In statistical inference for linear regression models with fixed X values, residuals from the fitted line estimate population error variance to construct confidence intervals and perform hypothesis tests for slope and intercept coefficients. These inferential procedures rely on the Student's t-distribution with $n-2$ degrees of freedom due to the estimation of two parameters (slope and intercept), defining margins of error via critical values multiplied by standard errors derived from residual deviation relative to X variability. This framework establishes a rigorous probabilistic boundary for parameter uncertainty, linking regression geometry directly to hypothesis testing mechanisms within mathematical statistics.
In statistical inference for linear regression models with fixed X values, residuals from the fitted line estimate population error variance to construct confidence intervals and perform hypothesis t…