## Problem Statement

Conduct a Factorial Analysis of Variance (ANOVA) to examine the differences in total points earned by students in a course based on various factors such as gender, course section, and division (lower or upper). The goal is to determine whether these factors have a statistically significant impact on the total scores.

## Solution

**Table 1: Summary Statistics
**

Table 1 provides a summary of descriptive statistics for the dependent variable, "Total Points." This variable is categorized by gender, course section, and whether students are in the lower or upper division. The table showcases the means, standard deviations, and sample sizes for each category.

## Null Hypotheses

**H_01:**There is no statistically significant difference in total scores by gender (Male, Female).**H_02:**There is no statistically significant difference in total scores by course section (1, 2, & 3).**H_03:**There is no statistically significant difference in total scores by division (Lower, Upper).**H_04:**Any differences in total scores among students grouped by gender do not statistically significantly vary as a function of course section.**H_05:**Any differences in total scores among students grouped by gender do not statistically significantly vary as a function of division.**H_06:**Any differences in total scores among students grouped by course section do not statistically significantly vary as a function of division.**H_07:**The difference in total scores between students based on gender and section remains statistically significantly constant regardless of division.

## Homogeneity of Variance Assumption

The homogeneity of variance assumption is examined using Levene's test. The result indicates that the assumption is violated since the p-value (0.003) is less than the significance level (0.05). This implies that the variance of total points differs significantly across the groups.

### Normality Assumption

To assess the normality of total points earned by students, both the Kolmogorov-Smirnov and Shapiro-Wilk tests were employed. Both tests reject the null hypothesis of normality since the p-values (0.001 and 0.047) are below the significance level (0.05).

### Detection of Outliers

Outliers are detected using the Interquartile Range (IQR) method. Values below Q1 - 1.5IQR and above Q3 + 1.5IQR are considered outliers.

### Test of Significance

Table 5 contains the results of the analysis of variance, which addresses the seven hypotheses stated earlier. Notably, all three main effects (gender, section, and division) are not statistically significant, as their p-values exceed the significance level (0.05). However, among the interaction effects, only one of the 2-way interactions is statistically significant, with a p-value of 0.027. This suggests that the interaction effect between course section and division has a significant impact on total scores.

**Figure 1: Interaction Effect between Course Section and Division
**

This figure illustrates the significant interaction effect mentioned above. Additionally, effect size and observed power are presented in Table 5 to provide a comprehensive overview of the analysis results.