1. **State the assumptions for a valid ANOVA and how to check their validity:** - The assumptions are: 1. Independence of observations. 2. Normality of residuals within each group. 3. Homogeneity of variances (equal variances) across groups. - To check these: - Independence: ensured by proper experimental design. - Normality: check residual plots, QQ-plots, or perform normality tests like Shapiro-Wilk. - Homogeneity: use Levene's test or Bartlett's test. 2. **Two reasons for running an experiment in blocks:** - To control or reduce variability from nuisance factors. - To increase the precision of comparisons among treatments by eliminating block effects. 3. **Define and explain importance:** - Randomization: randomly assigning experimental units to treatments to avoid bias. - Blocking: grouping similar experimental units to reduce variability from known sources. - Replication: repeating the experiment or treatments to estimate variability and improve reliability. 4. Given ANOVA table: | Source | DF | SS | MS | F | |-------------|----|------|-----|------| | Treatments | 4 | 14.2 | ? | ? | | Blocks | ? | 18.9 | ? | ? | | Error | 24 | ? | ? | | | Total | 34 | 41.9 | | | **a. Fill in the blanks:** - DF for Blocks: total DF - Treatments DF - Error DF = 34 - 4 - 24 = 6 - SS for Error: Total SS - Treatments SS - Blocks SS = 41.9 - 14.2 - 18.9 = 8.8 - MS calculations: - MS Treatments = SS Treatments / DF Treatments = $14.2 / 4 = 3.55$ - MS Blocks = $18.9 / 6 = 3.15$ - MS Error = $8.8 / 24 \approx 0.3667$ - F statistics: - F Treatments = MS Treatments / MS Error = $3.55 / 0.3667 \approx 9.68$ - F Blocks = MS Blocks / MS Error = $3.15 / 0.3667 \approx 8.59$ **b. Number of blocks:** - Number of blocks = DF Blocks + 1 = $6 + 1 = 7$ **c. Number of observations in each treatment total:** - Total observations = DF Total + 1 = $34 + 1 = 35$ - Number of treatments = 5 (DF Treatments + 1) - Thus, observations per treatment = Total observations / Number of treatments = $35 / 5 = 7$ **d. Test for differences among treatment means at $\alpha=0.05$:** - Critical F value for $df_1=4$, $df_2=24$ at 0.05 significance is about 2.78. - Since calculated F Treatments = 9.68 > 2.78, reject null hypothesis. - Conclusion: There is sufficient evidence to indicate differences among treatment means. **e. Test for differences among block means at $\alpha=0.05$:** - Critical F value for $df_1=6$, $df_2=24$ at 0.05 significance approx 2.51. - Calculated F Blocks = 8.59 > 2.51 - Conclusion: There is sufficient evidence to indicate differences among block means.