1. **Problem Statement:** We are given descriptive statistics for Higher Order Thinking (HOT) scores by mode of delivery and class size. We need to perform ANOVA and t-tests to analyze differences in effectiveness. 2. **ANOVA Table Completion:** - Total sample size $N=235$. - Grand mean $\bar{X} = 3.10$. - Calculate Sum of Squares Between (SSB), Within (SSW), and Total (SST). Formulas: $$SSB = \sum n_i (\bar{X}_i - \bar{X})^2$$ $$SSW = \sum (n_i - 1) s_i^2$$ $$SST = SSB + SSW$$ Degrees of freedom: $$df_{between} = k - 1$$ $$df_{within} = N - k$$ $$df_{total} = N - 1$$ Mean Squares: $$MSB = \frac{SSB}{df_{between}}$$ $$MSW = \frac{SSW}{df_{within}}$$ F-statistic: $$F = \frac{MSB}{MSW}$$ Calculations by class size groups (Small, Medium, Large): - Small: $n=30$, $\bar{X}=3.04$, $s=0.54$ - Medium: $n=87$, $\bar{X}=3.40$, $s=0.52$ - Large: $n=118$, $\bar{X}=2.89$, $s=0.75$ Calculate SSB: $$SSB = 30(3.04-3.10)^2 + 87(3.40-3.10)^2 + 118(2.89-3.10)^2$$ $$= 30(0.0036) + 87(0.09) + 118(0.0441)$$ $$= 0.108 + 7.83 + 5.204 = 13.142$$ Calculate SSW: $$SSW = (30-1)(0.54)^2 + (87-1)(0.52)^2 + (118-1)(0.75)^2$$ $$= 29(0.2916) + 86(0.2704) + 117(0.5625)$$ $$= 8.456 + 23.254 + 65.813 = 97.523$$ Calculate SST: $$SST = SSB + SSW = 13.142 + 97.523 = 110.665$$ Degrees of freedom: $$df_{between} = 3 - 1 = 2$$ $$df_{within} = 235 - 3 = 232$$ $$df_{total} = 235 - 1 = 234$$ Mean Squares: $$MSB = \frac{13.142}{2} = 6.571$$ $$MSW = \frac{97.523}{232} = 0.420$$ F-statistic: $$F = \frac{6.571}{0.420} = 15.64$$ ANOVA Table: | Source | SS | df | MS | F | |---------|--------|-----|-------|-------| | Between | 13.142 | 2 | 6.571 | 15.64 | | Within | 97.523 | 232 | 0.420 | | | Total | 110.665| 234 | | | 3. **Model:** $$Y_{ij} = \mu + \tau_i + \epsilon_{ij}$$ where $Y_{ij}$ is HOT score for $j$th observation in $i$th group, $\mu$ is overall mean, $\tau_i$ is effect of $i$th class size, $\epsilon_{ij}$ is random error $\sim N(0, \sigma^2)$. 4. **Hypothesis Testing for Class Size:** - Null hypothesis $H_0$: $\tau_1 = \tau_2 = \tau_3 = 0$ (no difference in means) - Alternative $H_a$: At least one $\tau_i \neq 0$ - Decision rule: Reject $H_0$ if $F > F_{critical}$ at $\alpha=0.05$ with df=(2,232). - Using F-table, $F_{critical} \approx 3.04$. - Since $F=15.64 > 3.04$, reject $H_0$. - Conclusion: Statistically significant differences in HOT by class size. 5. **One-way ANOVA for Mode of Delivery:** Groups: WEB ($n=115$, mean=2.85, SD=0.73), IVDL ($n=120$, mean=3.34, SD=0.54) Calculate SSB: $$SSB = 115(2.85 - 3.10)^2 + 120(3.34 - 3.10)^2 = 115(0.0625) + 120(0.0576) = 7.188 + 6.912 = 14.1$$ Calculate SSW: $$SSW = (115-1)(0.73)^2 + (120-1)(0.54)^2 = 114(0.5329) + 119(0.2916) = 60.7 + 34.7 = 95.4$$ Degrees of freedom: $$df_{between} = 2 - 1 = 1$$ $$df_{within} = 235 - 2 = 233$$ Mean Squares: $$MSB = 14.1 / 1 = 14.1$$ $$MSW = 95.4 / 233 = 0.409$$ F-statistic: $$F = 14.1 / 0.409 = 34.47$$ Decision: $F=34.47 > F_{critical}(1,233,0.05) \approx 3.84$, reject $H_0$. 6. **One-way ANOVA for Class Size:** Already done in step 2 and 4. 7. **Independent Sample t-test for Class Size (Medium vs Large):** - Medium: $n=87$, mean=3.40, SD=0.52 - Large: $n=118$, mean=2.89, SD=0.75 Calculate pooled standard deviation: $$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}} = \sqrt{\frac{86(0.2704) + 117(0.5625)}{203}} = \sqrt{\frac{23.254 + 65.813}{203}} = \sqrt{0.439} = 0.663$$ Calculate t-statistic: $$t = \frac{3.40 - 2.89}{s_p \sqrt{\frac{1}{87} + \frac{1}{118}}} = \frac{0.51}{0.663 \times \sqrt{0.0115 + 0.0085}} = \frac{0.51}{0.663 \times 0.141} = \frac{0.51}{0.0935} = 5.46$$ Degrees of freedom: $n_1 + n_2 - 2 = 203$ Critical t-value at $\alpha=0.05$ two-tailed is about 1.97. Since $5.46 > 1.97$, reject $H_0$. **Comparison:** Both ANOVA and t-test show significant differences between Medium and Large class sizes. **Final answers:** - ANOVA shows significant differences by class size and mode of delivery. - t-test confirms significant difference between Medium and Large class sizes.