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The James Webb Space Telescope's NIRSpec instrument will unveil the nature of exoplanet atmospheres across the wealth of planet types, from temperate terrestrial worlds to ultrahot Jupiters. In particular, the 0.6-5.3 micron PRISM mode is especially well-suited for efficient spectroscopic exoplanet observations spanning a number of important spectral features. We analyze a lab-measured NIRSpec PRISM mode Bright Object Time Series (BOTS) observation from the perspective of a JWST user to understand the instrument performance and detector properties. We create two realistic transiting exoplanet time series observations by performing injection-recovery tests on the lab-measured data to quantify the effects of real instrument jitter, drift, intrapixel sensitivity variations, and 1/$f$ noise on measured transmission spectra. By fitting the time series systematics simultaneously with the injected transit, we can obtain more realistic transit depth uncertainties that take into account noise sources that are currently not modeled by traditional exposure time calculators. We find that sources of systematic noise related to intrapixel sensitivity variations and PSF motions are apparent in the data at the level of a few hundred ppm, but can be effectively detrended using a low-order polynomial with detector position. We recover the injected spectral features of GJ 436 b and TRAPPIST-1 d, and place a 3-sigma upper limit on the detector noise floor of 14 ppm. We find that the noise floor is consistent with <10 ppm at the 1.7-sigma level, which bodes well for future observations of challenging targets with faint atmospheric signatures.