1. **State the problem:** We want to analyze if there is a significant difference in depression levels among three therapy groups using ANOVA. The null hypothesis $H_0$ states no difference among groups, and the alternative $H_1$ states at least one group differs. 2. **Given data:** - Group 1 scores: 7, 8, mean $M_1=11$ - Group 2 scores: 9, 12, mean $M_2=7$ - Group 3 scores: 12, 15, mean $M_3=17$ 3. **Calculate Grand Mean (GM):** $$GM = \frac{7 + 8 + 9 + 12 + 12 + 15}{6} = \frac{63}{6} = 10.5$$ 4. **Fill in (X - GM) and (X - GM)^2 for each score:** - Group 1: - $7 - 10.5 = -3.5$, $(-3.5)^2 = 12.25$ - $8 - 10.5 = -2.5$, $(-2.5)^2 = 6.25$ - Group 2: - $9 - 10.5 = -1.5$, $(-1.5)^2 = 2.25$ - $12 - 10.5 = 1.5$, $(1.5)^2 = 2.25$ - Group 3: - $12 - 10.5 = 1.5$, $(1.5)^2 = 2.25$ - $15 - 10.5 = 4.5$, $(4.5)^2 = 20.25$ 5. **Calculate (X - M) and (X - M)^2 for each score:** - Group 1 ($M_1=11$): - $7 - 11 = -4$, $16$ - $8 - 11 = -3$, $9$ - Group 2 ($M_2=7$): - $9 - 7 = 2$, $4$ - $12 - 7 = 5$, $25$ - Group 3 ($M_3=17$): - $12 - 17 = -5$, $25$ - $15 - 17 = -2$, $4$ 6. **Calculate sums of squares:** - Total Sum of Squares (SS_total): sum of all $(X - GM)^2$ $$SS_{total} = 12.25 + 6.25 + 2.25 + 2.25 + 2.25 + 20.25 = 45.5$$ - Within Groups Sum of Squares (SS_within): sum of all $(X - M)^2$ $$SS_{within} = 16 + 9 + 4 + 25 + 25 + 4 = 83$$ - Between Groups Sum of Squares (SS_between): $$SS_{between} = SS_{total} - SS_{within} = 45.5 - 83 = -37.5$$ This negative value indicates an inconsistency in the provided means or data; normally, $SS_{between}$ should be positive. 7. **Calculate degrees of freedom:** - $df_{between} = k - 1 = 3 - 1 = 2$ - $df_{within} = N - k = 6 - 3 = 3$ 8. **Calculate Mean Squares:** - $MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{-37.5}{2} = -18.75$ - $MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{83}{3} \approx 27.67$ 9. **Calculate F statistic:** $$F = \frac{MS_{between}}{MS_{within}} = \frac{-18.75}{27.67} \approx -0.68$$ 10. **Interpretation:** The negative $F$ value is not possible in ANOVA, indicating errors in the data or means provided. Please verify the group means or data. **Summary Table:** | Group | X | X-GM | (X-GM)^2 | X-M | (X-M)^2 | |-------|---|-------|----------|-----|---------| | 1 | 7 | -3.5 | 12.25 | -4 | 16 | | 1 | 8 | -2.5 | 6.25 | -3 | 9 | | 2 | 9 | -1.5 | 2.25 | 2 | 4 | | 2 | 12| 1.5 | 2.25 | 5 | 25 | | 3 | 12| 1.5 | 2.25 | -5 | 25 | | 3 | 15| 4.5 | 20.25 | -2 | 4 | **Final note:** Please check the group means as they do not align with the data points given, causing invalid ANOVA calculations.