To calculate the SSregression (Sum of Squares for the Regression Model) for the example in your screenshot, we use the relationship provided between the correlation coefficient, standard deviations, and mean values. Here's the process step-by-step:

1. Recall the Regression Formula for $SS_{regression}$: $SS_{regression} = r^2 \cdot SS_{total}$

2. Find $SS_{total}$: Using the formula for $SS_{total}$ with standard deviations ($SD_X, SD_Y$) and sample sizes: $SS_{total} = \sum (Y - \bar{Y})^2 = \text{Variance} \times (n-1)$

   Given $SD_Y = 0.36$, $n = 9$: $SS_{total} = (0.36^2) \times (9 - 1) = 0.1296 \times 8 = 1.0368$

3. Calculate $r^2$: The correlation coefficient $r = 0.24$: $r^2 = 0.24^2 = 0.0576$

4. Calculate $SS_{regression}$: Substitute into the formula: $SS_{regression} = r^2 \cdot SS_{total} = 0.0576 \cdot 1.0368 = 0.05975 \ (\approx 0.06)$

So, the SSregression for this example is approximately 0.06.

Let me know if you want further clarification or would like to work through related examples! Here are some additional questions for you:

1. How does the total sum of squares ($SS_{total}$) relate to model and error sums of squares?
2. Can you calculate the residual sum of squares ($SS_{error}$) from this data?
3. How does the correlation coefficient $r$ affect the variance explained by the model?
4. What is the interpretation of the regression effect size $R^2$ in this case?
5. Would this regression model be considered statistically significant based on an ANOVA F-test?

Tip: Always double-check sample sizes and variances when calculating $SS_{total}$, as these significantly influence the regression sums of squares!