  We present a search for the standard model Higgs boson production in
association with a $W$ boson in proton-antiproton collisions
($p\bar{p}\rightarrow W^\pm H \rightarrow \ell\nu b\bar{b}$) at a center of
mass energy of 1.96 TeV. The search employs data collected with the CDF II
detector which correspond to an integrated luminosity of approximately 2.7
fb$^{-1}$. We recorded this data with two kinds of triggers. The first kind
required high-p$_T$ charged leptons and the second required both missing
transverse energy and jets. The search selects events consistent with a
signature of a single lepton ($e^\pm/\mu^\pm$), missing transverse energy, and
two jets. Jets corresponding to bottom quarks are identified with a secondary
vertex tagging method and a jet probability tagging method. Kinematic
information is fed in an artificial neural network to improve discrimination
between signal and background. The search finds that both the observed number
of events and the neural network output distributions are consistent with the
standard model background expectations, and sets 95% confidence level upper
limits on the production cross section times branching ratio. The limits are
expressed as a ratio to the standard model production rate. The limits range
from 3.6 (4.3 expected) to 61.1 (43.2 expected) for Higgs masses from 100 to
150 GeV/$c^{2}$, respectively.
