Hypothesis Tests Using Pearson’s Product-Moment Correlation Coefficient

Introductory Overview

Hypothesis testing using the Pearson’s product-moment correlation coefficient involves evaluating hypotheses about the coefficient.
You conduct these tests based on a null hypothesis and an alternative hypothesis about the population correlation coefficient, represented by the Greek letter rho (ρ).

The null hypothesis (H0) often states that there is no correlation between the two variables. So, H0: ρ = 0.
The alternative hypothesis (H1) states the opposite, i.e., there is some correlation, either positive, negative, or of unspecified direction. So, H1: ρ <> 0.
The test statistic for this hypothesis test is the calculated r, the sample correlation coefficient. This statistic follows a t-distribution with n-2 degrees of freedom, where n refers to the sample size.

Calculate the correlation coefficient, r, from the given data.
Set up the null and alternative hypotheses.
Decide on the significance level, often denoted by an alpha level (α), for example, 0.05. The significance level represents the probability of rejecting a true null hypothesis.
Based on the sample size and the significance level, look up the critical value in the Student’s t-distribution table.
If r is greater than the critical value, reject the null hypothesis. If r is less than or equal to the critical value, you fail to reject the null hypothesis.

It is important to check the major assumptions—normality, linearity, and homoscedasticity—are met before performing the test.
The data should be collected from a pair of continuous variables; they must not be categories or ranks.
Keep in mind that rejecting the null hypothesis does not automatically imply a strong relationship; effect size should be referred to determine the degree of relationship between variables.

Hypothesis tests using correlation coefficients can provide valuable statistical evidence to support or refute assertions about relationships between variables.
Be careful to avoid common pitfalls, such as mistaking correlation for causation.
Remember, a significant correlation indicates the probability of a relationship existing, but it does not prove one.

Suppose a researcher is examining the relationship between hours studied and exam scores. The null hypothesis might be that there is no correlation between hours studied and exam scores (ρ = 0), while the alternative hypothesis would be that there is a correlation (ρ <> 0). Using this process, the researcher could statistically determine whether or not the amount of time spent studying has a legitimate effect on exam scores.