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The coefficient of determination, often denoted as \( R^2 \), plays a pivotal role in understanding how much of the variability in one variable can be explained by another variable. In the context of our exercise, where the correlation coefficient \( r \) between a car's engine size and its fuel economy is \(-0.774\), calculating \( R^2 \) involves squaring \( r \). So, \( R^2 = (-0.774)^2 = 0.599076 \).
This value, when expressed as a percentage, tells us that approximately 59.91% of the variance in fuel economy (measured in miles per gallon) can be explained by knowing the car's engine size.
In simpler terms, the engine size significantly impacts how fuel-efficient a car is. Therefore, understanding and calculating the coefficient of determination helps in assessing the predictive power of the linear relationship between engine size and fuel economy.

A linear relationship between two variables is like a straight line: as one variable changes, the other does so in a consistent manner. In statistical terms, this relationship is denoted by the correlation coefficient, \( r \). This value can vary between -1 and 1.

• An \( r \) of 1 implies a perfect positive linear relationship.
• An \( r \) of -1 implies a perfect negative linear relationship.
• An \( r \) around 0 suggests little to no linear relationship.

In our example, with \( r = -0.774 \), the negative sign indicates an inverse relationship: as the engine size increases, the fuel economy tends to decrease. The magnitude \( 0.774 \) suggests a strong relationship, although not perfect.
This is crucial for automakers and consumers alike because understanding this relationship helps in predicting and optimizing fuel efficiency based on engine size.

Variance is a statistical measure that describes the spread or dispersion of a set of values. When considering fuel economy in cars, variance tells us how much the miles per gallon (mpg) data tends to deviate from its mean.
In the context of engine size and fuel economy, variance helps us understand how diverse the fuel economy readings are for different engines. A higher variance indicates that the fuel economy readings vary widely across different cars, while a lower variance signifies that the values are clustered closely around the mean.
Using the coefficient of determination, we can see how much of this variability is explained by the engine size. Specifically, knowing that \( R^2 = 0.599076 \) means about 59.91% of the variance in fuel economy is accounted for by engine size, making it a significant factor in influencing fuel efficiency.

Fuel economy refers to how effectively a vehicle uses fuel to travel. It is typically measured in miles per gallon (mpg) or kilometers per liter, depending on regional preferences. The concept of fuel economy becomes important in discussions about automotive efficiency, cost-effectiveness, and environmental impact.
Modern vehicles strive for high fuel economy ratings to appeal to consumers who seek cost savings on fuel and a reduced carbon footprint. Several factors can influence a car's fuel economy, among which engine size is a notable one, as highlighted by our exercise.
A larger engine may offer more power but can consume more fuel, reducing the car's miles per gallon. Conversely, smaller engines may be less powerful but provide better fuel efficiency. Thus, understanding the balance between engine size and fuel economy is key for both manufacturers and consumers when making or purchasing vehicles.