Question

twenty different statistics students are randomly selected. for each of them, their body temperature (°c) is measured and their head circumference (cm) is measured. if it is found that r=0, does that indicate that there is no association betwen these two variables?

choose the correct answer below.

a. yes, because if r=0, the variables are completely unrelated.

b. no, because r does not measure the strength of the relationship, only its direction.

c. no, because if r=0, the variables are in a perfect linear relationship.

d. no, because while there is no linear correlation, there may be a relationship that is not linear.

Explanation:

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. A value of \( r = 0 \) indicates no linear correlation, but it does not rule out the possibility of a non - linear relationship (such as a quadratic, cubic, etc. relationship) between the two variables.

• Option A is incorrect because \( r = 0 \) only means no linear relationship, not that the variables are completely unrelated.
• Option B is incorrect because \( r \) does measure the strength (and direction) of the linear relationship.
• Option C is incorrect because \( r = 1 \) or \( r=- 1 \) indicates a perfect linear relationship, not \( r = 0 \).
• Option D correctly explains that \( r = 0 \) (no linear correlation) does not mean there is no relationship at all, as there could be a non - linear relationship.

Answer:

D. No, because while there is no linear correlation, there may be a relationship that is not linear.