1. **State the problem:** We need to compute the F ratio for three groups of scores: Group 1: 10, 10, 9, 8, 11 Group 2: 9, 8, 7, 7, 6 Group 3: 8, 10, 6, 8, 8 2. **Formula and explanation:** The F ratio in ANOVA is given by: $$F = \frac{\text{Mean Square Between Groups (MSB)}}{\text{Mean Square Within Groups (MSW)}}$$ Where: - MSB measures variance between group means. - MSW measures variance within each group. 3. **Step 1: Calculate group means** $$\bar{X}_1 = \frac{10 + 10 + 9 + 8 + 11}{5} = \frac{48}{5} = 9.6$$ $$\bar{X}_2 = \frac{9 + 8 + 7 + 7 + 6}{5} = \frac{37}{5} = 7.4$$ $$\bar{X}_3 = \frac{8 + 10 + 6 + 8 + 8}{5} = \frac{40}{5} = 8.0$$ 4. **Step 2: Calculate overall mean** $$\bar{X} = \frac{48 + 37 + 40}{15} = \frac{125}{15} = 8.3333$$ 5. **Step 3: Calculate Sum of Squares Between Groups (SSB)** $$SSB = n \sum (\bar{X}_i - \bar{X})^2 = 5[(9.6 - 8.3333)^2 + (7.4 - 8.3333)^2 + (8.0 - 8.3333)^2]$$ Calculate each term: $$(9.6 - 8.3333)^2 = 1.6044$$ $$(7.4 - 8.3333)^2 = 0.8711$$ $$(8.0 - 8.3333)^2 = 0.1111$$ So: $$SSB = 5(1.6044 + 0.8711 + 0.1111) = 5(2.5866) = 12.933$$ 6. **Step 4: Calculate Sum of Squares Within Groups (SSW)** Calculate variance within each group: Group 1: $$(10 - 9.6)^2 + (10 - 9.6)^2 + (9 - 9.6)^2 + (8 - 9.6)^2 + (11 - 9.6)^2$$ $$= 0.16 + 0.16 + 0.36 + 2.56 + 1.96 = 5.2$$ Group 2: $$(9 - 7.4)^2 + (8 - 7.4)^2 + (7 - 7.4)^2 + (7 - 7.4)^2 + (6 - 7.4)^2$$ $$= 2.56 + 0.36 + 0.16 + 0.16 + 1.96 = 5.2$$ Group 3: $$(8 - 8)^2 + (10 - 8)^2 + (6 - 8)^2 + (8 - 8)^2 + (8 - 8)^2$$ $$= 0 + 4 + 4 + 0 + 0 = 8$$ Sum all: $$SSW = 5.2 + 5.2 + 8 = 18.4$$ 7. **Step 5: Calculate degrees of freedom** Between groups: $$df_b = k - 1 = 3 - 1 = 2$$ Within groups: $$df_w = N - k = 15 - 3 = 12$$ 8. **Step 6: Calculate Mean Squares** $$MSB = \frac{SSB}{df_b} = \frac{12.933}{2} = 6.4665$$ $$MSW = \frac{SSW}{df_w} = \frac{18.4}{12} = 1.5333$$ 9. **Step 7: Calculate F ratio** $$F = \frac{MSB}{MSW} = \frac{6.4665}{1.5333} = 4.22$$ **Final answer:** The F ratio is approximately **4.22**.