1. **State the problem:** We have data for $x$ (thousands of automatic weapons) and $y$ (murders per 100,000 residents). We want to find the linear regression equation in the form $y = ax + b$ and then use it to predict murders for given $x$ values. 2. **Formula for linear regression coefficients:** $$a = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - a \sum x}{n}$$ where $n$ is the number of data points. 3. **Calculate sums:** Given data: $x = [11.5, 8.5, 7.2, 3.5, 2.9, 2.8, 2.3, 0.5]$ $y = [14, 11.3, 10.5, 6.9, 6.4, 6.1, 5.7, 4.6]$ Calculate: $$\sum x = 11.5 + 8.5 + 7.2 + 3.5 + 2.9 + 2.8 + 2.3 + 0.5 = 39.2$$ $$\sum y = 14 + 11.3 + 10.5 + 6.9 + 6.4 + 6.1 + 5.7 + 4.6 = 65.5$$ $$\sum x^2 = 11.5^2 + 8.5^2 + 7.2^2 + 3.5^2 + 2.9^2 + 2.8^2 + 2.3^2 + 0.5^2 = 132.25 + 72.25 + 51.84 + 12.25 + 8.41 + 7.84 + 5.29 + 0.25 = 290.38$$ $$\sum xy = (11.5)(14) + (8.5)(11.3) + (7.2)(10.5) + (3.5)(6.9) + (2.9)(6.4) + (2.8)(6.1) + (2.3)(5.7) + (0.5)(4.6) = 161 + 96.05 + 75.6 + 24.15 + 18.56 + 17.08 + 13.11 + 2.3 = 407.85$$ 4. **Calculate slope $a$:** $$a = \frac{8 \times 407.85 - 39.2 \times 65.5}{8 \times 290.38 - (39.2)^2} = \frac{3262.8 - 2567.6}{2323.04 - 1536.64} = \frac{695.2}{786.4} \approx 0.884$$ 5. **Calculate intercept $b$:** $$b = \frac{65.5 - 0.884 \times 39.2}{8} = \frac{65.5 - 34.65}{8} = \frac{30.85}{8} = 3.856$$ 6. **Regression equation:** $$y = 0.88x + 3.86$$ (rounded to 2 decimal places) 7. **Part A: Predict $y$ for $x=4.6$:** $$y = 0.884 \times 4.6 + 3.856 = 4.0664 + 3.856 = 7.9224$$ Rounded to 3 decimal places: $7.922$ 8. **Part B: Predict $y$ for $x=7.9$:** $$y = 0.884 \times 7.9 + 3.856 = 6.9836 + 3.856 = 10.8396$$ Rounded to 3 decimal places: $10.840$