What is the null hypothesis in a correlation analysis?

What is the null hypothesis in a correlation analysis?

• A. r = 0, indicating no relationship
• B. r = 1, indicating a perfect relationship
• C. r = -1, indicating a perfect inverse relationship
• D. r is not used in correlation

Answer: (A)

In correlation analysis we’re asking whether two variables move together in a linear way in the population. The quantity that expresses that relationship is the population correlation coefficient, rho (ρ). The null hypothesis is that ρ equals zero, meaning there is no linear relationship between the variables. In practice we use the sample correlation r to test this: if the data give a small p-value, we reject the null and conclude there is evidence of a linear relationship; if not, we don’t have enough evidence of a linear association. It’s important to note that r = 0 in the sample does not prove there is no relationship at all—there could be a nonlinear pattern—just not a linear one detected by this test. Conversely, r equals 1 or -1 would indicate a perfect linear relationship, which is not what the null asserts. And r is indeed the quantity used in correlation to measure and test the strength of linear association.

In correlation analysis we’re asking whether two variables move together in a linear way in the population. The quantity that expresses that relationship is the population correlation coefficient, rho (ρ). The null hypothesis is that ρ equals zero, meaning there is no linear relationship between the variables. In practice we use the sample correlation r to test this: if the data give a small p-value, we reject the null and conclude there is evidence of a linear relationship; if not, we don’t have enough evidence of a linear association.

It’s important to note that r = 0 in the sample does not prove there is no relationship at all—there could be a nonlinear pattern—just not a linear one detected by this test. Conversely, r equals 1 or -1 would indicate a perfect linear relationship, which is not what the null asserts. And r is indeed the quantity used in correlation to measure and test the strength of linear association.