Illustration of transmission spectrum injection and recovery testing procedure using measured χ² (left panels) and Δχ² (right panels) distributions. The J. Lustig-Yaeger et al. (2019) TRAPPIST-1b 10 bar CO₂ atmospheric transmission spectrum is scaled by 0.6, 1.2, and 1.8 times and injected into the precorrection spectrum of all integrations. Bootstrapped χ² values are recorded for linear and model atmosphere fits to the detrended mean flare spectrum under four scenarios: (i) a linear model fit to the spectrum with the injected signal; (ii) an atmosphere model fit to the spectrum with the injected signal; (iii) a linear model fit to the spectrum in the absence of any injected signals; and (iv) an atmospheric model fit to the spectrum in the absence of any injected signals. The presence of correlated noise in the no-atmosphere samples is highlighted by comparison to the random-variable χ² function with the same 13 degrees of freedom. Hypothesis tests for atmosphere versus linear fits are therefore computed using Δχ² distributions where the correlated noise baselines have been subtracted.