Simple linear regression is a statistical method that allows us to estimate the relationship between two variables. A predictor variable and a response variable. In the context of this project, the predictor is the amount of attended office hours by student, and the response is their respective performance (grade) on a midterm.

The simple linear regression model is given by: \[ y = \beta_0 + \beta_1 x + \epsilon \]

Where: - \(y\) is the dependent variable (e.g., test scores) - \(x\) is the independent variable (e.g., attendance) - \(\beta_0\) is the intercept - \(\beta_1\) is the slope - \(\epsilon\) is the error term

The slope \(\beta_1\) represents the change in \(y\) for a one-unit change in \(x\).

The formula for the least squares estimates of \(\beta_0\) and \(\beta_1\) are: \[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] \[ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \]

The coefficient of determination (\(R^2\)) is calculated as: \[ R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \]

Pre-Processing

scores_filtered=scores %>%
  mutate(full_name = paste(`First Name`, `Last Name`, sep = " "))

attendance_filtered=office_hours %>%
  mutate(full_name = paste(`First Name`, `Last Name`, sep = " "))

# Merge the data frames based on the 'full_name' column
combined_data=inner_join(scores_filtered, attendance_filtered, by = "full_name") %>%
  select(full_name, Total, `Total Score`)

# Categorize students based on attendance (0 to 6 hours)
combined_data=combined_data %>%
  mutate(attendance_category = cut(Total, breaks = c(-Inf, 3, 6), labels = c("0-3 hours", "4-6 hours")))

# Calculate average score for each attendance category
score_summary=combined_data %>%
  group_by(attendance_category) %>%
  summarise(average_score = mean(`Total Score`, na.rm = TRUE))

This histogram allows me to visualize whether or not there was a certain bias towards high attendance or low attendance. These bin counts are very similar. This will make it easier to effectively dive into the performance of students who fall in one of these two categories.

Just Based on this histogram, there a clear difference between the high attendance students and the low attendance students. The students that attend more office hours, seem to outscore the students who attended less. In order to determine whether or not this is statistically significant, we can look into creating a linear model that estimates the relationship between these two variables.

`geom_smooth()` using formula = 'y ~ x'

[1] "The p-value of the slope is:  0.28"

With a traditional significance level of 0.05, the relationship between a students attendance of office hours and their performance is not statistically significant. This does not necessarily mean that students do not benefit from attending these extra study hours. As we saw with the histogram, students who attended more office hours (in the high bin category) did have a higher test average. However, there are many factors that can predict a students performance on a midterm.