1. **Problem Statement:** Test whether there is an interaction between vegetable varieties and fertilizers on the number of leaves per plant at a 95% confidence level using the given data. 2. **Data Setup:** We have three vegetable varieties (V1, V2, V3), three fertilizers (F1, F2, F3), and three groups (Group I, II, III). The data is the number of leaves per plant. 3. **Statistical Method:** We use a two-way ANOVA with interaction to test if the interaction effect between vegetable varieties and fertilizers is significant. 4. **ANOVA Model:** $$Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \gamma_k + \epsilon_{ijk}$$ where: - $\mu$ is the overall mean, - $\alpha_i$ is the effect of the $i^{th}$ vegetable variety, - $\beta_j$ is the effect of the $j^{th}$ fertilizer, - $(\alpha\beta)_{ij}$ is the interaction effect between variety $i$ and fertilizer $j$, - $\gamma_k$ is the block (group) effect, - $\epsilon_{ijk}$ is the random error. 5. **Calculate Means:** Calculate the cell means, row means (varieties), column means (fertilizers), and overall mean. 6. **Sum of Squares:** Calculate the following sums of squares: - Total Sum of Squares (SST) - Sum of Squares for Variety (SSV) - Sum of Squares for Fertilizer (SSF) - Sum of Squares for Interaction (SSI) - Sum of Squares for Blocks (SSB) - Sum of Squares for Error (SSE) 7. **Degrees of Freedom:** - $df_{Variety} = 3 - 1 = 2$ - $df_{Fertilizer} = 3 - 1 = 2$ - $df_{Interaction} = (3-1)(3-1) = 4$ - $df_{Blocks} = 3 - 1 = 2$ - $df_{Error} = (3)(3)(3) - (2+2+4+2+1) = 27 - 11 = 16$ 8. **Mean Squares:** Calculate mean squares by dividing sums of squares by their respective degrees of freedom. 9. **F-Statistics:** Calculate F-values for interaction: $$F_{Interaction} = \frac{MSI}{MSE}$$ 10. **Decision Rule:** Compare $F_{Interaction}$ with critical value $F_{2,16,0.05}$. If $F_{Interaction} > F_{critical}$, reject null hypothesis of no interaction. 11. **Conclusion:** Based on the calculated F-value and critical value, conclude whether there is a significant interaction between vegetable varieties and fertilizers at 95% confidence level. **Note:** Detailed numerical calculations require summing and squaring the data values, which can be done using statistical software or by hand for exact values.