1. **Problem Statement:** We are given data points $(x_i, y_i)$ and want to fit a quadratic model $$y = b_0 + b_1 x + b_2 x^2$$ using the least squares method. 2. **Model and Design Matrix:** The design matrix $X$ for $n=6$ data points is: $$ X = \begin{bmatrix} 1 & 1.239 & 1.239^2 \\ 1 & 1.2832 & 1.2832^2 \\ 1 & 1.6028 & 1.6028^2 \\ 1 & 1.5438 & 1.5438^2 \\ 1 & 1.8079 & 1.8079^2 \\ 1 & 1.6972 & 1.6972^2 \end{bmatrix} $$ 3. **Vector of observations:** $$ \mathbf{y} = \begin{bmatrix}1.4579 \\ 1.7721 \\ 1.7147 \\ 1.9725 \\ 2.2892 \\ 2.4576\end{bmatrix} $$ 4. **Calculate $X^T X$ and $X^T y$:** Calculate each element by summing over data points: - $\sum 1 = 6$ - $\sum x_i = 1.239 + 1.2832 + 1.6028 + 1.5438 + 1.8079 + 1.6972 = 9.1739$ - $\sum x_i^2 = 1.239^2 + 1.2832^2 + 1.6028^2 + 1.5438^2 + 1.8079^2 + 1.6972^2 = 14.3717$ - $\sum x_i^3 = \sum x_i \cdot x_i^2 = 22.3467$ - $\sum x_i^4 = \sum (x_i^2)^2 = 35.1231$ - $\sum y_i = 1.4579 + 1.7721 + 1.7147 + 1.9725 + 2.2892 + 2.4576 = 11.664$ - $\sum x_i y_i = 1.239\times1.4579 + 1.2832\times1.7721 + ... = 18.1993$ - $\sum x_i^2 y_i = 1.239^2\times1.4579 + 1.2832^2\times1.7721 + ... = 28.3497$ Thus, $$ X^T X = \begin{bmatrix}6 & 9.1739 & 14.3717 \\ 9.1739 & 14.3717 & 22.3467 \\ 14.3717 & 22.3467 & 35.1231\end{bmatrix}, \quad X^T y = \begin{bmatrix}11.664 \\ 18.1993 \\ 28.3497\end{bmatrix} $$ 5. **Solve for $\hat{\beta} = (X^T X)^{-1} X^T y$:** Using matrix inversion and multiplication (rounded to 4 decimals): $$ \hat{\beta} = \begin{bmatrix}b_0 \\ b_1 \\ b_2\end{bmatrix} = \begin{bmatrix}-1.9693 \\ 3.0347 \\ -0.7412\end{bmatrix} $$ 6. **Final model:** $$ \hat{y} = -1.9693 + 3.0347 x - 0.7412 x^2 $$ 7. **Interpolation:** Calculate $y$ for given $x$ values: - For $x=1.2536$: $$ \hat{y} = -1.9693 + 3.0347(1.2536) - 0.7412(1.2536)^2 = -1.9693 + 3.8001 - 1.1637 = 0.6671 $$ - For $x=1.9697$: $$ \hat{y} = -1.9693 + 3.0347(1.9697) - 0.7412(1.9697)^2 = -1.9693 + 5.9743 - 2.8743 = 1.1307 $$ All answers are rounded to 4 decimal places as requested.