Question

The correlation between a car's horsepower and its fuel economy (in mpg) is \(r=-0.869\). What fraction of the variability in fuel economy is accounted for by the horsepower?

Short answer

75.52% of variability in fuel economy is accounted for by horsepower.

Step-by-step solution

Identify the Given Correlation

The problem provides the correlation coefficient, denoted as \( r \), which is \( r = -0.869 \). This represents the strength and direction of the linear relationship between horsepower and fuel economy.

Calculate the Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), represents the fraction of variability in the dependent variable that is explained by the independent variable. It is calculated as the square of the correlation coefficient: \[ r^2 = (-0.869)^2 = 0.755161 \].

Express the Result as a Percentage

To express \( r^2 \) as a percentage, multiply by 100: \[ 0.755161 \times 100 = 75.5161\% \]. This means that approximately 75.52% of the variability in fuel economy is accounted for by horsepower.

Key concepts

Understanding the Correlation Coefficient

The correlation coefficient, represented by the symbol \( r \), is a statistical measure that indicates the strength and direction of a linear relationship between two variables. In the context of the problem, the correlation coefficient \( r = -0.869 \) tells us about the relationship between a car's horsepower and its fuel economy (measured in miles per gallon, mpg). The value of \( r \) can range from -1 to 1, with negative values indicating an inverse relationship, positive values indicating a direct relationship, and a value of zero suggesting no linear relationship.

Here, since \( r = -0.869 \), the negative sign shows an inverse relationship. This means that as the horsepower increases, the fuel economy tends to decrease. More importantly, the absolute value of \( r \) (i.e., 0.869) is close to 1, indicating a strong linear relationship between horsepower and fuel economy.

When looking at real-world data, determining the correlation coefficient helps in making predictions and understanding how one variable might affect another.

Explaining Variability in Data

In statistics, variability refers to how spread out the data points in a dataset are. When examining two related variables, like horsepower and fuel economy, understanding how much of the variation in one variable (fuel economy) can be explained by the other (horsepower) is crucial. This is where the concept of the coefficient of determination comes in; it provides a measure of this explanatory power.

The formula for calculating the coefficient of determination is simply squaring the correlation coefficient, \( r^2 \). In our exercise, squaring the correlation coefficient \(-0.869\) gives us \( r^2 = 0.755161 \). This value, expressed in percentage terms, tells us that approximately 75.52% of the variability in fuel economy can be explained by horsepower. In practical terms, this means the majority of the changes in fuel economy across cars can be attributed to differences in their horsepower.

Understanding variability and what explains it is critical for data analysis, as it helps to identify key factors and predict future outcomes.

Grasping the Linear Relationship

A linear relationship between two variables is one where changes in one variable are associated with proportional changes in the other variable. In simpler terms, if you were to plot these variables on a graph, a linear relationship would appear as a straight line. This foundational concept is essential for interpreting the correlation coefficient.

In the exercise context, the linear relationship is between a car's horsepower and its fuel economy. The strong negative correlation of \( -0.869 \) signifies a sloping line downwards; as horsepower increases, fuel economy decreases. This is a visual representation of how these two factors interact.

Recognizing a linear relationship is essential in many domains, especially in fields like economics, engineering, and natural sciences, as it allows for straightforward interpretation and application of results. Models built on such fundamentals can provide clear insights and aid in decision-making processes.