When testing a correlation coefficient by hand, what does it mean if the calculated sample correlation coefficient is more extreme than the critical value retrieved from a statistical table?

In psychological research, when testing a correlation coefficient by hand, researchers use critical values to make statistical decisions. Match each scenario or set of parameters with its corresponding role or outcome in this process.

A researcher studying the relationship between noise levels and concentration calculates a Pearson's $r = .32$. The critical value found in a statistical table for their sample size at the $\alpha = .05$ level is $r = .35$. In this scenario, the researcher should reject the null hypothesis and conclude that a statistically significant relationship exists in the population.

A psychology researcher is analyzing the relationship between sleep duration and stress levels. After collecting data, they must determine if their results are statistically significant. Arrange the steps of the manual significance testing process in the correct logical order to determine if a relationship exists in the population.

When conducting a test of the correlation coefficient by hand, which two pieces of information are required to locate the appropriate critical value of $r$ in a statistical table?

When testing a correlation by hand, the critical value of $r$ acts as a threshold; for a result to be statistically significant, the calculated sample correlation must be more extreme than this value.

A psychology researcher calculates a Pearson's $r = .18$ for a study exploring the relationship between study hours and exam performance. The statistical table for their sample size indicates a critical value of $r = .25$ at the $\alpha = .05$ level. Because the calculated value is less extreme than the critical value, the researcher must conclude there is _____ evidence to reject the null hypothesis.

A researcher is preparing to test the correlation coefficient by hand for four different studies. Match each study's sample size to the correct degrees of freedom that should be used when consulting the critical-value table at $\alpha = .05$.

A researcher studies the correlation between exercise frequency and mood in 22 participants and obtains $r = .38$. Consulting a critical-value table at $\alpha = .05$ for $df = 20$ gives a critical value of .444. Because the sample $r$ of .38 is less extreme than the critical value of .444, the $p$-value is _____ .05, and the null hypothesis should be retained.

A peer reviewer is judging whether a researcher correctly concluded statistical significance after testing the correlation coefficient by hand. Place the following evaluation steps in the order the reviewer should perform them to assess the validity of the researcher's conclusion.

Describe the steps required to determine statistical significance when testing a correlation coefficient by hand using a critical value table. Your response should outline the parameters needed to find the critical value and the rule used to make a decision about the null hypothesis.

Explain whether the student should reject or retain the null hypothesis based on this critical value. What does this decision indicate about the population relationship between the variables, and what can be concluded about the $p$-value?

A researcher calculates a Pearson's correlation of $r = -.52$ with a sample of 22 participants ($df = 20$). The critical value table lists the critical value for 20 degrees of freedom as .444 at the $\alpha = .05$ level. Apply the decision rule for critical values of $r$ to state whether the researcher can reject the null hypothesis and explain why.