Question: In the context of the study investigating the correlation between calorie estimates and weight, explain the statistical logic behind why a researcher can use either the software-provided $p$-value or a hand-calculated critical value comparison to make a decision about the null hypothesis. Describe how the sample values from this study are evaluated under both methods.

Sample answer: In hypothesis testing for a correlation coefficient, both the $p$-value method and the critical value comparison method evaluate whether the observed sample correlation is extreme enough to reject the null hypothesis (which states that the population correlation is zero). In the software method, the calculated $p$-value of $.348$ is directly compared to the significance level alpha ($.05$). Because $.348 > .05$, the researcher retains the null hypothesis. In the hand-calculation method, the researcher determines the degrees of freedom ($df = 22 - 2 = 20$) and finds the corresponding two-tailed critical value of $.444$ from a reference table. The absolute value of the sample correlation coefficient ($|-.21| = .21$) is compared to this critical value. Because $.21 < .444$, the result is less extreme than the critical value, which mathematically confirms that the probability of obtaining this correlation by chance is greater than $.05$ ($p > .05$). Thus, both methods lead to the identical statistical conclusion to retain the null hypothesis.

Key points:

• Explain that the software method involves comparing the calculated $p$-value ($.348$) to the standard alpha level ($.05$) to retain the null hypothesis.
• Explain that the hand-calculation method involves finding the critical value ($.444$) for the degrees of freedom ($df = 20$).
• State that the absolute value of the correlation coefficient ($.21$) is compared to the critical value ($.444$) and found to be less extreme.
• Explain that both methods yield the identical conclusion because a correlation less extreme than the critical value mathematically corresponds to a $p$-value greater than $.05$.

Rubric: A complete response must: 1) Explain that both methods assess if the sample correlation is statistically significant under the null hypothesis; 2) Detail the software comparison showing $p = .348$ is compared to alpha = $.05$ and since $.348 > .05$, the null hypothesis is retained; 3) Detail the hand-calculation comparison showing that for $df = 20$, the absolute sample correlation of $.21$ is compared to the critical value of $.444$ and is less extreme; 4) Explain that these two methods are mathematically equivalent because a sample correlation less extreme than the critical value corresponds to a $p$-value greater than the significance level.