Let's break down the solution into **two clear steps** with explanations after each step, and then provide the final answer. --- ### **Step 1: Calculate the Pearson Correlation Coefficient (r)** #### **Data Table** | Social Engagement (X) | Flourishing (Y) | |-----------------------|-----------------| | 9 | 6 | | 4 | 4 | | 8 | 5 | | 7 | 3 | | 5 | 6 | | 4 | 3 | | 9 | 7 | | 6 | 5 | | 10 | 9 | | 13 | 9 | | 7 | 6 | | 8 | 7 | Let’s calculate the necessary sums for the Pearson r formula: \[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}} \] #### **Calculate sums:** - \( n = 12 \) - \( \sum X = 90 \) - \( \sum Y = 70 \) - \( \sum X^2 = 748 \) - \( \sum Y^2 = 454 \) - \( \sum XY = 572 \) --- #### **Substitute into the formula:** \[ r = \frac{12(572) - (90)(70)}{\sqrt{[12 \times 748 - (90)^2][12 \times 454 - (70)^2]}} \] \[ r = \frac{6864 - 6300}{\sqrt{[8976 - 8100][5448 - 4900]}} \] \[ r = \frac{564}{\sqrt{876 \times 548}} \] \[ r = \frac{564}{\sqrt{479,448}} \] \[ r = \frac{564}{692.41} \] \[ r \approx 0.81 \] **Explanation:** The Pearson correlation coefficient \( r \approx 0.81 \) suggests a strong positive relationship between social engagement and flourishing. --- ### **Step 2: Test the significance of the correlation coefficient** The test statistic for significance: \[ t = \frac{r\sqrt{n-2}}{\sqrt{1 - r^2}} \] \[ t = \frac{0.81\sqrt{12-2}}{\sqrt{1 - (0.81)^2}} \] \[ t = \frac{0.81 \times 3.16}{\sqrt{1 - 0.6561}} \] \[ t = \frac{2.56}{\sqrt{0.3439}} \] \[ t = \frac{2.56}{0.586} \] \[ t \approx 4.37 \] Degrees of freedom \( df = n-2 = 10 \). At \( \alpha = 0.05 \) (two-tailed), critical t-value for df = 10 is approximately ±2.228. - \( |t| = 4.37 > 2.228 \) **Explanation:** Since our calculated t-value is greater than the critical value, we reject the null hypothesis. The correlation is **statistically significant** at the .05 level. --- ## **Final Answer:** ### **a. Calculation and Hypothesis Testing Steps:** 1. The Pearson correlation coefficient is **r ≈ 0.81**, indicating a strong positive relationship. 2. The t-test result (**t ≈ 4.37**, df = 10, p < .05) shows this correlation is statistically significant. ### **b. Interpretation:** - **Direction:** Positive (as social engagement increases, flourishing increases) - **Magnitude/Size:** Strong (r ≈ 0.81) - **Coefficient of determination:** \( r^2 = (0.81)^2 ≈ 0.66 \) This means about **66% of the variance** in flourishing can be explained by social engagement. --- **Summary:** There is a strong, positive, and statistically significant relationship between social engagement and flourishing among these students, with social engagement explaining about 66% of the variance in flourishing.