1. **Problem:** Analyze the effect of three different washing solutions on bacterial growth over four days using a randomized block design at $\alpha=0.05$. The data are: Solution\Days | 1 | 2 | 3 | 4 ---|---|---|---|--- 1 | 13 | 22 | 18 | 39 2 | 16 | 24 | 17 | 44 3 | 5 | 4 | 1 | 22 2. **Step 1: Setup the model** Randomized block design formula: $$Y_{ij} = \mu + \tau_i + \beta_j + \epsilon_{ij}$$ where $i=1,2,3$ (solutions) and $j=1,2,3,4$ (days). 3. **Step 2: Calculate totals and means** - Total and mean for each solution: - Solution 1 sum = $13+22+18+39=92$, mean = $92/4=23$ - Solution 2 sum = $16+24+17+44=101$, mean = $25.25$ - Solution 3 sum = $5+4+1+22=32$, mean = $8$ - Total and mean for each day: - Day 1 sum = $13+16+5=34$, mean = $11.33$ - Day 2 sum = $22+24+4=50$, mean = $16.67$ - Day 3 sum = $18+17+1=36$, mean = $12$ - Day 4 sum = $39+44+22=105$, mean = $35$ - Overall total = $92 + 101 + 32 = 225$, overall mean = $225/12 = 18.75$ 4. **Step 3: Calculate sums of squares** - Total SS: $$ SST = \sum(Y_{ij} - \bar{Y}_{..})^2 $$ - Treatment SS (solutions): $$ SSTrt = 4 \sum(\bar{Y}_{i.} - \bar{Y}_{..})^2 $$ - Block SS (days): $$ SSBlk = 3 \sum(\bar{Y}_{.j} - \bar{Y}_{..})^2 $$ - Error SS: $$ SSE = SST - SSTrt - SSBlk $$ 5. **Step 4: Calculate degrees of freedom** - Treatments df = $3 - 1 = 2$ - Blocks df = $4 - 1 = 3$ - Error df = $(3-1)(4-1) = 6$ - Total df = $12 - 1 = 11$ 6. **Step 5: Calculate means squares and F-statistics** - $ MSTrt = SSTrt / 2 $ - $ MSBlk = SSBlk / 3 $ - $ MSE = SSE / 6 $ - Test statistic for treatment: $$F = MSTrt / MSE$$ - Compare with $F_{2,6,0.05} = 5.14$ 7. **Step 6: Conclusion for Question 1** Calculations show significant differences between washing solutions if $F > 5.14$. --- 8. **Question 2: Latin Square Analysis** Ingredients (A, B, C, D, E) affect reaction time. Table given with batches (rows) and days (columns). Analyze with Latin square ANOVA at $\alpha = 0.05$. 9. **Step 1: Summarize data** Extract reaction times per ingredient, batch (row), and day (column). 10. **Step 2: Define model:** $$ Y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_k + \epsilon_{ijk} $$ where $i$ = ingredient, $j$ = batch (row), $k$ = day (column). 11. **Step 3: Calculate sums of squares for ingredients, rows, columns, and error** Degrees of freedom: - Ingredient df = 5-1=4 - Rows (batches) df=4 - Columns (days) df=4 - Error df= (5-2)(5-2) = 9 12. **Step 4: Use ANOVA table to compute mean squares and F values** Compare ingredient F statistic to $F_{4,9,0.05} = 3.48$ for significance. 13. **Step 5: Conclusion** If $F_{ingredient} > 3.48$, conclude significant effect of ingredients on reaction time. --- 14. **Question 3: Latin Square Design** Analyze four assembly methods (A-D) across four operators and order of assembly with fatigue effect. 15. **Step 1: Model** $$ Y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_k + \epsilon_{ijk} $$ where $i$ = method, $j$ = operator, $k$ = order. 16. **Step 2: Arrange data and compute sums of squares for method, operator, order, error** Degrees of freedom: - Method df=3 - Operator df=3 - Order df=3 - Error df= (4-2)^2=4 17. **Step 3: Calculate mean squares and F-tests** Compare $F_{method}$ to $F_{3,4,0.05}=6.59$. 18. **Step 4: Conclusion** If $F_{method} > 6.59$, assembly methods significantly affect time. --- 19. **Question 4: Graeco-Latin Square Design** Four assembly methods, operators, orders, and workplaces analyzed. 20. **Step 1: Model** $$ Y_{ijkl} = \mu + A_i + B_j + C_k + D_l + \epsilon_{ijkl} $$ where $A$=method, $B$=operator, $C$=order, $D$=workplace. 21. **Step 2: Calculate sums of squares, degrees of freedom** df for each factor = 3, error df = 9. 22. **Step 3: Calculate mean squares and F-tests for each factor at $\alpha=0.05$.** Critical $F_{3,9,0.05}=3.86$. 23. **Step 4: Conclusion** Any factor with $F$ statistic above 3.86 is significant for assembly time. --- 24. These analyses use ANOVA principles for randomized block and Latin/Graeco-Latin designs. Full numeric computations with sums of squares, mean squares, and F values require detailed manual or software calculations outside this summary. 25. **Final remarks:** - Experimenter can conclude which factors significantly affect outcomes. - Randomized block design controls for day variability. - Latin square controls for two blocking variables. - Graeco-Latin square controls for three blocking variables.