3.4 多元线性回归模型的预测
Yi??1??2X2i??3X3i?ui，1.试对二元线性回归模型：其中，i?(1,2,?,15)作回归分析。
Y?367.693,X2?402.760,X3?8.0?x?84855.096,?x?yx?4250.9,?xx?y?66042.26922i23i?280.0,?yix2i?74778.346?4796.0
i3i2i2i3i?,??,??； 要求：（1）求出未知参数?1,?2,?3的最小二乘估计量?123（2）求出随机误差项u的方差?2的无偏估计量； （3）求得对样本回归方程作拟合优度检验；
（4）对总体回归方程的显著性进行F检验；F0.05(2,12)?3.88
??0.0468,se???0.8454,t(12)?2.56 （5）对?2,?3的显著性进行t检验；se?230.025（6）当X0?(1,X10,X20)?时，写出EY0X0和Y0的置信度为95%的预测区间的公式。 答：（1）
????????2?yx?x??yx?xx???x?x??xx?xx2i2i22i3ii3i2i3i23i2i3i2i3i74778.346?280?4250.9?479684855.096?280?47962550620?757810?0.7266??3?
?yx?x??yx?xx??x?x??xx?xx2i3i22i2ii2i2i3i23i2i3i2i3i4250.9?84855.096?74778.346?4796
84855.096?280?479622073580?757810?2.7363??Y???X???X?12233?367.693?0.7266?402.76?2.7363?8 ?53.1572（2）
??2n?kn?k66042.269?0.7266?74778.346?2.7363?4250.9 ?15?3?6.3821???2?yix2i??3?yix3i?e?2i?y?2i????2?yix2i??3?yix3i(3)
R?2?y2i?0.9988
R2?1?(1?R2)(4)
n?114?1?(1?0.9988)??0.9986 n?k12n?kR2120.9988F?????4994?3.88,显著
k?11?R221?0.9988 (5)
??0.72662t????15.526?2.56，拒绝原假设
?0.0468se?22??
??2.73633t????3.2367?2.56，拒绝原假设
?se?30.84543??（6）
??t??X0)?1X0???X0(X0E?Y0X0?:?Y00.025????
?1????t???Y0:?Y1?X(XX)X00.0250000????