In the example of testing the correlation between calorie estimates and weight, why does the researcher retain the null hypothesis?

When testing a correlation coefficient by hand, if the absolute value of the computed r is smaller than the critical value found in a statistical table, the researcher should reject the null hypothesis and conclude that a relationship exists between the two variables.

In the example study investigating the relationship between calorie estimates and weight, a health psychologist calculated several values to test her hypothesis. Match each statistical component from this study with its correct value or logical role in the process.

In a psychological study with a sample size of 22 participants, a researcher calculates a Pearson's correlation coefficient of -0.21. To determine if this relationship is statistically significant using a hand-calculation method, a critical value of 0.444 is used. Arrange the following steps to reflect the logical sequence of analysis required to derive the statistical conclusion for this study.

Imagine you are constructing a formal summary for a health psychology presentation based on a study of calorie estimates and weight ($N = 22$). Which of the following concluding statements correctly synthesizes both the manual comparison (using the critical value of $.444$) and the software output (providing a $p$-value of $.348$) to support the researcher's final decision?

In the health psychologist's example study investigating calorie estimates and weight, the statistical software provides a $p$-value of $.348$, which leads the researcher to reject the null hypothesis.

In the health psychologist's study examining the correlation between calorie estimates and weight, several statistical values are used to evaluate the hypothesis. Match each statistical value with the statement that best describes its meaning or role in this test.

A health psychologist investigating the correlation between calorie estimates and weight for a sample of $22$ students ($df = 20$) calculates a Pearson's $r$ of $-.21$. Upon evaluating this result against a critical value of $.444$, the researcher concludes that the correlation is not extreme enough to exceed the threshold for significance. To finalize this statistical judgment, the researcher must _____ the null hypothesis.

In the health psychologist's study, the degrees of freedom equal $20$ (computed as $22 - 2$). The subtraction of $2$ accounts for the number of _____ that must be estimated from the sample data in order to compute Pearson's $r$.

A peer reviewer is critically evaluating the health psychologist's null hypothesis test of the correlation between calorie estimates and weight. Arrange the following evaluative checks in the correct logical order for conducting a rigorous methodological critique.

In the health psychologist's study investigating calorie estimates and weight, describe the manual steps and values required to test the correlation coefficient by hand, including how degrees of freedom are calculated and how the final decision is reached.

Explain how the software's output and the hand-calculation method are conceptually aligned to lead to the identical statistical conclusion of retaining the null hypothesis.

If a researcher replicates the calorie estimate and weight study with the same sample size of $22$ ($df = 20$) and critical value of $.444$, but calculates a Pearson's $r$ of $-.50$, apply the hand-calculation decision rule to determine and justify whether they should reject or retain the null hypothesis.