Researchers articulate the expected relationships between the variables by the _______.

The correct choice is the null hypothesis. Researchers articulate the expected relationships between the variables by formulating a null hypothesis, which typically posits that there is no effect or no difference between groups being studied. It serves as a foundational element of statistical testing by providing a baseline or a point of reference from which the alternative hypothesis can be explored.

The null hypothesis is essential for statistical analyses as it allows researchers to use various statistical tests to determine if the observed data significantly deviates from what would be expected if the null hypothesis were true. If the results of the study show statistically significant evidence against the null hypothesis, it suggests that there is a relationship or effect present, leading to the consideration of the alternative hypothesis.

In contrast, while alternative hypotheses express the expected effect or difference, they build upon the null hypothesis rather than replace it. Parameters and statistics are tools and metrics used in analyses but do not directly articulate the expected relationships in the same manner that the null hypothesis does.