Question

which scatterplot best represents the relationship between a cars speed and its fuel efficiency?
a linear model would best represent the relationship because scatterplot a is fairly linear.
an exponential model would best represent the relationship because scatterplot b is fairly linear.
a power model would best represent the relationship because scatterplot c is fairly linear.
a linear model would best represent the relationship because scatterplot c is fairly linear.

Explanation:

Step1: Analyze the relationship between speed and fuel - efficiency

The relationship between a car's speed and its fuel efficiency is not a simple linear relationship. As speed increases, fuel efficiency first increases (up to a certain speed) and then may start to decrease. However, when we take the natural - logarithm of speed (\(x = \ln(\text{Speed})\)) and assume a power - model \(y=ax^{b}\) (by taking the natural - logarithm of both sides \(\ln(y)=\ln(a)+b\ln(x)\)), a linear relationship can be approximated.

Step2: Check the scatter - plots

Scatterplot A: If we assume a linear model \(y = mx + c\) (where \(y\) is fuel efficiency and \(x\) is speed), the relationship between fuel efficiency and speed is not a straight - line (linear) relationship in the non - transformed variables. As speed increases, the rate of change of fuel efficiency is not constant.

Scatterplot B: If we assume a linear model \(y = mx + c\) (where \(y\) is fuel efficiency and \(x\) is speed), the relationship is not linear.

Scatterplot C: When we use a power - model \(y = ax^{b}\) (by transforming \(x\) to \(\ln(x)\) and \(y\) to \(\ln(y)\) or just \(x=\ln(\text{Speed})\) and \(y = \text{Fuel Efficiency}\)), the scatter - plot of \(y\) (fuel efficiency) versus \(x=\ln(\text{Speed})\) is fairly linear. A power model \(y = ax^{b}\) (where \(x\) is speed) can be rewritten in a linear form for regression purposes. If \(y = ax^{b}\), then \(\ln(y)=\ln(a)+b\ln(x)\). If we consider \(x\) as \(\ln(\text{Speed})\) (a transformation of the original speed variable) and \(y\) as fuel efficiency, a linear - like pattern is observed in scatterplot C.

Answer:

A power model would best represent the relationship because scatterplot C is fairly linear.