Find the least-squares line given the matrix products \( A^{T} A=\left[\begin{array}{cc}4 & 58 \\ 58 & 1050\end{array}\right] \) and \( A^{T} \mathbf{b}=\left[\begin{array}{c}608.8 \\ 9669.2\end{array}\right] \) \( y= \) Ex:1.2 + Ex: 1.2 \( x \)

To find the least-squares line, we need to solve the normal equations given by \( A^T A \mathbf{w} = A^T \mathbf{b} \). Here, the matrix \( A^T A \) is a \( 2 \times 2 \) matrix, and \( A^T \mathbf{b} \) is a \( 2 \times 1 \) matrix. We have: \[ A^T A = \left[\begin{array}{cc} 4 & 58 \\ 58 & 1050 \end{array}\right] \] \[ A^T \mathbf{b} = \left[\begin{array}{c} 608.8 \\ 9669.2 \end{array}\right] \] Now, let's denote \( \mathbf{w} = \left[\begin{array}{c} w_0 \\ w_1 \end{array}\right] \), where \( w_0 \) is the y-intercept and \( w_1 \) is the slope. To find \( \mathbf{w} \), we can use the inverse of \( A^T A \): 1. Compute the determinant of \( A^T A \): \[ \text{det}(A^T A) = (4)(1050) - (58)(58) = 4200 - 3364 = 836 \] 2. Find the inverse of \( A^T A \): \[ (A^T A)^{-1} = \frac{1}{836} \left[\begin{array}{cc} 1050 & -58 \\ -58 & 4 \end{array}\right] \] 3. Now multiply by \( A^T \mathbf{b} \): \[ \mathbf{w} = (A^T A)^{-1} A^T \mathbf{b} = \frac{1}{836} \left[\begin{array}{cc} 1050 & -58 \\ -58 & 4 \end{array}\right] \left[\begin{array}{c} 608.8 \\ 9669.2 \end{array}\right] \] Calculating: \[ \left[\begin{array}{cc} 1050 & -58 \\ -58 & 4 \end{array}\right] \left[\begin{array}{c} 608.8 \\ 9669.2 \end{array}\right] = \left[\begin{array}{c} 1050 \times 608.8 - 58 \times 9669.2 \\ -58 \times 608.8 + 4 \times 9669.2 \end{array}\right] \] Calculating the first component: \[ 1050 \times 608.8 = 638040 \quad \text{and} \quad 58 \times 9669.2 = 560650.6 \] Thus, \[ 1050 \times 608.8 - 58 \times 9669.2 = 638040 - 560650.6 = 77389.4 \] Now calculating the second component: \[ -58 \times 608.8 + 4 \times 9669.2 = -35310.4 + 38676.8 = 3366.4 \] Putting it all together gives us: \[ \mathbf{w} = \frac{1}{836} \left[\begin{array}{c} 77389.4 \\ 3366.4 \end{array}\right] = \left[\begin{array}{c} 92.6 \\ 4.0 \end{array}\right] \] Thus, the least-squares line is: \[ y = 92.6 + 4.0x \]