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The Standard Model prediction on the forward-backward asymmetry for $b\bar b$ production ($A^b_{FB}$) is well consistent with the data of LEP I at the Z-pole, but deviates from the data at $\sqrt s=89\; 93$ GeV which are slightly away from the pole. This deviation implies that there still is room for new physics. We calculate the $A^b_{FB}$ at vicinity of Z-pole in the Little Higgs Model as well as other measurable parameters such as $R_b$ and $R_c$, by which we may constrain the parameter space of the Little Higgs Model. This can be tested in the newly proposed tera-Z factory. With the fitted parameters we further make predictions on $A^b_{FB}$ and $A^t_{FB}$ for $t\bar t$ production at the ILC energy., Comment: 18 pages, 8 figures and 3 tables. Radical changes including the title of the work are made and more reference added

The Standard Model prediction on the forward-backward asymmetry for $b\bar b$ production ($A^b_{FB}$) is well consistent with the data of LEP I at the Z-pole, but deviates from the data at $\sqrt s=89\; 93$ GeV which are slightly away from the pole. This deviation implies that there still is room for new physics. We calculate the $A^b_{FB}$ at vicinity of Z-pole in the Little Higgs Model as well as other measurable parameters such as $R_b$ and $R_c$, by which we may constrain the parameter space of the Little Higgs Model. This can be tested in the newly proposed tera-Z factory. With the fitted parameters we further make predictions on $A^b_{FB}$ and $A^t_{FB}$ for $t\bar t$ production at the ILC energy., Comment: 18 pages, 8 figures and 3 tables. Radical changes including the title of the work are made and more reference added