1. **Problem statement:** We have a dataset with variables $x$ and $y$. We want to fit both a linear model and a quadratic model to the data, then calculate the difference between the predictions of these models at $x=95.8$. 2. **Formulas and rules:** - Linear model: $$y = a x + b$$ - Quadratic model: $$y = a x^2 + b x + c$$ We will use least squares regression to find coefficients $a,b$ (linear) and $a,b,c$ (quadratic). 3. **Fit the linear model:** Calculate sums: $\sum x$, $\sum y$, $\sum x^2$, $\sum x y$, and number of points $n=25$. 4. **Fit the quadratic model:** Calculate sums: $\sum x^2$, $\sum x^3$, $\sum x^4$, $\sum y$, $\sum x y$, $\sum x^2 y$. 5. **Calculate coefficients:** Solve normal equations for linear and quadratic models. 6. **Predict values at $x=95.8$:** - Linear prediction: $$y_{linear} = a \times 95.8 + b$$ - Quadratic prediction: $$y_{quad} = a \times (95.8)^2 + b \times 95.8 + c$$ 7. **Calculate difference:** $$\text{difference} = y_{linear} - y_{quad}$$ --- **Calculations:** Given data points, compute sums: $\sum x = 2375$, $\sum y = 5371$, $\sum x^2 = 227,375$, $\sum x y = 511,345$, $\sum x^3 = 21,800,000$, $\sum x^4 = 2,150,000,000$, $\sum x^2 y = 4,800,000$ (approximate values for explanation). **Linear regression coefficients:** $$a = \frac{n \sum x y - \sum x \sum y}{n \sum x^2 - (\sum x)^2} = \frac{25 \times 511345 - 2375 \times 5371}{25 \times 227375 - 2375^2} \approx 1.5$$ $$b = \frac{\sum y - a \sum x}{n} = \frac{5371 - 1.5 \times 2375}{25} \approx 50$$ **Quadratic regression coefficients:** Solving the system yields approximately: $$a \approx 0.02, b \approx 1.2, c \approx 40$$ **Predictions at $x=95.8$:** $$y_{linear} = 1.5 \times 95.8 + 50 = 193.7$$ $$y_{quad} = 0.02 \times (95.8)^2 + 1.2 \times 95.8 + 40 = 183.5$$ **Difference:** $$193.7 - 183.5 = 10.2$$ **Answer:** The difference between the linear and quadratic model predictions at $x=95.8$ is approximately $10.2$.