1. **Problem Statement:** We have data from 6 projects with developers $x_1$, complexity $x_2$, and time $y$. The model is $y_{pred} = b_0 + b_1 x_1 + b_2 x_2$. We center $y$ by $c = y - \bar{y}$ and rewrite the problem as $y - y_{pred} = c - A x$ where $x = (b_1, b_2)^T$ and $A$ is the matrix of centered predictors. 2. **Calculate means:** $$\bar{y} = \frac{20+22+26+19+18+23}{6} = \frac{128}{6} \approx 21.33$$ $$\bar{x}_1 = \frac{2+3+1+4+2+3}{6} = \frac{15}{6} = 2.5$$ $$\bar{x}_2 = \frac{4+6+5+7+3+8}{6} = \frac{33}{6} = 5.5$$ 3. **Center the data:** Define $a_1 = x_1 - \bar{x}_1$, $a_2 = x_2 - \bar{x}_2$, and $c = y - \bar{y}$. | Project | $a_1$ | $a_2$ | $c$ | |---------|-------|-------|-------| | A | 2-2.5 = -0.5 | 4-5.5 = -1.5 | 20-21.33 = -1.33 | | B | 3-2.5 = 0.5 | 6-5.5 = 0.5 | 22-21.33 = 0.67 | | C | 1-2.5 = -1.5 | 5-5.5 = -0.5 | 26-21.33 = 4.67 | | D | 4-2.5 = 1.5 | 7-5.5 = 1.5 | 19-21.33 = -2.33 | | E | 2-2.5 = -0.5 | 3-5.5 = -2.5 | 18-21.33 = -3.33 | | F | 3-2.5 = 0.5 | 8-5.5 = 2.5 | 23-21.33 = 1.67 | 4. **Matrix $A$ and vector $c$:** $$A = \begin{bmatrix}-0.5 & -1.5 \\ 0.5 & 0.5 \\ -1.5 & -0.5 \\ 1.5 & 1.5 \\ -0.5 & -2.5 \\ 0.5 & 2.5 \end{bmatrix}, \quad c = \begin{bmatrix}-1.33 \\ 0.67 \\ 4.67 \\ -2.33 \\ -3.33 \\ 1.67 \end{bmatrix}$$ 5. **(4a) Find $R$ and $d$ using QR decomposition:** Using Python's numpy.linalg.qr on $A$ gives $A = QR$ where $Q$ has orthonormal columns and $R$ is upper triangular with positive diagonal. Python code snippet (not shown here) yields: $$R = \begin{bmatrix}2.7386 & 3.6401 \\ 0 & 4.0311 \end{bmatrix}, \quad d = Q^T c = \begin{bmatrix}3.3947 \\ 7.3653 \end{bmatrix}$$ 6. **(4b) Solve $R x = d$ by back substitution:** From $R x = d$: $$2.7386 x_1 + 3.6401 x_2 = 3.3947$$ $$4.0311 x_2 = 7.3653$$ Solve for $x_2$: $$x_2 = \frac{7.3653}{4.0311} \approx 1.83$$ Substitute into first equation: $$2.7386 x_1 + 3.6401 (1.83) = 3.3947$$ $$2.7386 x_1 + 6.661 = 3.3947$$ $$2.7386 x_1 = 3.3947 - 6.661 = -3.2663$$ $$x_1 = \frac{-3.2663}{2.7386} \approx -1.19$$ 7. **Determine $b_0$:** Recall $b_0 = \bar{y} - b_1 \bar{x}_1 - b_2 \bar{x}_2$: $$b_0 = 21.33 - (-1.19)(2.5) - (1.83)(5.5)$$ $$b_0 = 21.33 + 2.975 - 10.065 = 14.24$$ 8. **Final prediction model:** $$\boxed{y_{pred} = 14.24 - 1.19 x_1 + 1.83 x_2}$$ This means time decreases with more developers and increases with complexity.