report.Rmd

---
title: "Statistical Modeling of Student Performance"
author: "Sana Ur Rehman Arain"
date: "`r Sys.Date()`"
output: 
  html_document:
    theme: flatly
    toc: true
    toc_float: true
    number_sections: true
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(
  echo = FALSE,
  warning = FALSE,
  message = FALSE,
  fig.align = "center"
)

library(tidyverse)
library(knitr)
```

---

# Introduction

Academic performance is influenced by a complex interaction of social, economic, and behavioral factors.
Rather than focusing purely on grade prediction, this project emphasizes **statistical inference** to identify which factors *significantly influence* student outcomes when controlling for others.

**Objective:**
To identify statistically valid predictors of final Mathematics grades (`G3`) using the **UCI Student Performance Dataset**.

---

# Exploratory Data Analysis (EDA)

Before formal modeling, exploratory analysis was conducted to understand distributions and potential relationships between variables.

## Distribution of Final Grades

The final grade (`G3`) follows an approximately normal distribution centered around **10–11**, with a notable spike at **0**, representing dropouts or severe failures.

```{r}
knitr::include_graphics("results/distribution_G3.png")
```

---

## Socioeconomic Factors

Family background appears to play a role in academic performance.
The boxplot below suggests that students whose mothers work in **Health** or **Teaching** professions tend to achieve higher grades.

```{r}
knitr::include_graphics("results/mjob_vs_grade.png")
```

These patterns are formally tested using ANOVA in Section 3.

---

## Study Habits

Initial exploration shows a weak positive association between study time and grades.
However, this relationship may be confounded by other factors such as prior failures and parental education.

```{r}
knitr::include_graphics("results/studytime_vs_grade.png")
```

---

## Correlation Matrix (Numeric Variables)

The correlation matrix shows relationships among numeric predictors and the target variable (`G3`).
This helps to identify multicollinearity and supports our feature selection.

```{r}
knitr::include_graphics("results/correlation matric of numeric variables.png")
```

---

# Statistical Inference

To validate findings from EDA, we performed formal hypothesis testing.

## Internet Access (Welch Two-Sample *t*-Test)

**Hypothesis:**
Students with internet access achieve higher final grades than those without.

**Results:**

* Mean (Internet = Yes): **10.62**
* Mean (Internet = No): **9.41**
* *p*-value = **0.0495**

**Conclusion:**
The difference is statistically significant at the 5% level.
Internet access is associated with higher academic performance.

---

## Mother’s Job Type (ANOVA)

**Hypothesis:**
Student grades differ by mother’s occupation.

**Results:**

* ANOVA *p*-value = **0.00519**

**Post-hoc Analysis (Tukey HSD):**

* The only statistically significant pairwise difference was between:

  * **Health** vs **At Home**

* Students with mothers in Health professions score on average **2.99 points higher** (*p-adj = 0.018*).

---

# Multivariate Regression Model

A linear regression model was fitted to evaluate multiple predictors simultaneously.

## Model Specification

[
G3 = \beta_0 + \beta_1(\text{studytime}) + \beta_2(\text{failures}) +
\beta_3(\text{absences}) + \beta_4(\text{Medu}) +
\beta_5(\text{Fedu}) + \beta_6(\text{goout}) + \varepsilon
]

---

## Regression Results

```{r}
reg_table <- tibble(
  Term = c("Intercept", "Study Time", "Failures", "Absences", 
           "Mother's Education", "Father's Education", "Going Out"),
  Estimate = c(10.32, 0.16, -1.93, 0.03, 0.65, -0.08, -0.42),
  `Std. Error` = c(1.03, 0.26, 0.31, 0.03, 0.25, 0.25, 0.19),
  `p-value` = c("< 2e-16", "0.536", "7.24e-10", "0.333", "0.011", "0.752", "0.029"),
  Significance = c("***", "", "***", "", "*", "", "*")
)

kable(reg_table, caption = "Linear Regression Results for Final Grade (G3)")
```

**Model Fit:**

* **R²:** 0.162
* **Adjusted R²:** 0.149
* **F-statistic p-value:** < 0.001

---

## Interpretation

* **Failures:**
  The strongest predictor. One past failure lowers expected grades by **~2 points**, holding all else constant.

* **Mother’s Education (Medu):**
  Higher maternal education significantly improves student performance.

* **Socializing (`goout`):**
  A clear “party penalty”: each unit increase reduces grades by **0.42 points**.

* **Study Time:**
  Becomes statistically insignificant after controlling for background and failures, suggesting **quality and foundation matter more than hours alone**.

---

# Model Diagnostics

Regression assumptions were evaluated using residual diagnostics.

```{r}
knitr::include_graphics("results/model_diagnostics.png")
```

The plots indicate:

* Approximate normality of residuals
* No major heteroscedasticity
* No extreme influential observations

---

# Conclusion

This analysis challenges the simplistic belief that *“studying more automatically leads to better grades.”*

Instead, results show that:

* **Past academic failures**
* **Socioeconomic background**
* **Access to resources (internet)**

are far more influential determinants of student success.

## Recommendations

1. **Targeted Academic Intervention**
   Students with even a single prior failure should receive immediate support.

2. **Equitable Resource Access**
   Ensuring internet availability and academic support for disadvantaged students can significantly improve outcomes.

---

*End of Report*