  We present a search for the standard model Higgs boson produced in
association with a $W^{\pm}$ boson. This search uses data corresponding to an
integrated luminosity of 7.5 fb$^{-1}$ collected by the CDF detector at the
Tevatron. We select $WH \to \ell\nu b \bar{b}$ candidate events with two jets,
large missing transverse energy, and exactly one charged lepton. We further
require that at least one jet be identified to originate from a bottom quark.
Discrimination between the signal and the large background is achieved through
the use of a Bayesian artificial neural network. The number of tagged events
and their distributions are consistent with the standard model expectations. We
observe no evidence for a Higgs boson signal and set 95% C.L. upper limits on
the $WH$ production cross section times the branching ratio to decay to $b\bar
b$ pairs, $\sigma(p\bar p \rightarrow W^{\pm} H) \times {\cal B}(H\rightarrow
b\bar b)$, relative to the rate predicted by the standard model. For the Higgs
boson mass range of 100 GeV/c$^2$ to 150 GeV/c$^2$ we set observed (expected)
upper limits from 1.34 (1.83) to 38.8 (23.4). For 115 GeV/c$^2$ the upper limit
is 3.64 (2.78). The combination of the present search with an independent
analysis that selects events with three jets yields more stringent limits
ranging from 1.12 (1.79) to 34.4 (21.6) in the same mass range. For 115
GeV/c$^2$ the upper limit is 2.65 (2.60).
