Question

one of the following is a correct statement involving correlation. the other three contain blunders. which one is correct? the correlation between the gas mileage of a car and its weight is r = -0.71 gallon - pounds. the correlation between gpa and hours of sleep per night is r = 1.06. there is a correlation of r = 0.54 between the position a football player plays and his or her weight. the correlation between amount of fertilizer and pounds of tomatoes harvested was found to be r = 0.33.

Explanation:

Step1: Recall correlation rules

The correlation coefficient \(r\) ranges from - 1 to 1. A value close to - 1 indicates a strong negative linear relationship, a value close to 1 indicates a strong positive linear relationship, and a value close to 0 indicates a weak or no linear relationship. Also, correlation makes sense only for quantitative variables.

Step2: Analyze each statement

• For the statement about car mileage and weight (\(r=-0.71\)), weight and gas - mileage are quantitative variables, and a negative \(r\) value makes sense as heavier cars generally have lower gas - mileage.
• For the statement about GPA and hours of sleep (\(r = 1.06\)), since \(r\) must be in the range \([-1,1]\), a value of \(r = 1.06\) is incorrect.
• For the statement about football player position and weight (\(r=0.54\)), position is a categorical variable (e.g., quarterback, linebacker), and correlation is not defined for categorical and quantitative variables in this way.
• For the statement about fertilizer and tomatoes harvested (\(r = 0.33\)), if there is a non - zero linear relationship between the amount of fertilizer (quantitative) and pounds of tomatoes harvested (quantitative), a positive \(r\) value is plausible, but we need to check the context carefully. However, the main incorrect one is the \(r = 1.06\) case.

Answer:

The statement "The correlation between GPA and hours of sleep per night is \(r = 1.06\)" is incorrect because the correlation coefficient \(r\) must be in the interval \([-1,1]\).