Expanding on the work of Nuyts et. al [1], Bai et. al. [2], and Bai and Shao [3], who all studied the effects of attenuation and attenuation correction on tumor-to-background ratios and signal detection, we have derived a general expression for the tumor-to-background ratio (TBR) for SPECT attenuated data that have been reconstructed with a linear, non-iterative reconstruction operator O. A special case of this is when O represents discrete filtered back-projection (FBP). The TBR of the reconstructed, uncorrected attenuated data (TBR(no-AC)) can be written as a weighted sum of the TBR of the FBP-reconstructed unattenuated data (TBR(FBP)) and the TBR of the FBP-reconstructed "difference" projection data (TBR(diff)). We evaluated the expression for TBR(no-AC) for a variety of objects and attenuation conditions. The ideal observer signal-to-noise ratio (SNR(ideal)) was also computed in projection space, in order to obtain an upper bound on signal detectability for a signal-known-exactly/background-known-exactly (SKE/BKE) detection task. The results generally show that SNR(ideal) is lower for tumors located deeper within the attenuating medium and increases for tumors nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient mu lead to lower values for SNR(ideal). The TBR for FBP-reconstructed, uncorrected attenuated data can both under- and over-estimate the true TBR, depending on several properties of the attenuating medium, including the shape of the attenuator, the uniformity of the attenuator, and the degree to which the data are attenuated.