1. **Stating the problem:** We have yield data from three farms (A, B, C) with two factors: irrigation (whole plot factor) and fertilizer (sub-plot factor). We want to perform a split-plot ANOVA to analyze the effects of irrigation and fertilizer on yield. 2. **Understanding split-plot ANOVA:** This design has two factors: a whole-plot factor (irrigation) applied to large plots and a sub-plot factor (fertilizer) applied within those plots. The analysis partitions variance into components due to irrigation, fertilizer, their interaction, and error. 3. **Data summary:** - Irrigation levels: 1, 2 - Fertilizer levels: 1, 2, 3, 4 - Farms: A, B, C (replicates) 4. **Step 1: Calculate means for each factor level and interaction.** Calculate the mean yield for each irrigation level, fertilizer level, and their combinations across farms. 5. **Step 2: Calculate Sum of Squares (SS):** - SSTotal: Total variation in data - SS_Irrigation: Variation due to irrigation - SS_Fertilizer: Variation due to fertilizer - SS_Interaction: Variation due to interaction between irrigation and fertilizer - SS_Error: Residual variation 6. **Step 3: Calculate degrees of freedom (df):** - df_Irrigation = number of irrigation levels - 1 = 2 - 1 = 1 - df_Fertilizer = number of fertilizer levels - 1 = 4 - 1 = 3 - df_Interaction = df_Irrigation * df_Fertilizer = 1 * 3 = 3 - df_Error = total observations - number of groups 7. **Step 4: Calculate Mean Squares (MS):** $$MS = \frac{SS}{df}$$ 8. **Step 5: Calculate F-statistics:** $$F = \frac{MS_{factor}}{MS_{error}}$$ 9. **Step 6: Compare F-values to critical F-values from F-distribution tables to determine significance.** 10. **Note:** Due to the complexity and length of manual calculations, statistical software (e.g., R, SAS) is recommended for exact F-test values and p-values. **Final answer:** The split-plot ANOVA partitions variance into irrigation, fertilizer, interaction, and error components, and the F-tests determine which factors significantly affect yield. The detailed calculations require software for precise results.