1. **Problem:** Analyze an experiment comparing three washing solutions (1, 2, 3) over four days to study their effectiveness in retarding bacterial growth using a randomized block design with significance level $\alpha=0.05$. Step 1: Identify factors: Treatments (solutions) = 3, Blocks (days) = 4, observations = 12. Step 2: Organize data in matrix: $$\begin{matrix}\text{Day}\rightarrow &1 &2 &3 &4 \\ \text{Solution} & & & & \\ 1 &13 &22 &18 &39 \\ 2 &16 &24 &17 &44 \\ 3 &5 &4 &1 &22 \end{matrix}$$ Step 3: Calculate sums and means for total, treatments, and blocks. Step 4: Perform two-way ANOVA with blocks. Step 5: Calculate total sum of squares (SST), treatment sum of squares (SSTrt), block sum of squares (SSBlk), error sum of squares (SSE), degrees of freedom. Step 6: Compute mean squares and F-ratios; compare F-values against critical F ($F_{2,6}$ for treatments). Step 7: Conclude if there is a significant effect of washing solutions on bacterial growth at $\alpha=0.05$. 2. **Problem:** Analyze effect of five ingredients (A to E) on reaction time using a Latin square design controlling for day and batch effects, $\alpha=0.05$. Step 1: Extract reaction time data replacing letters with numeric times: $$\begin{matrix}\text{Batch}\rightarrow &1 &2 &3 &4 &5 \\ \text{Day} & & & & & \\ 1 &8 &7 &1 &7 &3 \\ 2 &11 &2 &7 &3 &8 \\ 3 &4 &9 &10 &1 &5 \\ 4 &6 &8 &6 &6 &10 \\ 5 &4 &2 &3 &8 &8 \end{matrix}$$ Step 2: Perform Latin square ANOVA accounting for rows (days), columns (batches), and treatments (ingredients). Step 3: Calculate sum of squares components, degrees of freedom, mean squares, F-values. Step 4: Compare F-treatment against critical value for $\alpha=0.05$ and conclude. 3. **Problem:** Analyze effect of four assembly methods (A-D) on assembly time using Latin square design controlling for operators and order of assembly, with $\alpha=0.05$. Step 1: Extract assembly times: $$\begin{matrix}\text{Operator}\rightarrow &1 &2 &3 &4 \\ \text{Order} & & & & \\ 1 &10 &14 &7 &8 \\ 2 &7 &18 &11 &8 \\ 3 &5 &10 &11 &9 \\ 4 &10 &10 &12 &14 \end{matrix}$$ Step 2: Perform Latin square ANOVA with rows = orders, columns = operators, treatments = methods. Step 3: Calculate sums of squares, degrees of freedom, mean squares, F-statistics. Step 4: Compare with critical F at $\alpha=0.05$ and conclude. 4. **Problem:** Analyze Graeco-Latin square experiment with assembly methods and workplaces as factors plus operators and orders, $\alpha=0.05$. Step 1: Extract data: $$\begin{matrix}\text{Operator}\rightarrow &1 &2 &3 &4 \\ \text{Order} & & & & \\ 1 &11 &10 &14 &8 \\ 2 &8 &12 &10 &12 \\ 3 &9 &11 &7 &15 \\ 4 &9 &8 &18 &6 \end{matrix}$$ Step 2: Conduct analysis of variance for Graeco-Latin square considering four factors. Step 3: Calculate sums of squares and corresponding F-tests. Step 4: Conclude significance of factors at $\alpha=0.05$. **Summary:** Each problem requires applying the respective ANOVA or design-specific test, calculating sums of squares, mean squares, and F-tests, then comparing against critical values to conclude significance of treatments or factors.