A new small-diameter Monitored Drift Tube (sMDT) chamber has been developed for the muon spectrometer of the ATLAS experiment to handle the higher collision rates expected at the CERN High Luminosity Large Hadron Collider (HL-LHC). This paper presents measurements of the tracking resolution and hit efficiency of two prototype sMDT chambers constructed at the University of Michigan. Using cosmic-ray muons the sMDT tracking resolution of 103.7$pm8.1$ textmu m was measured for one chamber and 101.8$pm$7.8 textmu m for the other, compared with a design resolution of 106 textmu m. A further tracking resolution improvement to 83.4$pm$7.8 textmu m was obtained by using new high-gain readout electronics which will be added for HL-LHC. An average tracking efficiency of (98.5$pm$0.2)% was found for both chambers. The methodology used to determine the detector tracking resolution and efficiency, including reconstruction of sMDT data and a Geant4 simulation of the test chamber, is presented in detail.
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