Wing Sheung Chan

108 Statistical interpretation and results is used instead of the simple p -value p µ s = Z ∞ q obs f ( q µ s | µ s ) d q µ s (6.7) alone. The CL s value conservatively quantifies the strength of statistical evidence for the alternative hypothesis as against the null hypothesis. The probability distributions of the test statistics are approximated using the asymptotic formulae described in Reference [127] . 6.2. Uncertainties As for any scientific research, careful consideration of possible uncertainties is what makes a truthful and trustworthy result. In the following, the sources and handling of uncertainties considered in the maximum-likelihood fits will be discussed. 6.2.1. Prefit uncertainty estimations In the likelihood function, systematic uncertainties are parameterised by Gaussian- constrained NPs (denoted as α ). Their prefit values and uncertainties are set to be 0 ± 1 . Reconstructed objects Systematic uncertainties associated to the reconstructed objects are estimated by the respective ATLAS working groups based on various measurements. These include the electron-related uncertainties (resolution, scale, and trigger, reconstruction, isolation and identification efficiencies) [84, 85] , muon-related uncertainties (resolution, scale, and trigger, reconstruction, isolation and track-to-vertex-association efficiencies) [86, 87] , τ had - vis -related uncertainties (energy scale, and reconstruction and identification efficiencies) [81, 82] , E miss T -related uncertainties (track soft term resolution and scale) [88, 89] , jet-related uncertainties (energy resolution and scale) [128] and flavour-tagging-related uncertainties ( b -tagging efficiencies) [75] , The effect of these uncertainties are modelled by varying the corresponding kinematics or efficiency scale factors by ± 1 standard deviations ( σ ) and interpolating/extrapolating their effects on event yields in each bin. Tau energy scale Since both the m coll ( `, τ ) distribution in CRZ τ τ and the combined NN output distribution in the SR are sensitive to the reconstructed τ had - vis energy, uncertainties in the TES are one of the dominant sources of systematic uncertainties and deserve careful considerations. These uncertainties mainly affect the Z → τ τ background modelling.