数学建模,教师的薪金

数学建模,教师的薪金数学建模,教师的薪金数学建模,教师的薪金数学建模,教师的薪金编制仅供参考审核批准生效日期地址：电话：传真:邮编:昆明理工大学第六届大学生数学建模竞赛承诺书我们仔细阅读了昆明理工大学大学生数学建模竞赛的竞赛规则。我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人研究、讨论与赛题有关的问题。我们知道，抄袭别人的成果是违反竞赛规则的。如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。如有违反竞赛规则的行为，我们将受到严肃处理。评阅编号（由组委会评阅前进行编号）：

昆明理工大学第六届大学生数学建模竞赛评阅专用页评阅编号（由组委会评阅前进行编号）：评阅记录（供评阅时使用）：评阅人评分备注总分大学教师薪金模型一、摘要：模型一：利用Matlab建立X1--X7与Z（薪金）的线性关系，得到散点图与一般的线性回归模型（模型一），模型如下：模型二：由模型一整体效果与StepwiseTable图推断部分变量对Z（薪金）的影响并不显著，由残差分析法筛选出影响明显的变量X1、X4，将他们的平方项与交互项加入建立新的回归模型（模型二），模型如下：得到R2，F,P，与模型一相比较。模型一调用原始参数较全，但回归性差，模型二只启用影响明显的变量，回归模型更为显著，可靠度更高。于是得出：教师的薪金与工作时间、学历关系明显，与性别、是否受雇于重点大学、是否接受过培训的关系较小，即女教师没有受到不公正的待遇，婚姻状况也不会影响收入。关键词：回归分析，互交作用，图形结合，残值分析法。

二、问题重述：某地人事部门为研究中学教师的薪金与他们的资历、性别、教育程度及培训情况等因素之间的关系，要建立一个数学模型，分析人事策略的合理性，特别是考察女教师是否受到不公平的待遇，以及她们的婚姻状况是否会影响收入。要求从当地教师中随机选了3414位进行观察后，从中所保留的90个观察对象的数据进行分析。（1）进行变量，建立变量与的回归模型的关系，说明教师薪金与哪些变量的关系密切，是否存在性别和婚姻状况上的差异。（2）除了变量本身之外，尝试将它们的平方项或交叉项加到模型中，建立更好的模型。三、模型假设：1、该地区的人事部门对中学教师的薪金调查是可信的；2、各参数对薪金的影响呈线性关系；3、工作时间、性别、教育程度及培训情况之间相互独立，没有交互作用；四、符号说明：Z：月薪（元）；X1：工作时间（月）；X2：1男性，女性；X3：1男性或单身女性，0已婚女性；X4：学历（取值0-6，值越大表示学历越高）；X5：1受雇于重点大学，0其它；X6：1受过培训的毕业生，0未受过培训的毕业生或受过培训的肄业生；X7：1已两年以上未从事教学工作，0其它。五、分析与建立模型：首先,调用所有相关变量,运用Matlab分别得到，Z与X1--X7之间的关系及散点图,由此知Z与各变化量呈线性关系，于是可以建立线性回归模型：Z（薪金）为因变量,X1--X7分别表示对Z的值产生影响的各个变量,表示回归系数,表示随机变量.用Matlab求解模型（见附录），得到的值与置信区间如下：参数参数估计值置信区间1.1311[1.02681.2353]0.0027[0.00230.0031]-0.0229[-0.14320.0974]0.0094[-0.10050.1193]0.1089[0.02960.1882]0.0385[-0.06700.1440]0.1817[-0.05070.4142]0[00]=0.7889F=51.6934P=0表一由上表中=0.7889可以知道薪金（）的78.89%可由模型确定,由P=0远小于0.05,模型从整体上看是成立的,但是还可以看到一个问题，即些部分的置信区间包含0点,因此我们可以知道这些变量对因变量的影响是不显著的.在Matlab中运行stepwise命令得到下图：由图可知，在模型中对因变量的影响是不显著的.于是只保留，并将它们的交互项和平方项加入，建立逐步线性回归方程如下：然后使用Matlab求解模型（程序见附录），得到值与置信区间如下：参数参数估计值置信区间6.9026[6.85576.9496]0.0043[0.00370.0049]0.1746[0.10730.2419]-0.0001[-0.00030.0001]-0.0000[-0.0000-0.0000]-0.0228[-0.0371-0.0085]=0.9008F=152.6081P=0通过新建模型中得到的数据,可以看到明显提高,薪金的90.08%可由模型确定.远小于0.05,F远超过临界值，回归模型更为显著，可靠度增高.然后进行残差分析，在Matlab中运行命令rcoplot得到残差图如下：由图可知,除个别数据外,其他数据的残差离零点均较近,且残差的置信区间都包含零点.这说明回归模型能较好地符合原始数据，而个别异常点可以忽略.六、模型的评价优点:该方案实用简单，可行性强，模型简单，易于理解。模型一首先用简单的线性规划进行分析.结构简单，计算方便，有利于对相似问题进行求解和对模型进行扩充。模型二的建立是从一般问题到特殊问题的发展过程.根据已知的数据，从常识和经验进行初步分析，并运用了逐步线性回归方法以及辅作散点图，决定取那几个回归变量及它们的函数形式.把对影响不显著的变量（）予以排除，又运用残值分析法建立新的回归模型.使得精确值增高，模型更合理.缺点：1该模型在处理此问题时有假设与理想化的思想，与实际问题的求解还有所差距.比如所求模型结果只达到了模型设想的80%左右.七、参考文献【01】赵静，数学建模与数学实验，北京，高等教育出版社，2003【02】苏彦华，MATLAB7.0从入门到精通，北京，人民邮电出版社，2010

八、附录：薪金模型数据表：编号ZX1X2X3X4X5X6X7199870000002101514110000310281811010041250191100005102819010100610281900000071018270000018107230000000912903011000010120430010000111352310120101212043100010013110438000000141118411100001511274200000016125942110100171127421100001811274200010019109547000001201113520000012114625201201022118254110000231404540001002411825400000025159455112110261459660001002712376711010028123767010100291496750100003014247811010031142479010000321347911101003313429200000134131094000100351814103002110361534103000000371430103110000381439111110100391946114113110402216114114110411834114114111421416117000001432052139110100442087140002111452264154002111462201158114011472992159115111481695162010000491792167110100501690173000001511827174000001522604175112110531720199010000541720209000000552159209014100561852210010000572104213110100581852220000001591852222000000602210222110000612266223010000622027223110000631852227000100641852232000001651995235000001662616245113110672324253110100681852257010001692054260000000702617284113110711948287110000721720290010001732604308112110741852309110101751942319000100762027325110000771942326110100781720329110100792048337000000802334346112111811720355000001821942357110000832117380110001842742387112111852740403112111861942406110100872266437010000882436453010000892067458010000902000464112110与Z的关系及散点图：>>x1=[7141819191927303030313138414242424247525254545455666767757879919294103103103111114114114117139140154158159162167173174175199209209210213220222222223223227232235245253257260284287290308309319325326329337346355357380387403406437453458464]';|>>X1=[ones(90,1)x1];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';>>x2=[011100001000010110000100101001010000111110100110100100001001010001100110110111010111110001]';>>X2=[ones(90,1)x2];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';x3=[011110001110010110001100101111110000111110100111100110111001110001110111110111010111111111x3=[011110001110010110001100101111110000111110100111100110111001110001110111110111010111111111]';>>X3=[ones(90,1)x3];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';>>x4=[000000000020000000002000200000000020003440022450000200400000000003000300200000020002200002]';>>X4=[ones(90,1)x4];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';x5=[001010000001000101000010111101010110011110111010100100101000001001100100111011010001110001]';>>X5=[ones(90,1)x5];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';>>x6=[000000000010000000001000100000000010001110011110000100000000000001000100100000010001100001]';>>X6=[ones(90,1)x6];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';>>x7=[000000100000000000110000000000001000000011011110011000000100000110010001010000011011100000]';>>X7=[ones(90,1)x7];>>Z=[99810151028125010281028101810721290120413521204110411181127125911271127109511131462118214041182159414591237123714961424142413471342131018141534143014391946221618341416205220872264220129921695179216901827260417201720215918522104185218522210226620271852185219952616232418522054261719481720260418521942202719421720204823341720194221172742274019422266243620672000]';模型一程序：>>x1;>>x2;>>x3;>>x4;>>x5;>>x6;>>x7;>>x=[x1,x2,x3,x4,x5,x6,x7]; >>stepwise(x,z)>>[b,bint,r,rint,stats]=regress(z,x)Warning:R-squareandtheFstatisticarenotwell-definedunlessXhasacolumnofones.Type"helpregress"formoreinformation.>Inregressat162b=1.0e+003*1.13110.0027-0.0229-0.02290.10890.03850.1817bint=1.0e+003*1.02681.23530.00230.0031-0.14320.0974-0.10050.11930.02960.1882-0.06700.1440-0.05070.41420.00000.0000r=1.0e+003*1.65450.84440.18550.8335-0.13031.62840.87311.60450.80960.49590.75090.95221.58710.78571.57840.13340.78350.92830.82960.81870.70530.75750.90221.55240.36230.87620.0791-0.23460.39810.05520.38940.02690.73180.81531.05301.44590.6510-0.01650.34330.4525-0.28550.6775-0.07730.23460.20420.2691-0.27410.2091-0.13820.55590.55370.10160.12871.2156-0.10650.1048-0.23810.45381.18740.39250.07660.39030.52640.42770.42120.0587-0.3250-0.73521.1048-0.02600.2513-0.8069-0.1873-1.18460.32650.1687-0.4836-0.49010.9376-1.00780.16050.0992-0.6887-1.0969-1.1316-0.6574-0.3883-0.4231-0.4339-0.5262rint=1.0e+003*0.17483.1341-0.62322.3119-1.28301.6541-0.63502.3020-1.54681.28610.14763.1092-0.59152.33760.12273.0863-0.66092.2801-0.96481.9566-0.63062.1323-0.51462.41900.10473.0695-0.68672.25810.09573.0611-1.33991.6067-0.68902.2561-0.53952.3961-0.63832.2975-0.65002.2874-0.68032.0908-0.71702.2319-0.56652.37100.06903.0358-1.08891.8135-0.59342.3457-1.39841.5565-1.66071.1915-1.06991.8662-1.42381.5342-1.07921.8580-1.45381.5077-0.74192.2056-0.65542.2860-0.36922.4751-0.03852.9304-0.82912.1311-1.49941.4664-1.12311.8097-0.96791.8728-1.69541.1244-0.79842.1535-1.56241.4078-1.19141.6607-1.22271.6311-1.10371.6418-1.59581.0476-1.26601.6842-1.62451.3482-0.92212.0338-0.92432.0317-1.36171.5650-1.34651.6040-0.26102.6923-1.12020.9073-1.37011.5798-1.72441.2482-1.02291.9305-0.28732.6621-1.08921.8742-1.39791.5511-1.09131.8719-0.93641.9892-1.04821.9036-1.05451.8969-1.40651.5240-1.80921.1592-2.16470.6943-0.36302.5727-1.48671.4347-1.22391.7264-2.23590.6221-1.64301.2684-2.61240.2432-1.11771.7708-1.30011.6376-1.95880.9916-1.96480.9846-0.51082.3859-2.44630.4306-1.29721.6182-1.36301.5614-2.12750.7501-2.53050.3368-2.5628