Question

,The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). Make a scatterplot of mileage versus speed. Show that the correlation between speed and mileage is

r=0. Which statement explains why the correlation is r=0 even though there is a strong relationship between speed and mileage?
A) Correlation does not describe curved relationships between variables, no matter how strong they are.
B) Correlation is never a measure of the strength of the relationship between two variables.
C) Since the correlation is r=0, there is no strong relationship between speed and mileage; the scatterplot is misleading.
D) The correlation can be r=0 even if there is a strong linear relationship between the variables.
E) Correlation only measures the strength of the relationship between two variables when the units of the two variables are the same.

Answer 1

The correlation coefficient of 0 does not mean there is no relationship between speed and mileage, but rather that the relationship is not linear, which is why A) is the correct statement. Correlation does not effectively describe non-linear relationships.

Step-by-step explanation:

The question asks us to consider the relationship between an automobile's speed and its gas mileage, and why a scatterplot might show a correlation coefficient (r) of 0. If the gas mileage first increases with speed and then decreases, we might expect to see a curve in the scatterplot that goes up and then comes down, resembling an inverted U. This type of relationship is non-linear, which is why it can yield an r value of 0 even if there is a strong relationship between the two variables.

The correct statement that explains why r=0 even though there is a strong relationship is:

A) Correlation does not describe curved relationships between variables, no matter how strong they are.

The correlation coefficient measures the linear association between two variables, not non-linear ones. Therefore, when the relationship is curved, as it typically is with speed and mileage where there's an optimal speed for fuel efficiency before efficiency begins to decrease, the correlation might be close to zero. This does not mean that there's no relationship, just that the relationship isn't linear.