Linear Fit 433725

1. **State the problem:** We need to find the linear function $y = mx + b$ that best fits the given data points $(1,7), (3,12), (5,17), (7,22), (9,27)$. 2. **Formula for linear regression slope and intercept:** $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ where $n$ is the number of points. 3. **Calculate sums:** - $n = 5$ - $\sum x = 1 + 3 + 5 + 7 + 9 = 25$ - $\sum y = 7 + 12 + 17 + 22 + 27 = 85$ - $\sum xy = (1)(7) + (3)(12) + (5)(17) + (7)(22) + (9)(27) = 7 + 36 + 85 + 154 + 243 = 525$ - $\sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165$ 4. **Calculate slope $m$:** $$m = \frac{5 \times 525 - 25 \times 85}{5 \times 165 - 25^2} = \frac{2625 - 2125}{825 - 625} = \frac{500}{200} = 2.5$$ 5. **Calculate intercept $b$:** $$b = \frac{85 - 2.5 \times 25}{5} = \frac{85 - 62.5}{5} = \frac{22.5}{5} = 4.5$$ 6. **Final linear function:** $$y = 2.5x + 4.5$$ This function best fits the data points given.