When determining the statistical significance of a manually calculated Pearson's $r$, what condition must be met when comparing the sample value to a reference table?

In manual statistical testing, a Pearson's $r$ value is considered statistically significant if the calculated sample correlation is closer to zero than the critical value found in a reference table.

Using a reference table for a two-tailed Pearson's $r$ at $\alpha = .05$, match each research study's data to the correct statistical conclusion.

Reference Table Data:

• For $df = 18$, critical value = $0.444$
• For $df = 28$, critical value = $0.361$

(Remember: $df = n - 2$)

A researcher is consulting a reference table to determine the statistical significance of a manually calculated Pearson's $r$. Based on the structural relationship between sample size ($n$), alpha level ($\alpha$), and critical values, rank the following study setups from the one that will have the highest absolute critical value (requiring the strongest correlation to reach significance) to the one with the lowest absolute critical value (requiring the weakest correlation to reach significance).

A student researcher calculates a Pearson's correlation of $r = -0.52$ and finds a critical value of $0.48$ in a reference table for their sample size. The student argues that because $-0.52$ is numerically less than $0.48$, the result must be nonsignificant. Based on the standard criteria for hypothesis testing where significance is determined by the magnitude of the correlation, the student's evaluation of this result is ________.

When evaluating a manually calculated Pearson's $r$, what condition indicates that the result is statistically significant?

A psychology researcher is planning several studies and needs to understand how sample size ($N$) and alpha levels interact to determine the threshold for statistical significance. Arrange the following research scenarios in order of the magnitude of the critical value of Pearson's $r$ they require, starting with the scenario that requires the lowest (easiest to exceed) correlation coefficient and ending with the one requiring the highest (hardest to exceed) coefficient.

When a researcher manually calculates a Pearson's $r$, the correlation is deemed statistically significant if the calculated sample value is less extreme than the critical value found in a reference table.

A psychology research department is developing a new standardized laboratory manual for manual data analysis. You are tasked with creating the formal procedural workflow that research assistants must follow to determine the statistical significance of a manually calculated Pearson's $r$ result. Arrange the following steps in the correct order to construct a valid and reliable significance-testing protocol.

A psychology researcher is manually evaluating the results of a study. Given that the critical value for their sample size and alpha level is $0.45$, match each calculated sample correlation coefficient to its correct statistical interpretation.

A researcher manually calculates a Pearson's $r$ of $-0.55$ and compares it to a critical value of $0.50$. The researcher concludes that the result is not statistically significant because $-0.55$ is numerically 'less than' the positive threshold of $0.50$. This reasoning is _____ because statistical significance depends on how extreme the correlation is relative to the critical value, regardless of its sign.

A researcher wants to demonstrate how increasing sample size can turn a non-significant correlation into a statistically significant one. They plan to select a single correlation coefficient that is not significant at $n = 15$ but becomes significant at $n = 25$, using $\alpha = .05$. The relevant critical values from a standard reference table are: - $n = 15$ ($df = 13$): critical $r = 0.514$ - $n = 25$ ($df = 23$): critical $r = 0.396$ Which correlation coefficient should the researcher select to create this demonstration?

A student researcher is manually calculating a correlation coefficient. Explain how they would apply the concept of critical values to determine the statistical significance of their sample correlation, including the specific steps to locate the necessary value in a reference table and the decision rule used to draw their conclusion.

Analyze the student's scenario. What should they conclude about the statistical significance of their calculated correlation? In your analysis, explain the relationship between the sample value, the critical value, and the concept of a value being 'more extreme'.

Evaluate the validity of the following statement: 'When manually testing a correlation, if a calculated sample Pearson's $r$ is closer to zero than the critical value listed for that sample size and alpha level, the result is statistically significant because it indicates the relationship is close to the expected value.' Explain your evaluation in one to three sentences.