1. **State the assumptions for a valid ANOVA and how to check them:** - Assumption 1: Independence of observations. Check by ensuring random sampling and experimental design. - Assumption 2: Normality of residuals. Check using normal probability plots or Shapiro-Wilk test on residuals. - Assumption 3: Homogeneity of variances (equal variances among groups). Check using Levene's test or Bartlett's test. 2. **Two reasons for running an experiment in blocks:** - To reduce variability caused by known nuisance factors by grouping similar experimental units. - To increase precision of treatment comparisons by removing block-to-block variability. 3. **Definitions and importance:** - Randomization: Assigning experimental units to treatments randomly to avoid bias and confounding. - Blocking: Grouping similar experimental units (blocks) to control variability from nuisance factors. - Replication: Repeating treatments on multiple experimental units to estimate experimental error and increase reliability. 4. **Complete the ANOVA table and answer subquestions:** Given: - DF(Treatments) = 4, SS(Treatments) = 14.2 - SS(Blocks) = 18.9, DF(Error) = 24 - DF(Total) = 34, SS(Total) = 41.9 Calculate missing degrees of freedom: $$DF_{Blocks} = DF_{Total} - DF_{Treatments} - DF_{Error} = 34 - 4 - 24 = 6$$ Calculate SS(Error): $$SS_{Error} = SS_{Total} - SS_{Treatments} - SS_{Blocks} = 41.9 - 14.2 - 18.9 = 8.8$$ Calculate Mean Squares: $$MS_{Treatments} = \frac{SS_{Treatments}}{DF_{Treatments}} = \frac{14.2}{4} = 3.55$$ $$MS_{Blocks} = \frac{SS_{Blocks}}{DF_{Blocks}} = \frac{18.9}{6} = 3.15$$ $$MS_{Error} = \frac{SS_{Error}}{DF_{Error}} = \frac{8.8}{24} \approx 0.3667$$ Calculate F-values: $$F_{Treatments} = \frac{MS_{Treatments}}{MS_{Error}} = \frac{3.55}{0.3667} \approx 9.68$$ $$F_{Blocks} = \frac{MS_{Blocks}}{MS_{Error}} = \frac{3.15}{0.3667} \approx 8.59$$ **Answers:** b. Number of blocks $= DF_{Blocks} + 1 = 6 + 1 = 7$ c. Number of observations $= DF_{Total} + 1 = 34 + 1 = 35$ Since there are 5 treatments ($DF_{Treatments} + 1 = 4 + 1$), observations per treatment: $$\frac{35}{5} = 7$$ d. Test for treatment differences at $\alpha =0.05$ with $(4,24)$ df: Critical $F$ from table $\approx 2.78$ Since $F_{Treatments} = 9.68 > 2.78$, reject null hypothesis, treatments differ significantly. e. Test for block differences at $\alpha=0.05$ with $(6,24)$ df: Critical $F \approx 2.51$ Since $F_{Blocks} = 8.59 > 2.51$, reject null hypothesis, blocks differ significantly.