1. **Problem statement:** We have petrol consumption data (miles per gallon, m.p.g.) for cars with the same engine size at different speeds (m.p.h.). We want to: a. Plot a scatter graph and draw a line of best fit. b. Estimate petrol consumption at 45 m.p.h. c. Estimate speed when petrol consumption is 27 m.p.g. 2. **Data:** Speeds $x = \{30, 62, 40, 80, 70, 55, 75\}$ Consumption $y = \{38, 25, 35, 20, 26, 34, 22\}$ 3. **Scatter plot and line of best fit:** We plot the points $(x, y)$ on a graph with $x$ as speed and $y$ as petrol consumption. To find the line of best fit $y = mx + c$, calculate: - Mean of $x$: $\bar{x} = \frac{30+62+40+80+70+55+75}{7} = \frac{412}{7} \approx 58.86$ - Mean of $y$: $\bar{y} = \frac{38+25+35+20+26+34+22}{7} = \frac{200}{7} \approx 28.57$ - Compute slope $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$ Calculate: $\sum (x_i - \bar{x})(y_i - \bar{y}) = (30-58.86)(38-28.57)+(62-58.86)(25-28.57)+(40-58.86)(35-28.57)+(80-58.86)(20-28.57)+(70-58.86)(26-28.57)+(55-58.86)(34-28.57)+(75-58.86)(22-28.57)$ $= (-28.86)(9.43)+(3.14)(-3.57)+(-18.86)(6.43)+(21.14)(-8.57)+(11.14)(-2.57)+(-3.86)(5.43)+(16.14)(-6.57)$ $= -272.16 -11.21 -121.27 -181.19 -28.62 -20.95 -106.06 = -741.46$ $\sum (x_i - \bar{x})^2 = (-28.86)^2 + 3.14^2 + (-18.86)^2 + 21.14^2 + 11.14^2 + (-3.86)^2 + 16.14^2$ $= 833.31 + 9.86 + 355.67 + 447.91 + 124.12 + 14.90 + 260.50 = 2046.27$ So, $m = \frac{-741.46}{2046.27} \approx -0.3623$ 4. Find intercept $c$ using $\bar{y} = m \bar{x} + c$: $c = \bar{y} - m\bar{x} = 28.57 - (-0.3623)(58.86) = 28.57 + 21.33 = 49.90$ 5. **Equation of line of best fit:** $$ y = -0.3623x + 49.90 $$ 6. **Estimate petrol consumption at 45 m.p.h.:** $$ y = -0.3623 \times 45 + 49.90 = -16.30 + 49.90 = 33.60 $$ So, estimated consumption is about 33.6 m.p.g. 7. **Estimate speed for consumption 27 m.p.g.:** Set $y = 27$, solve for $x$: $$ 27 = -0.3623x + 49.90 \Rightarrow -0.3623x = 27 - 49.90 = -22.90 $$ $$ x = \frac{-22.90}{-0.3623} \approx 63.25 $$ So, estimated speed is about 63.3 m.p.h.