📘 Section 12.3 – Confidence Intervals in Multiple Linear Regression

🎯 Purpose of Confidence Intervals

Confidence intervals provide a range of plausible values for regression coefficients. They help assess the precision of the estimated coefficients and support inference about the population parameters.

🧮 Confidence Interval Formula

The 100(1−α)% confidence interval for the regression coefficient β_j is:

β̂_j ± t_{α/2, n−k−1} × se(β̂_j)

β̂_j: Least squares estimate of coefficient
se(β̂_j): Estimated standard error
t_{α/2, n−k−1}: t critical value with (n − k − 1) degrees of freedom

📊 GPA Model Example

Let’s consider the GPA model:

GPA = β₀ + β₁×Study + β₂×Sleep + β₃×Attendance + ε

Suppose we have the following estimates and standard errors from a regression output:

Coefficient	Estimate	Standard Error	95% CI
β₀	1.256	0.315	0.529 to 1.983
β₁ (Study)	0.082	0.018	0.038 to 0.126
β₂ (Sleep)	0.291	0.045	0.188 to 0.394
β₃ (Attendance)	0.014	0.006	0.000 to 0.028

These intervals are calculated using the t-distribution with degrees of freedom n − k − 1, where n = 5 students and k = 3 predictors, so df = 1.

🔍 Interpretation

We are 95% confident that the true slope for Study Hours is between 0.038 and 0.126.
The interval for Attendance just barely includes 0, suggesting a weak but borderline significant effect.
Shorter intervals indicate more precise estimates.

🧠 Summary

Confidence intervals allow inference about true regression parameters.
They are influenced by sample size, variability, and multicollinearity.
Wide intervals signal uncertainty; narrow ones reflect strong evidence.