1. **Problem Statement:** We need to understand why blocking is done to maximize differences between blocks and then test if drug dosage levels (Low, Medium, High) significantly affect systolic blood pressure while accounting for individual patient differences (blocking on patients). 2. **Part (a) Explanation:** Blocking is done to maximize the difference between blocks to control for known sources of variability—in this case, individual patient differences. By grouping similar experimental units (patients) into blocks, we remove variability between blocks from the error term, making it easier to detect treatment effects (dosages) more precisely. 3. **Part (b) Analysis:** We have repeated measures data: each patient receives all three dosages (Low, Medium, High). Data Table: Patient 1: Low = 130, Medium = 125, High = 120 Patient 2: Low = 140, Medium = 138, High = 132 Patient 3: Low = 135, Medium = 130, High = 125 4. **Statistical Approach:** Use a one-way repeated measures ANOVA with blocking by patient. 5. **Calculate Means:** - Mean per dosage: $$\overline{X}_{Low} = \frac{130 + 140 + 135}{3} = 135$$ $$\overline{X}_{Medium} = \frac{125 + 138 + 130}{3} \approx 131$$ $$\overline{X}_{High} = \frac{120 + 132 + 125}{3} \approx 125.67$$ - Mean per patient: $$\overline{X}_{P1} = \frac{130 + 125 + 120}{3} = 125$$ $$\overline{X}_{P2} = \frac{140 + 138 + 132}{3} \approx 136.67$$ $$\overline{X}_{P3} = \frac{135 + 130 + 125}{3} = 130$$ - Grand mean: $$\overline{X} = \frac{130 + 125 + 120 + 140 + 138 + 132 + 135 + 130 + 125}{9} = 130.22$$ 6. **Sum of Squares (SS):** - Total SS: $$SS_{Total} = \sum (X_{ij} - \overline{X})^{2}$$ Calculate each squared difference and sum: Patient 1 Low: $(130 - 130.22)^2 = 0.05$ Patient 1 Medium: $(125 - 130.22)^2 = 27.25$ Patient 1 High: $(120 - 130.22)^2 = 104.88$ Patient 2 Low: $(140 - 130.22)^2 = 95.59$ Patient 2 Medium: $(138 - 130.22)^2 = 60.56$ Patient 2 High: $(132 - 130.22)^2 = 3.11$ Patient 3 Low: $(135 - 130.22)^2 = 22.89$ Patient 3 Medium: $(130 - 130.22)^2 = 0.05$ Patient 3 High: $(125 - 130.22)^2 = 27.25$ Sum: $SS_{Total} = 341.63$ - SS Between Patients (Blocks): $$SS_{Blocks} = k \sum (\overline{X}_{P_i} - \overline{X})^{2}, k=3$$ Calculation: $3[(125-130.22)^2 + (136.67-130.22)^2 + (130-130.22)^2] = 3[27.25 + 41.79 + 0.05] = 3 * 69.09 = 207.27$ - SS Between Treatments (Dosages): $$SS_{Treatment} = b \sum (\overline{X}_{T_j} - \overline{X})^{2}, b=3$$ Calculation: $3[(135 - 130.22)^2 + (131 - 130.22)^2 + (125.67 - 130.22)^2] = 3[22.89 + 0.61 + 20.83] = 3 * 44.33 = 133.00$ - SS Error: $$SS_{Error} = SS_{Total} - SS_{Blocks} - SS_{Treatment} = 341.63 - 207.27 - 133.00 = 1.36$$ 7. **Degrees of Freedom:** - $df_{Total} = N - 1 = 9 - 1 = 8$ - $df_{Blocks} = b - 1 = 3 - 1 = 2$ - $df_{Treatment} = k - 1 = 3 - 1 = 2$ - $df_{Error} = (b - 1)(k - 1) = 2 * 2 = 4$ 8. **Mean Squares:** - $MS_{Blocks} = SS_{Blocks} / df_{Blocks} = 207.27 / 2 = 103.64$ - $MS_{Treatment} = SS_{Treatment} / df_{Treatment} = 133.00 / 2 = 66.50$ - $MS_{Error} = SS_{Error} / df_{Error} = 1.36 / 4 = 0.34$ 9. **F-Statistic for Treatment:** $$F = \frac{MS_{Treatment}}{MS_{Error}} = \frac{66.50}{0.34} \approx 195.59$$ 10. **Decision:** Test at $\alpha=0.05$, and from F-tables for $df_1=2$ and $df_2=4$, critical $F \approx 6.94$. Since $195.59 > 6.94$, we reject the null hypothesis. 11. **Conclusion:** There is a significant effect of drug dosage on systolic blood pressure after accounting for patient differences. **Final answers:** - (a) Blocking reduces variability and increases test sensitivity by grouping similar experimental units. - (b) Drug dosage significantly affects systolic blood pressure ($F(2,4) = 195.59, p < 0.05$).