1. **State the problem:** We want to test if the level of depression varies by marital status using an ANOVA F-test. 2. **Data groups:** - Married: $8, 10, 12, 8, 7$ - Unmarried: $12, 11, 9, 14, 4$ - Divorced: $18, 12, 16, 6, 8$ - Widowed: $13, 9, 12, 16, 15$ 3. **ANOVA test formula:** $$F = \frac{\text{Between-group variance}}{\text{Within-group variance}} = \frac{SSB/(k-1)}{SSW/(N-k)}$$ where $k$ is number of groups, $N$ total observations, $SSB$ sum of squares between groups, $SSW$ sum of squares within groups. 4. **Calculate group means:** - Married mean $= \frac{8+10+12+8+7}{5} = 9$ - Unmarried mean $= \frac{12+11+9+14+4}{5} = 10$ - Divorced mean $= \frac{18+12+16+6+8}{5} = 12$ - Widowed mean $= \frac{13+9+12+16+15}{5} = 13$ 5. **Calculate overall mean:** $$\bar{X} = \frac{(9*5)+(10*5)+(12*5)+(13*5)}{20} = \frac{(45+50+60+65)}{20} = \frac{220}{20} = 11$$ 6. **Calculate SSB:** $$SSB = 5[(9-11)^2 + (10-11)^2 + (12-11)^2 + (13-11)^2] = 5[4 + 1 + 1 + 4] = 5 \times 10 = 50$$ 7. **Calculate SSW:** Sum of squared deviations within each group: - Married: $(8-9)^2+(10-9)^2+(12-9)^2+(8-9)^2+(7-9)^2 = 1+1+9+1+4=16$ - Unmarried: $(12-10)^2+(11-10)^2+(9-10)^2+(14-10)^2+(4-10)^2 = 4+1+1+16+36=58$ - Divorced: $(18-12)^2+(12-12)^2+(16-12)^2+(6-12)^2+(8-12)^2 = 36+0+16+36+16=104$ - Widowed: $(13-13)^2+(9-13)^2+(12-13)^2+(16-13)^2+(15-13)^2 = 0+16+1+9+4=30$ Total $SSW = 16 + 58 + 104 + 30 = 208$ 8. **Degrees of freedom:** - Between groups $df_1 = k-1 = 4-1=3$ - Within groups $df_2 = N-k = 20-4=16$ 9. **Calculate mean squares:** - $MSB = \frac{SSB}{df_1} = \frac{50}{3} \approx 16.67$ - $MSW = \frac{SSW}{df_2} = \frac{208}{16} = 13$ 10. **Calculate F-statistic:** $$F = \frac{MSB}{MSW} = \frac{16.67}{13} \approx 1.28$$ 11. **Decision:** Compare $F$ to critical value $F_{3,16}$ at significance level (e.g., 0.05). The critical value is about 3.24. Since $1.28 < 3.24$, we **fail to reject** the null hypothesis. **Conclusion:** There is no significant evidence that depression levels vary by marital status at the 5% significance level.