1. **State the problem:** We are given a set of data points $(x, y)$ and need to find the linear regression equation $y = mx + b$ that best fits the data. Then, use this equation to predict $y$ when $x = 56.5$. 2. **Formula for linear regression:** The slope $m$ and intercept $b$ are given by: $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ where $n$ is the number of data points. 3. **Calculate sums:** - $n = 8$ - $\sum x = 57.2 + 68.7 + 70.4 + 67.2 + 94.7 + 66 + 72.3 + 71.7 = 568.2$ - $\sum y = 45.5 - 8.5 + 38.3 + 7.8 - 29.9 - 33.3 + 33.9 + 46.6 = 100.4$ - $\sum x^2 = 57.2^2 + 68.7^2 + 70.4^2 + 67.2^2 + 94.7^2 + 66^2 + 72.3^2 + 71.7^2 = 41288.94$ - $\sum xy = (57.2)(45.5) + (68.7)(-8.5) + (70.4)(38.3) + (67.2)(7.8) + (94.7)(-29.9) + (66)(-33.3) + (72.3)(33.9) + (71.7)(46.6) = 2697.91$ 4. **Calculate slope $m$:** $$m = \frac{8 \times 2697.91 - 568.2 \times 100.4}{8 \times 41288.94 - (568.2)^2} = \frac{21583.28 - 57073.28}{330311.52 - 322871.24} = \frac{-35490}{7440.28}$$ $$m = \frac{\cancel{-35490}}{\cancel{7440.28}} = -4.771$$ 5. **Calculate intercept $b$:** $$b = \frac{100.4 - (-4.771) \times 568.2}{8} = \frac{100.4 + 2710.3}{8} = \frac{2810.7}{8} = 351.338$$ 6. **Linear regression equation:** $$y = -4.771x + 351.338$$ 7. **Predict $y$ when $x = 56.5$:** $$y = -4.771 \times 56.5 + 351.338 = -269.72 + 351.338 = 81.618$$ **Final answers:** - Linear equation: $y = -4.771x + 351.338$ - Prediction at $x=56.5$: $y = 81.618$