  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.
