1. **Problem Statement:** We are given data on the weight (in kg) and fuel efficiency (in miles per gallon, MPG) of 8 cars. We want to understand the relationship between weight and fuel efficiency. 2. **Understanding the Problem:** Fuel efficiency tends to decrease as the weight of the car increases. This suggests a negative correlation. 3. **Formula and Approach:** To analyze this, we can use the concept of correlation or fit a linear regression model: $$\text{MPG} = m \times \text{Weight} + b$$ where $m$ is the slope and $b$ is the intercept. 4. **Data Points:** | Car | Weight (kg) | MPG | |-----|-------------|-----| | A | 1549 | 25 | | B | 1610 | 20 | | C | 1737 | 21 | | D | 1777 | 20 | | E | 1486 | 23 | | F | 1962 | 16 | | G | 2384 | 16 | | H | 1957 | 19 | 5. **Calculate the slope $m$ and intercept $b$ using least squares method:** Let $x_i$ be weights and $y_i$ be MPG values. Calculate means: $$\bar{x} = \frac{1549 + 1610 + 1737 + 1777 + 1486 + 1962 + 2384 + 1957}{8} = \frac{14462}{8} = 1807.75$$ $$\bar{y} = \frac{25 + 20 + 21 + 20 + 23 + 16 + 16 + 19}{8} = \frac{160}{8} = 20$$ Calculate numerator and denominator for slope: $$\sum (x_i - \bar{x})(y_i - \bar{y}) = (1549 - 1807.75)(25 - 20) + (1610 - 1807.75)(20 - 20) + \ldots + (1957 - 1807.75)(19 - 20)$$ Calculate each term: $(1549 - 1807.75)(5) = (-258.75)(5) = -1293.75$ $(1610 - 1807.75)(0) = (-197.75)(0) = 0$ $(1737 - 1807.75)(1) = (-70.75)(1) = -70.75$ $(1777 - 1807.75)(0) = (-30.75)(0) = 0$ $(1486 - 1807.75)(3) = (-321.75)(3) = -965.25$ $(1962 - 1807.75)(-4) = (154.25)(-4) = -617$ $(2384 - 1807.75)(-4) = (576.25)(-4) = -2305$ $(1957 - 1807.75)(-1) = (149.25)(-1) = -149.25$ Sum numerator: $$-1293.75 + 0 - 70.75 + 0 - 965.25 - 617 - 2305 - 149.25 = -5401$$ Calculate denominator: $$\sum (x_i - \bar{x})^2 = (-258.75)^2 + (-197.75)^2 + (-70.75)^2 + (-30.75)^2 + (-321.75)^2 + 154.25^2 + 576.25^2 + 149.25^2$$ Calculate each: $66968.06 + 39099.06 + 5005.56 + 945.56 + 103522.06 + 23787.06 + 332059.06 + 22275.56 = 588662.98$ 6. **Calculate slope $m$:** $$m = \frac{-5401}{588662.98} \approx -0.00918$$ 7. **Calculate intercept $b$:** $$b = \bar{y} - m \times \bar{x} = 20 - (-0.00918)(1807.75) = 20 + 16.59 = 36.59$$ 8. **Final linear model:** $$\boxed{\text{MPG} = -0.00918 \times \text{Weight} + 36.59}$$ This means for every additional kg, MPG decreases by about 0.00918. 9. **Interpretation:** Heavier cars tend to have lower fuel efficiency, as shown by the negative slope.