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We have performed a systematic analysis to compute the one-loop electroweakcorrections to the Z->b b-bar vertex in terms of $\epsilon_b$ and $R_b$ in thecontext of the minimal $SU(5)$ and no-scale $SU(5)\times U(1)$ supergravitymodels. With the measured top mass, $m_t=174\pm 10^{+13}_{-12}$ $\GeV$ ,recently announced by CDF, we use the latest LEP data on $\epsilon_b$ and $R_b$($\equiv{\Gamma(Z->b b-bar)/{\Gamma(Z->hadrons)}}$) in order to constrainfurther the two models. We find that the present LEP data on $\epsilon_b$ and$R_b$ constrain the two models rather severely. Especially, the low-$\tan\beta$region is constrained more severely. $\tan\beta\lsim 2.5$ $(4.0)$ is excludedby $\epsilon_b$ at $90\%$ C.~L. for $m_t\gsim 170$ $(180)$ $\GeV$ in theminimal $SU(5)$ (no-scale $SU(5)\times U(1)$) supergravity. Even more stringentconstraint comes from $R_b$. It excludes $\tan\beta\lsim 4.0$ at $90\%$ C.~L.for $m_t\gsim 160$ $(170)$ $\GeV$ in the minimal $SU(5)$ (no-scale $SU(5)\timesU(1)$) supergravity. We also find that the sign on $\mu$ in the two models canbe determined from $\epsilon_b$ and $R_b$ at $90\%$ C.~L.Comment: RevTeX 3.0, 15 Pages+2 figures(not included but available as a  uuencoded file from gtpark@phyb.snu.ac.kr), SNUTP-94-6

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">ε</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">b</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>constraints on the minimal SU(5) and SU(5)×U(1) supergravity models

εbconstraints on the minimal SU(5) and SU(5)×U(1) supergravity models