Linear Model 53D797

1. **State the problem:** We need to find a linear model of the form $$y = mx + b$$ that fits the given data points: $$x: 4, 7, 10, 13, 16, 19$$ $$y: 6, 16, 20, 29, 38, 43$$ 2. **Formula used:** The linear model is $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. The slope $$m$$ is calculated by: $$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ where $$\bar{x}$$ and $$\bar{y}$$ are the means of $$x$$ and $$y$$ values respectively. 3. **Calculate means:** $$\bar{x} = \frac{4 + 7 + 10 + 13 + 16 + 19}{6} = \frac{69}{6} = 11.5$$ $$\bar{y} = \frac{6 + 16 + 20 + 29 + 38 + 43}{6} = \frac{152}{6} \approx 25.333$$ 4. **Calculate numerator for slope:** Calculate each $$ (x_i - \bar{x})(y_i - \bar{y}) $$: - $$(4 - 11.5)(6 - 25.333) = (-7.5)(-19.333) = 144.999$$ - $$(7 - 11.5)(16 - 25.333) = (-4.5)(-9.333) = 42.0$$ - $$(10 - 11.5)(20 - 25.333) = (-1.5)(-5.333) = 8.0$$ - $$(13 - 11.5)(29 - 25.333) = (1.5)(3.667) = 5.5$$ - $$(16 - 11.5)(38 - 25.333) = (4.5)(12.667) = 57.0$$ - $$(19 - 11.5)(43 - 25.333) = (7.5)(17.667) = 132.5$$ Sum numerator: $$144.999 + 42.0 + 8.0 + 5.5 + 57.0 + 132.5 = 389.999$$ 5. **Calculate denominator for slope:** Calculate each $$ (x_i - \bar{x})^2 $$: - $$(4 - 11.5)^2 = (-7.5)^2 = 56.25$$ - $$(7 - 11.5)^2 = (-4.5)^2 = 20.25$$ - $$(10 - 11.5)^2 = (-1.5)^2 = 2.25$$ - $$(13 - 11.5)^2 = (1.5)^2 = 2.25$$ - $$(16 - 11.5)^2 = (4.5)^2 = 20.25$$ - $$(19 - 11.5)^2 = (7.5)^2 = 56.25$$ Sum denominator: $$56.25 + 20.25 + 2.25 + 2.25 + 20.25 + 56.25 = 157.5$$ 6. **Calculate slope $$m$$:** $$m = \frac{389.999}{157.5} \approx 2.476$$ 7. **Calculate y-intercept $$b$$:** Use formula $$b = \bar{y} - m \bar{x}$$: $$b = 25.333 - 2.476 \times 11.5 = 25.333 - 28.474 = -3.141$$ 8. **Write the linear model:** $$y = 2.476x - 3.141$$ **Final answer:** $$\boxed{y = 2.476x - 3.141}$$