1. **Problem Statement:** We want to determine if server type (A, B, C), region (North, South), and their interaction significantly affect response time. 2. **Method:** Use Two-Way ANOVA to analyze the effects of two categorical independent variables (server type and region) on a continuous dependent variable (response time). 3. **Data Summary:** - Server A North: 120, 130, 125 - Server A South: 140, 135, 138 - Server B North: 110, 115, 112 - Server B South: 150, 145, 148 - Server C North: 100, 105, 102 - Server C South: 160, 155, 158 4. **Calculate group means:** - $\bar{X}_{A,N} = \frac{120+130+125}{3} = 125$ - $\bar{X}_{A,S} = \frac{140+135+138}{3} = 137.67$ - $\bar{X}_{B,N} = \frac{110+115+112}{3} = 112.33$ - $\bar{X}_{B,S} = \frac{150+145+148}{3} = 147.67$ - $\bar{X}_{C,N} = \frac{100+105+102}{3} = 102.33$ - $\bar{X}_{C,S} = \frac{160+155+158}{3} = 157.67$ 5. **Calculate overall mean:** $$\bar{X} = \frac{\sum \text{all values}}{18} = \frac{120+130+125+140+135+138+110+115+112+150+145+148+100+105+102+160+155+158}{18} = 132.5$$ 6. **Calculate main effects means:** - Server type means: - $\bar{X}_A = \frac{125 + 137.67}{2} = 131.33$ - $\bar{X}_B = \frac{112.33 + 147.67}{2} = 130$ - $\bar{X}_C = \frac{102.33 + 157.67}{2} = 130$ - Region means: - $\bar{X}_N = \frac{125 + 112.33 + 102.33}{3} = 113.89$ - $\bar{X}_S = \frac{137.67 + 147.67 + 157.67}{3} = 147.67$ 7. **ANOVA model:** Response time $= \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}$ - $\mu$: overall mean - $\alpha_i$: effect of server type $i$ - $\beta_j$: effect of region $j$ - $(\alpha\beta)_{ij}$: interaction effect - $\epsilon_{ijk}$: random error 8. **Interpretation:** - Large differences in means suggest server type and region affect response time. - Interaction is suggested by different differences between North and South across server types. 9. **Conclusion:** Performing the full ANOVA test (calculating sums of squares, degrees of freedom, mean squares, and F-statistics) will confirm significance. Since the question is about evaluation, the data shows clear differences in response times by server type and region, and interaction likely exists. Final answer: Server type, region, and their interaction significantly affect response time based on observed means.