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The coefficient of determination is a useful statistic often denoted as \(r^2\). It tells us what proportion of the variability in one variable is accounted for by its relationship with another variable. In this context, we're looking at the relationship between a car's horsepower and its fuel economy. Understanding \(r^2\) helps to clarify the degree of influence one variable has over another, offering a clearer picture of their interaction. When we square the correlation coefficient \(r\), we get the coefficient of determination. For example, if \(r= -0.869\), then \(r^2 = (-0.869)^2 = 0.755161\). This calculation shows that around 75.5% of the variability in the car's fuel economy can indeed be explained by its horsepower. This tells us a lot about the strength of the relationship between these two variables, giving valuable insights into how changes in horsepower could impact fuel economy.

Fuel economy is a crucial factor for many car buyers. It refers to how efficiently a vehicle converts fuel into miles driven, typically measured in miles per gallon (mpg). A higher mpg means more distance is covered using less fuel. For instance, when the correlation coefficient showed a strong negative relationship between horsepower and fuel economy, it suggested that as horsepower increases, fuel consumption efficiency tends to decrease. This means more powerful cars, which often have higher horsepower, may consume more fuel per mile. Understanding this relationship can significantly affect consumer choices, especially for those looking to minimize fuel costs or reduce environmental impact. Hence, knowing how different factors like horsepower affect fuel economy can lead to more informed decisions.

Horsepower measures an engine's power output, and its variability can have significant implications for a vehicle's performance and fuel efficiency. In simple terms, horsepower impacts how quickly and powerfully a car can move, which in turn influences fuel consumption. When we talk about horsepower variability, we're considering how differences in horsepower affect other performance aspects such as acceleration and speed. However, not all horsepower increases lead to proportional performance gains, partly because of other influencing factors like vehicle weight and aerodynamics. The strong correlation between horsepower and fuel economy as shown by the correlation coefficient suggests that variations in horsepower are closely tied to shifts in fuel economy. Therefore, understanding how variations in horsepower can lead to changes in driving cost and environmental impact becomes crucial for both manufacturers and consumers alike. Ultimately, while high horsepower might be appealing for speed enthusiasts, it's essential to strike a balance to ensure practical fuel usage.