Random Dataset for Regression graph:

set.seed(123)
study_time <- seq(1, 10, by = 1)
test_score <- 50 + 5 * study_time + rnorm(10, mean = 0, sd = 5)
data <- data.frame(study_time, test_score)
invisible(head(data))
model <- lm(test_score ~ study_time, data = data)
invisible(summary(model))
invisible(library(ggplot2))

ggplot(data, aes(x = study_time, y = test_score)) +
  geom_point(color = "#8C1D40", size = 3) +
  invisible(geom_smooth(method = "lm", se = FALSE, color = "black")) +
  theme_minimal() +
  invisible(labs(title = "Study Time vs Test Score", x = "Hours Studied", y = "Test Score"))

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

residuals <- resid(model)
fitted <- fitted(model)
res_data <- data.frame(fitted, residuals)

ggplot(res_data, aes(x = fitted, y = residuals)) +
  geom_point(color = "#003366", size = 3) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  theme_minimal() +
  labs(title = "Residual Plot", x = "Fitted Values", y = "Residuals")

This interactive plot visualizes the relationship between:

• \(x\): Hours Studied
  
• \(y\): Test Score
  
• Added variation (noise) in the data

Even though the data roughly follows a linear trend, each point is affected by random noise \(\varepsilon\), which represents:

\[ \varepsilon \sim \mathbb{N}(0, \sigma^2) \]

This means that actual scores vary due to unpredictable factors, even if more study time generally leads to higher scores.

Key Takeaway:

• The pattern shows a positive linear relationship: as \(x\) increases, \(y\) tends to increase.
• The spread around reflects residual variation in real-world observations.