高中数学：线性回归方程
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">线性回归是利用数理统计中的回归分析来确定两种或两种以上变数间相互依赖的定量关系的一种统计分析方法，是变量间的相关关系中最重要的一部分，主要考查概率与统计知识，考察学生的阅读能力、数据处理能力及运算能力，题目难度中等，应用广泛．</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">一 线性回归方程公式</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JZZrE3Hh3IK~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=lB6kQZ1HStZI3i2dBieldZp%2FRXg%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">二 规律总结</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JZaL3GG5Hjs~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=g0UjDWg8QAeOLlYUf%2B2t9wAvnZ8%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JZac9Stz8Dw~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=yL1aO%2FKRm%2FpE5HRcj%2FVIXn9fnp4%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">(3)回归分析是处理变量相关关系的一种数学方法．主要用来解决：</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">①确定特定量之间是否有相关关系，如果有就找出它们之间贴近的数学表达式；</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">②根据一组观察值，预测变量的取值及判断变量取值的变化趋势；</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">③求线性回归方程．</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">三 线性回归方程的求法</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例1 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JZaqCCxNFqv~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=7IMmUC40%2FK4p%2BjWq3cd%2F3d1%2FUMg%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JZb69ECVxzk~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=OlXfWYRCq3dCfIDsWXfvp9oM8dY%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">四 线性回归方程的应用</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例2 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Ja9WG9DvgG9~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=JcTH2wrYY%2FIyJxgymFE%2FYDJZ8Qc%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Ja9oGCabtiF~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=CsHT11gmnBH9owq%2BNy07lqU0m3w%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例3 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JaA2IPAJvfr~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=S5MCydO0WNNqRI0p4e1z6YU1O1Q%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JaAMGyGhnTJ~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=sVCcgdVa7jGJ91SmEglrKRNWLLI%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例4 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JaAd8nCeo0L~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=Rc8TJBXLs95nr403nXb%2FS1Z%2BzKU%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jakp40OEdja~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=8w6BGIEGJOSPAR8JRPfianGvG90%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例5 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JalBDzSxpx0~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=wR7nZBkHqxrHV8TZVuuUAh09JOI%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JalYEOLBLNR~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=CHGraCdMVQHyTw9mgHmgxSL2RKo%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例6 </p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jals5n3RL5m~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=FK1unso30PpbitRXyfGvT1WL%2Fw8%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jam9hZO0W2~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=zE21UdieSwelr8JpK7UJVeRYfEY%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><strong style="color: blue;">推导2个样本点的线性回归方程</strong></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例7 设有两个点A（x1，y1），B（x2，y2），用最小二乘法推导其线性回归方程并进行分析。</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">解：由最小二乘法，设<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbFdBo7dG5R~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=19B5DnX2pDd6qenE8WzUnPyiBCw%3D" style="width: 100%; margin-bottom: 20px;">，则样本点到该直线的“距离之和”为<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbFv7ukiDwt~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=JTz2Y%2Ff%2BrWO2980gY55P%2BBhMCa4%3D" style="width: 100%; margin-bottom: 20px;"></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">从而可知：当<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbGB3FGcP5h~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=0%2B%2BKnjeWeE%2Fh0rDqPX%2BEPPc4nx8%3D" style="width: 100%; margin-bottom: 20px;">时，b有最小值。将<img src="https://p6-sign.toutiaoimg.com/pgc-image/S64JbGLG6v6crG~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=uWhqi1xbE8XW%2BOnzETSrJXxHitI%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbGVEawef6n~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=K3f0pIlW8QGhDYwy8tNbFYKBXDs%3D" style="width: 100%; margin-bottom: 20px;">代入“距离和”计算式中，视其为关于b的二次函数，再用配方法，可知：<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbbX7nh7bcm~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=eg30csB6%2BlQfEexoVHUH8qiFv3s%3D" style="width: 100%; margin-bottom: 20px;"></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">此时直线方程为：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbbtHaNfgnO~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=Gwos9RUqx2lGbwzpSC9NV7S1yAA%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">设AB中点为M<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jbc44Dry2Nm~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=Vl5P%2B2bZW4igZhy6u1sPzx0AWWo%3D" style="width: 100%; margin-bottom: 20px;">，则上述线性回归方程为</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbcI3RNc5Ng~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=LMhLJm34UzkOyxov07RWzrJDz7M%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">可以看出，由两个样本点推导的线性回归方程即为过这两点的直线方程。这和我们的认识是一致的：对两个样本点，最好的拟合直线就是过这两点的直线。</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">上面我们是用最小二乘法对有两个样本点的线性回归直线方程进行了直接推导，主要是分别对关于a和b的二次函数进行研究，由配方法求其最值及所需条件。实际上，由线性回归系数计算公式：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbcXDQDvo7A~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=mLKF9aw3wraXJF7rLUY1RQBx50o%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">可得到线性回归方程为</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbvbGdU5mB8~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=ANytlOyVTb40R%2BbA4wLUF0%2BA%2Bng%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">设AB中点为M<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jbvq58d9svo~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=sgjeozENjkH%2F2l8iizMkkeQZpY4%3D" style="width: 100%; margin-bottom: 20px;">，则上述线性回归方程为</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64Jbw8RFOJ1P~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=R8uBBHVHyq8zmLAkd1ghgC1%2BRzU%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">。</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><strong style="color: blue;">求回归直线方程</strong></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例8 在硝酸钠的溶解试验中，测得在不同温度<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbwK18BpImL~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=yx8OzQNBemoaCdtLgEwueOdGBYE%3D" style="width: 100%; margin-bottom: 20px;">下，溶解于100份水中的硝酸钠份数<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JbwTBPIgqM8~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=kM8LIxWiqDCa%2Fda6%2FOVdwnwlKu8%3D" style="width: 100%; margin-bottom: 20px;">的数据如下</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">描出散点图并求其回归直线方程.</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">解：建立坐标系，绘出散点图如下：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcAk6nQL1Jo~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=CNgYltSvCW2iNffC76wwzHHQ3yc%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">由散点图可以看出：两组数据呈线性相关性。设回归直线方程为：<img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcAt2veh8JG~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=042up9AExdVPxl0%2BMz1hQLVUPGE%3D" style="width: 100%; margin-bottom: 20px;"></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">由回归系数计算公式：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcB7E6fN7KQ~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=hTfk4KHmK%2B1lkeETNJ1idq6FiiA%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcBLFGau8NN~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=RATI%2BuAVjoPXsjCJtfPyzU17Z2w%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">可求得：b=0.87，a=67.52，从而回归直线方程为：y=0.87x+67.52。</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">三、综合应用</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">例3、假设关于某设备的使用年限x和所支出的维修费用y（万元）有如下统计资料：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcBW3BDIwub~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=HddTkemyvNWCdHekkja7s5p6cyE%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">（1）求回归直线方程；（2）估计使用10年时，维修费用约是多少？</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">解：（1）设回归直线方程为：</p><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcZ2RboeHS~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=xOpdE5i6ynrRa4FQVLMs7QgINPQ%3D" style="width: 100%; margin-bottom: 20px;"><img src="https://p3-sign.toutiaoimg.com/pgc-image/S64JcZa1u9WcFX~noop.image?_iz=58558&from=article.pc_detail&x-expires=1664492639&x-signature=mv0FKmn6KbQweiBTuXALQT1HbfE%3D" style="width: 100%; margin-bottom: 20px;">
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">（2）将x = 10代入回归直线方程可得y = 12.38，即使用10年时的维修费用大约是12.38万元。</p>线性回归方程也是高考常考考点之一，希望同学们能认真学习，掌握线性回归方程的求法及应用，认真体会线性回归方程的求解过程，理解变量间的相关关系，从而体会统计思想在实际生活中的应用及重要．<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">▍ 来源：综合网络</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">▍ 编辑：Wordwuli</p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;">▍ 声明：如有侵权，请联系删除；若需转载，请注明出处。</p>
页:
[1]