Case context: A health psychologist is planning a follow-up study on the relationship between students' estimated daily calorie intake and their actual body weight. Drawing from the initial study of $22$ students where Pearson's $r$ was $-.21$ and the $p$-value was $.348$, she decides to recruit a larger sample of $102$ students to increase statistical power. She still has no prior expectation about the direction of the relationship.

Question: Based on this new research design, calculate the degrees of freedom for the correlation test. Then, identify whether the psychologist should use a one-tailed or two-tailed test, and state the mathematical decision rule she would use if she conducts the test by hand with a critical value.

Sample answer: For the new study with a sample size of $N = 102$ students, the degrees of freedom are computed as $df = N - 2 = 102 - 2 = 100$. Because the researcher has no prior expectation about the direction of the relationship between calorie estimates and weight, she must use a two-tailed test. If she conducts the test by hand, she must find the critical value for $df = 100$ at the $.05$ alpha level. Her decision rule will be: if the absolute value of the calculated Pearson's $r$ is greater than or equal to the critical value, she will reject the null hypothesis; if the absolute value of the calculated Pearson's $r$ is less than the critical value, she will retain the null hypothesis.

Key points:

• Calculate the degrees of freedom for the new study as $df = 100$.
• Specify the use of a two-tailed test since there is no directional hypothesis.
• State that the absolute value of the computed sample correlation coefficient must be compared to the critical value.
• Formulate the decision rule to reject the null hypothesis if the absolute correlation is greater than or equal to the critical value, and retain it if it is less than the critical value.

Rubric: To earn full credit, the response must: 1) Correctly calculate the degrees of freedom as $df = 100$; 2) Justify the use of a two-tailed test due to having no expectation about the relationship direction; 3) Correctly formulate the decision rule comparing the absolute value of Pearson's $r$ to the critical value for $100$ degrees of freedom.