1. **State the assumptions for a valid ANOVA and how you would check their validity.** - Assumptions: 1. Independence of observations. 2. Normality of residuals/errors. 3. Homogeneity of variances (equal variances across groups). - Checking validity: 1. Independence is ensured by proper experimental design. 2. Use normal probability plots (Q-Q plots) or Shapiro-Wilk test to check normality of residuals. 3. Use Levene's test or Bartlett's test to check equality of variances. 2. **Give two reasons for running an experiment in blocks.** - To reduce variability from nuisance factors by grouping similar experimental units. - To increase the precision of the experiment by controlling for block effects. 3. **Define and explain Randomization, Blocking, and Replication** - Randomization: Randomly assigning treatments to experimental units to avoid bias and ensure independence. - Blocking: Grouping similar experimental units together to account for variability due to nuisance factors. - Replication: Repeating treatment applications to estimate experimental error and increase reliability of results. 4. **Complete the ANOVA Table and answer questions:** Given: DF_Total = 34, SS_Total = 41.9, Treatments_DF = 4, Treatments_SS = 14.2, Blocks_SS = 18.9, Error_DF =24 a. Fill blanks: - Total DF is 34; so Error DF = Total DF - Treatments DF - Blocks DF. - First find Blocks DF: Since total DF = Treatments DF + Blocks DF + Error DF, $$34 = 4 + \text{Blocks DF} + 24 \Rightarrow \text{Blocks DF} = 34 - 4 - 24 = 6$$ - Now calculate SS_Error: $$SS_{Error} = SS_{Total} - SS_{Treatments} - SS_{Blocks} = 41.9 - 14.2 - 18.9 = 8.8$$ - Calculate Mean Squares (MS): $$MS_{Treatment} = \frac{SS_{Treatments}}{DF_{Treatments}} = \frac{14.2}{4} = 3.55$$ $$MS_{Blocks} = \frac{SS_{Blocks}}{DF_{Blocks}} = \frac{18.9}{6} = 3.15$$ $$MS_{Error} = \frac{SS_{Error}}{DF_{Error}} = \frac{8.8}{24} = 0.3667$$ - Calculate F values: $$F_{Treatment} = \frac{MS_{Treatment}}{MS_{Error}} = \frac{3.55}{0.3667} \approx 9.68$$ $$F_{Blocks} = \frac{MS_{Blocks}}{MS_{Error}} = \frac{3.15}{0.3667} \approx 8.59$$ Final ANOVA Table: | Source | DF | SS | MS | F | |-------------|----|------|-------|-------| | Treatments | 4 | 14.2 | 3.55 | 9.68 | | Blocks | 6 | 18.9 | 3.15 | 8.59 | | Error | 24 | 8.8 | 0.3667| | | Total | 34 | 41.9 | | | b. **Number of blocks in the design:** Since Blocks DF = Number of Blocks - 1 = 6, $$\Rightarrow \text{Number of Blocks} = 6 + 1 = 7$$ c. **Number of observations in each treatment total:** Total observations = Total DF +1 = 34 +1 = 35 observations. There are 5 treatments (DF 4 means 5 treatments). Assuming equal number of observations per treatment, $$n = \frac{35}{5} = 7$$ So, 7 observations per treatment. d. **Test for differences among treatment means at $\alpha = 0.05$.** - Null hypothesis $H_0$: Treatments means are equal. - Alternative hypothesis $H_a$: At least one treatment mean differs. - Find critical $F$ value with $df_1=4$ and $df_2=24$ at $\alpha=0.05$. Approximated $F_{crit} \approx 2.78$. - Since calculated $F_{Treatment} = 9.68 > 2.78$, reject $H_0$. - Conclusion: There is sufficient evidence to conclude differences among treatment means. e. **Test for differences among block means at $\alpha = 0.05$.** - Null hypothesis $H_0$: Block means are equal. - Alternative hypothesis $H_a$: At least one block mean differs. - Degrees of freedom: $df_1 = 6$, $df_2 = 24$. Critical $F_{crit} \approx 2.51$. - Since calculated $F_{Blocks} = 8.59 > 2.51$, reject $H_0$. - Conclusion: There is sufficient evidence to conclude differences among block means.