1. **State the problem:** Hannah needs to drive 312 miles home and wants to know how much it will cost to fill up her gas tank based on the regression equation derived from her gas fill-up log. 2. **Given data:** Miles driven ($x$): 290, 283, 288, 278, 282, 330, 344 Cost to fill up ($y$): 23.63, 23.06, 23.29, 21.93, 22.93, 26.04, 27.22 3. **Find the regression equation:** Using Hawkes formula (linear regression), the equation is generally: $$y = mx + b$$ where $m$ is the slope and $b$ is the intercept. 4. **Calculate slope $m$ and intercept $b$:** Calculate means: $$\bar{x} = \frac{290 + 283 + 288 + 278 + 282 + 330 + 344}{7} = \frac{2095}{7} = 299.29$$ $$\bar{y} = \frac{23.63 + 23.06 + 23.29 + 21.93 + 22.93 + 26.04 + 27.22}{7} = \frac{167.1}{7} = 23.87$$ Calculate slope $m$: $$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ Calculate numerator: $$(290 - 299.29)(23.63 - 23.87) + (283 - 299.29)(23.06 - 23.87) + \ldots + (344 - 299.29)(27.22 - 23.87) = 153.43$$ Calculate denominator: $$(290 - 299.29)^2 + (283 - 299.29)^2 + \ldots + (344 - 299.29)^2 = 1081.43$$ So, $$m = \frac{153.43}{1081.43} = 0.142$$ Calculate intercept $b$: $$b = \bar{y} - m \bar{x} = 23.87 - 0.142 \times 299.29 = 23.87 - 42.52 = -18.65$$ 5. **Regression equation:** $$y = 0.142x - 18.65$$ 6. **Predict cost for 312 miles:** $$y = 0.142 \times 312 - 18.65 = 44.30 - 18.65 = 25.65$$ 7. **Answer:** Hannah should budget approximately **25.65** to fill up when she gets home. All calculations are rounded to two decimal places as appropriate.