1. **State the problem:** Given a set of points $(x,y)$ and four candidate equations for the least-squares regression line $\hat{y} = mx + b$, determine which equation best fits the data. 2. **Recall the formula for the least-squares regression line:** $$\hat{y} = mx + b$$ where $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ and $$b = \frac{\sum y - m \sum x}{n}$$ 3. **Calculate the necessary sums from the data:** - $n = 10$ - $\sum x = (-5) + (-3) + 4 + 1 + (-1) + (-2) + 0 + 2 + 3 + (-4) = -5$ - $\sum y = (-10) + (-8) + 9 + 1 + (-2) + (-6) + (-1) + 3 + 6 + (-8) = -16$ - $\sum x^2 = (-5)^2 + (-3)^2 + 4^2 + 1^2 + (-1)^2 + (-2)^2 + 0^2 + 2^2 + 3^2 + (-4)^2 = 25 + 9 + 16 + 1 + 1 + 4 + 0 + 4 + 9 + 16 = 85$ - $\sum xy = (-5)(-10) + (-3)(-8) + 4(9) + 1(1) + (-1)(-2) + (-2)(-6) + 0(-1) + 2(3) + 3(6) + (-4)(-8) = 50 + 24 + 36 + 1 + 2 + 12 + 0 + 6 + 18 + 32 = 181$ 4. **Calculate the slope $m$:** $$m = \frac{10 \times 181 - (-5) \times (-16)}{10 \times 85 - (-5)^2} = \frac{1810 - 80}{850 - 25} = \frac{1730}{825}$$ 5. **Simplify the fraction:** $$m = \frac{\cancel{1730}}{\cancel{825}} = 2.097$$ 6. **Calculate the intercept $b$:** $$b = \frac{-16 - 2.097 \times (-5)}{10} = \frac{-16 + 10.485}{10} = \frac{-5.515}{10} = -0.552$$ 7. **Write the regression line equation:** $$\hat{y} = 2.097x - 0.552$$ **Final answer:** The best fit line is $\hat{y} = 2.097x - 0.552$.