1. **State the problem:** We want to use the Scheffe test to compare pairs of group means from three groups (A: no sleep apnea, B: untreated sleep apnea, C: treated sleep apnea) after an ANOVA showed significant differences. 2. **Recall formulas:** - Total sample size per group: $n=11$ - Treatment totals: $T_i = n \times \text{mean}_i$ - Sum of squares between two groups $i$ and $j$: $$SS_{between\,ij} = \frac{(T_i - T_j)^2}{n_i + n_j}$$ - Scheffe test statistic for groups $i$ and $j$: $$F_{i\,vs\,j} = \frac{SS_{between\,ij}}{MS_{within}} \times \frac{k-1}{1}$$ where $MS_{within} = 0.0330$ and $k=3$ groups. 3. **Calculate treatment totals:** - $T_A = 11 \times 0.42 = 4.62$ - $T_B = 11 \times 0.56 = 6.16$ - $T_C = 11 \times 0.30 = 3.30$ 4. **Calculate $SS_{between}$ and $F$ for pairs:** - For A vs B: $$SS_{between\,AB} = \frac{(4.62 - 6.16)^2}{11 + 11} = \frac{(-1.54)^2}{22} = \frac{2.3716}{22} = 0.1078$$ $$F_{A\,vs\,B} = \frac{0.1078}{0.0330} \times (3-1) = 3.266 \times 2 = 6.532$$ - For A vs C: $$SS_{between\,AC} = \frac{(4.62 - 3.30)^2}{22} = \frac{1.32^2}{22} = \frac{1.7424}{22} = 0.0792$$ $$F_{A\,vs\,C} = \frac{0.0792}{0.0330} \times 2 = 2.4 \times 2 = 4.8$$ - For B vs C: $$SS_{between\,BC} = \frac{(6.16 - 3.30)^2}{22} = \frac{2.86^2}{22} = \frac{8.1796}{22} = 0.3718$$ $$F_{B\,vs\,C} = \frac{0.3718}{0.0330} \times 2 = 11.27 \times 2 = 22.54$$ 5. **Interpret results at $\alpha=0.05$:** - Critical $F$ for Scheffe test is the same as ANOVA critical value $3.316$. - Since $F_{A\,vs\,B} = 6.532 > 3.316$, conclude means differ between no sleep apnea and untreated sleep apnea. - Since $F_{A\,vs\,C} = 4.8 > 3.316$, conclude means differ between no sleep apnea and treated sleep apnea. - Since $F_{B\,vs\,C} = 22.54 > 3.316$, conclude means differ between untreated and treated sleep apnea. **Final answers:** - $SS_{between\,AB} = 0.1078$, $F_{A\,vs\,B} = 6.532$ - $SS_{between\,AC} = 0.0792$, $F_{A\,vs\,C} = 4.8$ - $SS_{between\,BC} = 0.3718$, $F_{B\,vs\,C} = 22.54$ All pairs show significant differences at $\alpha=0.05$.