Let's break down the steps for this **two-factor ANOVA** problem. --- ## **Step 1: Organize the Data** **Given Sums of Squares:** - SST (Total) = 285 - SSA (Factor A) = 75 - SSB (Factor B) = 28 - SSAB (Interaction) = 32 - SSE (Error) = 150 **Degrees of Freedom:** - Factor A: \(a-1 = 2-1 = 1\) - Factor B: \(b-1 = 3-1 = 2\) - Interaction: \((a-1)(b-1) = 1 \times 2 = 2\) - Error: \(N-ab = 30 - 6 = 24\) - Total: \(N-1 = 30-1 = 29\) *(Assuming a = 2, b = 3, n = 5, so N = 2×3×5 = 30)* --- ## **Step 2: Build the ANOVA Table** | Source | Sum of Squares | df | Mean Square | F | p-value | |--------------|----------------|----|------------------|------------|---------| | Factor A | 75 | 1 | 75/1 = **75** | | | | Factor B | 28 | 2 | 28/2 = **14** | | | | Interaction | 32 | 2 | 32/2 = **16** | | | | Error | 150 | 24 | 150/24 = **6.25**| | | | Total | 285 | 29 | | | | --- ## **Step 3: Calculate F-values** - \( F_A = \frac{MS_A}{MS_E} = \frac{75}{6.25} = 12.00 \) - \( F_B = \frac{14}{6.25} = 2.24 \) - \( F_{AB} = \frac{16}{6.25} = 2.56 \) --- ## **Step 4: Find p-values** - Use **F-distribution tables or software**: - \( F_A \) (1,24): \( F = 12.00 \), p ≈ 0.002 - \( F_B \) (2,24): \( F = 2.24 \), p ≈ 0.13 - \( F_{AB} \) (2,24): \( F = 2.56 \), p ≈ 0.099 --- ## **Step 5: Fill Out Table** | Source | SS | df | MS | F | p-value | |--------------|-----|----|------|------|---------| | Factor A | 75 | 1 | 75 | 12.00| 0.002 | | Factor B | 28 | 2 | 14 | 2.24 | 0.13 | | Interaction | 32 | 2 | 16 | 2.56 | 0.099 | | Error | 150 | 24 | 6.25 | | | | Total | 285 | 29 | | | | --- ## **Step 6: Interpret Results** ### **a) Main Effect for Factor A** - p = 0.002 < 0.05 ⇒ **Significant** - **Conclusion:** There is a significant main effect for Factor A. ### **b) Main Effect for Factor B** - p = 0.13 > 0.05 ⇒ **Not Significant** - **Conclusion:** There is NOT a significant main effect for Factor B. ### **c) Interaction (A × B)** - p = 0.099 > 0.05 ⇒ **Not Significant** - **Conclusion:** There is NOT a significant interaction effect. --- ## **Final Answers (to two decimal places):** ### **Means Squares** - Factor A: **75.00** - Factor B: **14.00** - Interaction: **16.00** - Error: **6.25** ### **F-values** - Factor A: **12.00** - Factor B: **2.24** - Interaction: **2.56** ### **p-values** - Factor A: **0.002** - Factor B: **0.13** - Interaction: **0.099** --- ## **Summary Table** | Source | SS | df | MS | F | p-value | |--------------|-----|----|------|------|---------| | Factor A | 75 | 1 | 75.00|12.00 | 0.002 | | Factor B | 28 | 2 |14.00 | 2.24 | 0.13 | | Interaction | 32 | 2 |16.00 | 2.56 | 0.099 | | Error |150 |24 |6.25 | | | | Total |285 |29 | | | | - **Significant effect:** Factor A - **Not significant:** Factor B, Interaction --- **If you need the specific text answers for the blanks, let me know!**

# Step-by-Step Solution for Two-Factor ANOVA This guide explains the process of conducting a two-factor ANOVA using provided data. ## Step 1: Organize the Data ### Given Sums of Squares - **Total (SST)**: 285 - **Factor A (SSA)**: 75 - **Factor B (SSB)**: 28 - **Interaction (SSAB)**: 32 - **Error (SSE)**: 150 ### Degrees of Freedom - **Factor A**: \(a - 1 = 2 - 1 = 1\) - **Factor B**: \(b - 1 = 3 - 1 = 2\) - **Interaction**: \((a - 1)(b - 1) = 1 \times 2 = 2\) - **Error**: \(N - ab = 30 - 6 = 24\) - **Total**: \(N - 1 = 30 - 1 = 29\) *Assuming \(a = 2\), \(b = 3\), \(n = 5\) which gives \(N = 2 \times 3 \times 5 = 30\)* --- ## Step 2: Build the ANOVA Table | Source | Sum of Squares | df | Mean Square | F | p-value | |--------------|----------------|----|------------------|------------|---------| | Factor A | 75 | 1 | \( \frac{75}{1} = 75 \) | | | | Factor B | 28 | 2 | \( \frac{28}{2} = 14 \) | | | | Interaction | 32 | 2 | \( \frac{32}{2} = 16 \) | | | | Error | 150 | 24 | \( \frac{150}{24} \approx 6.25 \) | | | | Total | 285 | 29 | | | | --- ## Step 3: Calculate F-values ### Formulas - \( F_A = \frac{MS_A}{MS_E} \) - \( F_B = \frac{MS_B}{MS_E} \) - \( F_{AB} = \frac{MS_{AB}}{MS_E} \) ### Calculations - \( F_A = \frac{75}{6.25} = 12.00 \) - \( F_B = \frac{14}{6.25} = 2.24 \) - \( F_{AB} = \frac{16}{6.25} = 2.56 \) --- ## Step 4: Find p-values Use **F-distribution tables or statistical software** to find p-values: - \( F_A (1,24) \): \( F = 12.00 \), p ≈ 0.002 - \( F_B (2,24) \): \( F = 2.24 \), p ≈ 0.13 - \( F_{AB} (2,24) \): \( F = 2.56 \), p ≈ 0.099 --- ## Step 5: Fill Out Table | Source | SS | df | MS | F | p-value | |--------------|-----|----|------|------|---------| | Factor A | 75 | 1 | 75 | 12.00| 0.002 | | Factor B | 28 | 2 | 14 | 2.24 | 0.13 | | Interaction | 32 | 2 | 16 | 2.56 | 0.099 | | Error | 150 | 24 | 6.25 | | | | Total | 285 | 29 | | | | --- ## Step 6: Interpret Results ### a) Main Effect for Factor A - **p = 0.002 < 0.05** ⇒ **Significant** - **Conclusion:** There is a significant main effect for Factor A. ### b) Main Effect for Factor B - **p = 0.13 > 0.05** ⇒ **Not Significant** - **Conclusion:** There is NOT a significant main effect for Factor B. ### c) Interaction (A × B) - **p = 0.099 > 0.05** ⇒ **Not Significant** - **Conclusion:** There is NOT a significant interaction effect. --- ## Final Answers (to Two Decimal Places) ### Mean Squares - **Factor A**: 75.00 - **Factor B**: 14.00 - **Interaction**: 16.00 - **Error**: 6.25 ### F-values - **Factor A**: 12.00 - **Factor B**: 2.24 - **Interaction**: 2.56 ### p-values - **Factor A**: 0.002 - **Factor B**: 0.13 - **Interaction**: 0.099 --- ## Summary Table | Source | SS | df | MS | F | p-value | |--------------|-----|----|------|------|---------| | Factor A | 75 | 1 | 75.00| 12.00| 0.002 | | Factor B | 28 | 2 | 14.00| 2.24 | 0.13 | | Interaction | 32 | 2 | 16.00| 2.56 | 0.099 | | Error | 150 | 24 | 6.25 | | | | Total | 285 | 29 | | | | - **Significant effect:** Factor A - **Not significant:** Factor B, Interaction