Main idea: Good predictions require both a useful model and honest evaluation.
We simulate data for 120 students. Each observation has:
The dataset is synthetic, created only to demonstrate statistical methods. It does not establish a causal effect.
Rotate or zoom the 3D points to inspect how the two predictors relate to exam scores.
A multiple linear regression combines both predictors:
\[ \widehat{y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 x_{i1} + \widehat{\beta}_2 x_{i2} \]
Here, \(\widehat{y}_i\) is the predicted score, \(x_{i1}\) is study hours, and \(x_{i2}\) is practice problems.
The coefficients are estimated by minimizing the sum of squared residuals:
\[ \min_{\beta_0,\beta_1,\beta_2} \sum_{i=1}^{n}(y_i-\beta_0-\beta_1x_{i1}-\beta_2x_{i2})^2 \]
These are associations from simulated observations, not proof of cause and effect.
A residual is the difference between observed and predicted scores:
\[e_i = y_i - \widehat{y}_i\]
Root mean squared error summarizes typical prediction error in score points:
\[ \mathrm{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i-\widehat{y}_i)^2} \]
Training-set RMSE: 7.75 score points.
This is an in-sample result; a held-out test set would better measure performance on new students.
The following is the core R workflow used in this presentation:
# Fit a regression model using the simulated data
model <- lm(exam_score ~ study_hours + practice_problems,
data = students)
# Compute fitted values and training residuals
predicted <- predict(model, newdata = students)
errors <- students$exam_score - predicted
rmse <- sqrt(mean(errors^2))
summary(model)
rmse
Bottom line: Statistics provides the tools to make predictive models interpretable and testable.