasted Quantity Demand is approximately **9.97 thousand units** when Price is 8 and Income is 15.