If a chi square test of independence results in failing to reject the null hypothesis, how are the nominal variables related?

When a chi-square test of independence results in failing to reject the null hypothesis, this indicates that there is insufficient evidence to suggest that the two nominal variables are related in any systematic way. Thus, the correct answer highlights that these variables are independent of one another.

In statistical terms, independence means that the occurrence or value of one variable does not provide any information about the occurrence or value of the other. Failing to reject the null hypothesis implies that any observed differences in the frequency counts of the categories of these variables can be attributed to random variation rather than a meaningful association.

Understanding this concept is crucial because it allows researchers to determine whether two categories from different variables operate independently or whether they have some level of interaction. In this case, since the null hypothesis stands, the implication is that changes in one variable do not influence the other, indicating a lack of relationship between them.