Example of Computing Pearson's r

Arrange the steps for calculating Pearson's r using the computational formula in the correct order.

In the computational formula for Pearson's $r$ ($r = \frac{\sum(z_xz_y)}{N}$), how does the calculation distinguish between a positive and a negative relationship between two psychology variables?

In a study analyzing the relationship between 'Stress levels' and 'Sleep quality,' a psychology researcher uses the computational formula for Pearson's $r$. Match each component of the formula $r = \frac {\sum(z_xz_y)}{N}$ with its specific analytical role in processing this data.

A researcher concludes that the initial step of transforming raw scores into $z$-scores is optional when using the computational formula for Pearson's $r$ ($r = \frac{\sum(z_x z_y)}{N}$) and that using the original raw scores will yield the same correlation coefficient. This researcher's conclusion is correct.

According to its computational formula, Pearson's $r$ is defined as the mean of the ____.

According to the computational formula for Pearson's $r$ ($r = \frac{\sum(z_x z_y)}{N}$), the correlation coefficient represents the total sum of all participants' $z$-score cross-products, which causes the value of $r$ to grow larger as the sample size ($N$) increases.

A psychology researcher investigating the relationship between 'Social Media Use' and 'Subjective Well-being' in a sample of 25 participants ($N=25$) calculates that the sum of the $z$-score cross-products ($\sum z_x z_y$) equals -12.5. Applying the formula $r = \frac{\sum z_x z_y}{N}$, the Pearson's $r$ correlation coefficient for this data is _____.

A psychology student collects data on sleep hours ($X$) and cognitive performance test scores ($Y$) for a sample of students. To apply the computational formula for Pearson's $r$ ($r = \frac {\sum(z_xz_y)}{N}$), match each of the following mathematical terms to its corresponding role or representation in this study.

By analyzing the mathematical structure of the Pearson's $r$ formula ($r = \frac {\sum(z_xz_y)}{N}$), a researcher can determine that dividing the sum of the standardized cross-products by the sample size ($N$) calculates the _____ of all these cross-products.

A research assistant proposes a protocol to calculate Pearson's $r$ from raw data. Evaluate the logical and mathematical requirements of the computational formula ($r = \frac {\sum(z_xz_y)}{N}$) and arrange the steps in the correct chronological order to successfully compute the coefficient.

Define Pearson's $r$ in terms of its computational components and describe the step-by-step process required to calculate it from a set of raw scores for two variables.

Based on the computational formula for Pearson's $r$, diagnose the student's mathematical error and explain what the student actually calculated compared to what the formula requires.

A researcher studies the relationship between hours of sleep and memory recall. For a sample of 50 participants ($N = 50$), the researcher transforms all variables to $z$ scores and calculates that the sum of the cross-products is 25 ($\sum(z_xz_y) = 25$). Using the computational formula for Pearson's $r$, what is the correlation coefficient for this study? Briefly explain your calculation.