1. Problem: Analyze the randomized block design data to test if washing solutions affect bacterial growth (α = 0.05). Step 1: State hypotheses. - Null hypothesis ($H_0$): No difference among solutions. - Alternative hypothesis ($H_a$): At least one solution differs. Step 2: Organize data by solutions (treatments) and days (blocks): - Solutions: 1, 2, 3 - Blocks (days): 1, 2, 3, 4 Data: Solution 1: 13, 22, 18, 39 Solution 2: 16, 24, 17, 44 Solution 3: 5, 4, 1, 22 Step 3: Calculate totals and means (solutions and blocks), grand total $G$: - Total per solution: $T_1 = 13+22+18+39=92$ $T_2 = 16+24+17+44=101$ $T_3 = 5+4+1+22=32$ - Total per block: $B_1=13+16+5=34$ $B_2=22+24+4=50$ $B_3=18+17+1=36$ $B_4=39+44+22=105$ - Grand total $G=92+101+32=225$ Step 4: Calculate sums of squares: - Total sum of squares (SST): $SST = \sum y_{ij}^2 - \frac{G^2}{N}$, with $N=12$. Calculate $\sum y_{ij}^2 = 13^2+22^2+...+22^2$ - Treatment sum of squares (SS_T): $SS_T = \sum \frac{T_i^2}{b} - \frac{G^2}{N}$, where $b=4$ (blocks). - Block sum of squares (SS_B): $SS_B = \sum \frac{B_j^2}{t} - \frac{G^2}{N}$, where $t=3$ (treatments). - Residual sum of squares (SS_E): $SS_E = SST - SS_T - SS_B$ Step 5: Calculate mean squares (MS), degrees of freedom (df). - $df_T = t - 1 = 2$ - $df_B = b - 1 = 3$ - $df_E = (t-1)(b-1) = 6$ - $MS_T = SS_T / df_T$ - $MS_B = SS_B / df_B$ - $MS_E = SS_E / df_E$ Step 6: Calculate F-statistic for treatments: $$F = \frac{MS_T}{MS_E}$$ Step 7: Compare with critical $F_{\alpha, df_T, df_E}$ (at 0.05 level). - If $F > F_{critical}$, reject $H_0$ and conclude differences among solutions. 2. Problem: Analyze Latin square design to test ingredient effects on reaction time (α = 0.05). Step 1: Identify factors - treatments (ingredients A-E), rows (batches), and columns (days). Step 2: Compute totals and sums of squares for treatments, rows, and columns. Step 3: Calculate total sum of squares, treatment sum of squares, row sum of squares, and column sum of squares. Step 4: Calculate residual sum of squares. Step 5: Calculate mean squares and degrees of freedom. Step 6: Calculate F-statistic for treatments and compare with critical values. Step 7: Conclude whether ingredient affects reaction time. 3. Problem: Analyze Latin square design data for assembly methods on assembly time (α = 0.05). Repeat analysis as in problem 2 with treatments (assembly methods), rows (operators), and columns (orders). Calculate sums of squares, mean squares, F-statistics, and conclude. 4. Problem: Analyze Graeco-Latin square design data including workplaces as fourth factor (α = 0.05). Step 1: Recognize four factors: treatment (assembly methods), rows (order), columns (operator), and Greek letters (workplace). Step 2: Calculate sums of squares for all four factors and residual. Step 3: Calculate mean squares and appropriate F-tests for the main treatment effect. Step 4: Compare F-statistics to critical values at α = 0.05. Step 5: Conclude if assembly method and workplaces significantly affect assembly time. Final: Each design controls different sources of variability (blocks, batches, operators, workplace) to isolate treatment effects, confirming statistical differences if $F$ tests exceed critical values at 5% significance level.