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| | | +| 1977 | 5.0 | 5.0 | +| 1978 | 5.0 | 6.0 | +| 1979 | 4.0 | 7.0 | + +![1604831603005](assets/1604831603005.png) + +利用回归直线进行估计和预测: + +- 点估计:利用估计的回归方程,对于x的某一特定的值,求出y的一个估计值,就是点估计 +- 区间估计:利用估计的回归方程,对于x的一个特定值,求出y的一个估计值的区间就是区间估计 + +**估计标准误差的计算** + +为了度量回归方程的可靠性,通常计算估计标准误差。它度量观察值回绕着回归直线的变化程度或分散程度。 + +估计平均误差: + +![1604839042933](assets/1604839042933.png) + +- 公式中根号内的分母是n-2,而不是n,这是由于自由度为n-2。 +- 估计标准误差越大,则数据点围绕回归直线的分散程度就越大,回归方程的代表性越小。 +- 估计标准误差越小,则数据点围绕回归直线的分散程度越小,回归方程的代表愈大,其可靠性越高。 + +**置信区间估计** + +![1604839165325](assets/1604839165325.png) + +**在1—a置信水平下预测区间为**: + +![1604839285325](assets/1604839285325.png) + +**求一个具体的值** + +某企业从有关资料中发现广告投入和产品销售有密切的关系。近年该企业广告费和销售额资料如下表,若2003年广告费为120万,请用医院线性回归求2003年产品销售额的置信区间和预测区间(α=0.05) + +| 年份 | 广告费x(万元) | 销售额y(百万元) | +| ---- | --------------- | ----------------- | +| 1994 | 35 | 18 | +| 1995 | 52 | 25 | +| 1996 | 60 | 30 | +| 1997 | 72 | 38 | +| 1998 | 85 | 41 | +| 1999 | 80 | 44 | +| 2000 | 95 | 49 | +| 2001 | 100 | 52 | +| 2002 | 105 | 60 | + +求解如下 + +- ![1604839519431](assets/1604839519431.png) +- ![1604839547186](assets/1604839547186.png)=-3.65 + 0.57 ×120 = 64.75 +- ![1604839593386](assets/1604839593386.png) +- ![1604839622739](assets/1604839622739.png)=64.75±2.365 × 2.43 × 0.743=64.75 ± 4.2699 +- ![1604839693312](assets/1604839693312.png)=64.72 ± 2.365 × 2.43 ×1.2459 = 64.75 ± 4.3516 + +结果图 + +![1604839812051](assets/1604839812051.png) + +影响区间宽度的因素: +- 置信水平(1-a),区间宽度随置信水平的增大而增大 +- 数据的离散程度Se,区间宽度随离程度的增大而增大样本容量 +- 区间宽度随样本容量的增大而减小 +- X0与X均值之间的差异,随着差异程度的增大而增大 \ No newline at end of file