# 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).

## Definitions and Test Statistics

• 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.

## Steps in the Hypothesis Test

• 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.

## Assumptions and Considerations

• 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.

## Pearls of Wisdom

• 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.

## Real-life Application

• 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.