Let's answer each part step by step: ### (a) How many factors do the data have? **Answer:** There are **2 factors** in the data (Factor A and Factor B). --- ### (b) How many levels do each factor have? **Answer:** - **Factor A:** 2 levels (A1, A2) - **Factor B:** 2 levels (B1, B2) --- ### (c) Determine the number of treatments used? **Answer:** - Each unique combination of a level of Factor A and a level of Factor B is a treatment. - There are 2 levels for Factor A × 2 levels for Factor B = **4 treatments**. --- ### (d) Construct an ANOVA for this experiment and determine whether there is interaction between the two factors or not. Use α = 0.05. #### Step 1: Tabulate the Data | | B1 | B2 | |----------|------|------| | **A1** | 29.6 | 47.3 | | **A1** | 35.2 | 42.1 | | **A2** | 12.9 | 28.4 | | **A2** | 17.6 | 22.7 | #### Step 2: Layout the Data for ANOVA - A1B1: 29.6, 35.2 - A1B2: 47.3, 42.1 - A2B1: 12.9, 17.6 - A2B2: 28.4, 22.7 #### Step 3: Calculate Means - **Grand Mean (GM):** \( GM = \frac{29.6 + 35.2 + 47.3 + 42.1 + 12.9 + 17.6 + 28.4 + 22.7}{8} = \frac{235.8}{8} = 29.475 \) - **Means by group:** - A1B1: (29.6 + 35.2)/2 = 32.4 - A1B2: (47.3 + 42.1)/2 = 44.7 - A2B1: (12.9 + 17.6)/2 = 15.25 - A2B2: (28.4 + 22.7)/2 = 25.55 #### Step 4: ANOVA Table For a 2x2 factorial, you would calculate sums of squares for: - Factor A - Factor B - Interaction (A × B) - Error **Due to space, I'll outline the process (you can do calculations in Excel or by hand):** 1. **SS_Total:** Sum of squared differences from the grand mean. 2. **SS_A:** Difference between the means of A1 and A2, adjusted by the number of observations. 3. **SS_B:** Difference between the means of B1 and B2, adjusted by the number of observations. 4. **SS_AB (interaction):** Difference between cell means and what would be expected from A and B alone. 5. **SS_Error:** Difference between each observation and its cell mean. **Degrees of freedom:** - df_A = (levels of A - 1) = 1 - df_B = (levels of B - 1) = 1 - df_AB = (levels of A - 1) × (levels of B - 1) = 1 - df_Error = N - (number of groups) = 8 - 4 = 4 **Calculate mean squares and F-values:** - MS = SS / df - F = MS_effect / MS_error **Compare F values to critical value at α = 0.05 for df1 = 1, df2 = 4** #### Step 5: Conclusion If the F value for interaction is **greater** than the critical value (or p-value < 0.05), there **is significant interaction**. Otherwise, there is **no significant interaction**. Let me know if you want the explicit ANOVA calculations and I can compute them step by step!

Let's perform the detailed ANOVA calculations step by step for the given data. --- ### Step 1: Data Recap | **A / B** | **B1** | **B2** | **Row Mean** | |------------|---------|---------|--------------| | **A1** | 29.6 | 47.3 | (29.6+47.3)/2 = 38.45 | | **A1** | 35.2 | 42.1 | (35.2+42.1)/2 = 38.65 | | **A2** | 12.9 | 28.4 | (12.9+28.4)/2 = 20.65 | | **A2** | 17.6 | 22.7 | (17.6+22.7)/2 = 20.15 | ### Step 2: Calculate Grand Mean (GM) \[ GM = \frac{29.6 + 47.3 + 35.2 + 42.1 + 12.9 + 28.4 + 17.6 + 22.7}{8} = \frac{238.8}{8} = 29.85 \] ### Step 3: Calculate Main Effect Means - **A1 mean:** \[ \bar{A1} = \frac{29.6 + 47.3 + 35.2 + 42.1}{4} = \frac{154.2}{4} = 38.55 \] - **A2 mean:** \[ \bar{A2} = \frac{12.9 + 28.4 + 17.6 + 22.7}{4} = \frac{81.6}{4} = 20.4 \] - **B1 mean:** \[ \bar{B1} = \frac{29.6 + 35.2 + 12.9 + 17.6}{4} = \frac{95.3}{4} = 23.825 \] - **B2 mean:** \[ \bar{B2} = \frac{47.3 + 42.1 + 28.4 + 22.7}{4} = \frac{140.5}{4} = 35.125 \] ### Step 4: Calculate Sum of Squares Total observations \( n = 8 \). --- ### Sum of Squares Total (SS\_Total): \[ SS_{Total} = \sum (Y_{ij} - GM)^2 \] Compute each term: \[ \begin{aligned} (29.6 - 29.85)^2 &= 0.0625 \\ (47.3 - 29.85)^2 &= 301.5225 \\ (35.2 - 29.85)^2 &= 28.9225 \\ (42.1 - 29.85)^2 &= 147.1225 \\ (12.9 - 29.85)^2 &= 288.1225 \\ (28.4 - 29.85)^2 &= 2.1025 \\ (17.6 - 29.85)^2 &= 147.1225 \\ (22.7 - 29.85)^2 &= 50.1225 \\ \end{aligned} \] Sum: \[ SS_{Total} = 0.0625 + 301.5225 + 28.9225 + 147.1225 + 288.1225 + 2.1025 + 147.1225 + 50.1225 = \textbf{965.04} \] --- ### Sum of Squares for Factor A (SS\_A): \[ SS_A = n_b \times \sum_{a} (\bar{A}_a - GM)^2 \] Where \( n_b = 4 \) (number of B levels per A). \[ \begin{aligned} (\bar{A1} - GM)^2 &= (38.55 - 29.85)^2 = 74.9025 \\ (\bar{A2} - GM)^2 &= (20.4 - 29.85)^2 = 91.1225 \\ \end{aligned} \] Compute: \[ SS_A = 4 \times (74.9025 + 91.1225) = 4 \times 166.025 = 664.1 \] --- ### Sum of Squares for Factor B (SS\_B): \[ SS_B = n_a \times \sum_{b} (\bar{B}_b - GM)^2 \] Where \( n_a = 4 \): \[ \begin{aligned} (\bar{B1} - GM)^2 &= (23.825 - 29.85)^2 = 36.0625 \\ (\bar{B2} - GM)^2 &= (35.125 - 29.85)^2 = 28.5225 \\ \end{aligned} \] Compute: \[ SS_B = 4 \times (36.0625 + 28.5225) = 4 \times 64.585 = 258.34 \] --- ### Sum of Squares for Interaction (SS\_AB): \[ SS_{AB} = \sum_{a,b} n_{ab} (\bar{Y}_{ab} - \bar{A}_a - \bar{B}_b + GM)^2 \] Since each cell has 2 observations (\( n_{ab} = 2 \)): Calculate each cell's deviation: \[ \begin{aligned} A1B1:\quad & \bar{Y}_{A1B1} = (29.6 + 35.2)/2 = 32.4 \\ & \text{Expected } = \bar{A1} + \bar{B1} - GM = 38.55 + 23.825 - 29.85 = 32.525 \\ & \text{Difference} = 32.4 - 32.525 = -0.125 \\ \Rightarrow & \quad (-0.125)^2 = 0.0156 \\ \text{Contribution} &= 2 \times 0.0156 = 0.0312 \\ \end{aligned} \] Similarly for other cells: - **A1B2:** \[ \bar{Y}_{A1B2} = (47.3 + 42.1)/2 = 44.7 \] Expected: \[ 38.55 + 35.125 - 29.85 = 43.825 \] Difference: \[ 44.7 - 43.825 = 0.875 \] Squared: \[ 0.875^2 = 0.7656 \] Contribution: \[ 2 \times 0.7656 = 1.5312 \] - **A2B1:** \[ \bar{Y}_{A2B1} = (12.9 + 17.6)/2 = 15.25 \] Expected: \[ 20.4 + 23.825 - 29.85 = 14.375 \] Difference: \[ 15.25 - 14.375 = 0.875 \] Squared: \[ 0.875^2 = 0.7656 \] Contribution: \[ 2 \times 0.7656 = 1.5312 \] - **A2B2:** \[ \bar{Y}_{A2B2} = (28.4 + 22.7)/2 = 25.55 \] Expected: \[ 20.4 + 35.125 - 29.85 = 25.675 \] Difference: \[ 25.55 - 25.675 = -0.125 \] Squared: \[ (-0.125)^2=0.0156 \] Contribution: \[ 2 \times 0.0156=0.0312 \] Sum all contributions: \[ SS_{AB} = 0.0312 + 1.5312 + 1.5312 + 0.0312 = \boxed{3.124} \] --- ### Sum of Squares for Error (SS\_Error): \[ SS_{Error} = SS_{Total} - SS_A - SS_B - SS_{AB} \] \[ SS_{Error} = 965.04 - 664.1 - 258.34 - 3.124 = 965.04 - 925.564 = \boxed{39.476} \] --- ### Step 5: Degrees of Freedom (df) | Effect | df | |---------|----| | A | 1 | | B | 1 | | AB | 1 | | Error | 8 - 4 = 4 | --- ### Step 6: Mean Squares (MS) \[ MS_A = \frac{SS_A}{df_A} = \frac{664.1}{1} = 664.1 \] \[ MS_B = \frac{258.34}{1} = 258.34 \] \[ MS_{AB} = \frac{3.124}{1} = 3.124 \] \[ MS_{Error} = \frac{39.476}{4} = 9.869 \] --- ### Step 7: F-Statistics \[ F_A = \frac{MS_A}{MS_{Error}} = \frac{664.1}{9.869} \approx 67.33 \] \[ F_B = \frac{258.34}{9.869} \approx 26.19 \] \[ F_{AB} = \frac{3.124}{9.869} \approx 0.317 \] ### Step 8: Critical Value at α=0.05 Using F-distribution tables for df1=1 and df2=4: - \( F_{crit} \approx 6.16 \) --- ### **Final Conclusion:** - **Factor A:** \( F \approx 67.33 \) > 6.16 → **Significant** - **Factor B:** \( F \approx 26.19 \) > 6.16 → **Significant** - **Interaction (A×B):** \( F \approx 0.317 \) < 6.16 → **Not significant** **Thus, there is no significant interaction between the factors at α=0.05.** Both factors A and B significantly affect the response independently. --- **Summary:** | Effect | F-value | Significance at α=0.05 | Conclusion | |---------------|----------|------------------------|--------------------------| | Factor A | 67.33 | Yes | Significant | | Factor B | 26.19 | Yes | Significant | | Interaction | 0.317 | No | No significant interaction| --- Let me know if you'd like further details or interpretations!