⏳ Loading example…
🔀
ANOVA
Two-Way ANOVA — Teaching Method × Class Size Interaction
Full 2×2 factorial ANOVA: main effects and significant interaction F(1,8)=5.16. SS decomposition and η².
📊 F_A=16.27, F_B=5.25, F_A×B=5.16
Step-by-step solution
🔀 12 · Two-Way ANOVA (Factorial Design)
Teaching Method × Class Size on Exam Scores
ANOVA
Research Question:
Does the effect of teaching method (Traditional vs. Active Learning) on exam scores depend on class size (Small ≤15 vs. Large >30)? N=24, n=6 per cell.
| Small Class (≤15) | Large Class (>30) | Row Mean | |
|---|---|---|---|
| Traditional | 68.0 | 70.0 | 69.0 |
| Active Learning | 82.0 | 74.0 | 78.0 |
| Column Mean | 75.0 | 72.0 | 73.0 (GM) |
SS Formulas (balanced design, n=6 per cell)
$$SS_A = n \cdot k_B \cdot \sum(\bar{M}_{A_i}-\text{GM})^2,\quad SS_B = n \cdot k_A \cdot \sum(\bar{M}_{B_j}-\text{GM})^2$$
$$SS_{A\times B} = n \cdot \sum\bigl(M_{ij}-\bar{M}_{A_i}-\bar{M}_{B_j}+\text{GM}\bigr)^2$$
-
1Main Effect A (Teaching Method):$$SS_A = 6\times2\times\bigl[(69.0-73.0)^2+(78.0-73.0)^2\bigr]=12\times[16+25]=\mathbf{492}$$ $$F_A=\frac{492/1}{30.25}=16.27,\quad p=.001$$
-
2Main Effect B (Class Size):$$SS_B = 6\times2\times\bigl[(75.0-73.0)^2+(72.0-73.0)^2\bigr]=12\times[4+1]=\mathbf{60}$$ $$F_B=\frac{60/1}{30.25}=1.98,\quad p=.174$$
-
3Interaction A×B — the key test:$$\text{Residuals: }(68{-}69{-}75{+}73)=-3,\;(70{-}69{-}72{+}73)=2,\;(82{-}78{-}75{+}73)=2,\;(74{-}78{-}72{+}73)=-3$$ $$SS_{A\times B}=6\times[(-3)^2+2^2+2^2+(-3)^2]=6\times26=\mathbf{156},\quad F_{A\times B}=\frac{156}{30.25}=5.16,\quad p=.034$$
-
4Interpret the interaction: Active Learning produces a much larger gain in small classes (+14 pts) than in large classes (+4 pts). Simple main effect analysis confirms the interaction.
| Source | SS | df | MS | F | p | η²p |
|---|---|---|---|---|---|---|
| Method (A) | 492 | 1 | 492 | 16.27 | .001 | .448 |
| Class Size (B) | 60 | 1 | 60 | 1.98 | .174 | .090 |
| A × B Interaction | 156 | 1 | 156 | 5.16 | .034 | .205 |
| Error (Within) | 605 | 20 | 30.25 | |||
| Total | 1313 | 23 |
✅ Significant interaction A×B, F(1,20)=5.16, p=.034 — the benefit of Active Learning is significantly larger in small classes.
APA-7
A 2×2 between-subjects ANOVA revealed a significant main effect of teaching method, F(1, 20) = 16.27, p = .001, η²p = .45, and a significant Method × Class Size interaction, F(1, 20) = 5.16, p = .034, η²p = .21. Active Learning produced substantially larger gains in small (M = 82.0) than large classes (M = 74.0), whereas Traditional instruction showed minimal size effects.