数理统计课程设计.doc

题目3. 求正态总体的均值差和方差的置信区间. 通过实验加深对统计推断的基本概念的和基本思想的理解. 置信水平0.95，0.90。取不同值。模拟次数大于100次。、问题分析：1在Matlab中模拟均值、标准差的产生由于要求模拟次数大于100次，不妨就模拟120次，且要求，每次取不同的值，因此可以在Matlab中通过随机数产生器函数normrnd(mu,sigma,m,n)得到120个（，）的不同组合。2置信区间的求法总体参数的点估计作为待估参数的近似值给出了明确的数量描述,在统计分析中有多方面的应用. 但点估计没有给出这种近似的精确程度和可信程度,使其在实际应用中受到很大的限制,区间估计却可以弥补这一不足。所谓区间估计，就是用两个估计量与估计未知参数，使得随机区间能够包含未知参数的概率为指定的。即： 称满足上述条件的区间为的置信区间，称为置信水平。称为置信下限，称为置信上限。 标准差已知时正态总体均值的区间估计问题假设检验：H0 : = 0 H1 : ；（已知）检验统计量为，H0成立时拒绝域：查表求满足：对0于，。所以对于总体中的样本，的置信区间为： 其中可以用Matlab中的命令norminv(1-a /2)计算。 均值已知时正态总体标准差的区间估计问题假设检验：H0 : H1 : ； = 0（已知）检验统计量为:,当H0成立时，由此可查临界值表，构造拒绝域。由于，查表求临界值与，其中查表可用Matlab中的命令chi2inv求出：c1 = chi2inv(1-alpha/2,n), = chi2inv(alpha/2 ,n)，使得则的置信区间为 。 、算法步骤：下面程序会用到的命令和变量的含义:alpha为显著性水平 mu- 均值的取值 sigma-标准差的取值 muci-均值的区间估计 sigmaci-标准差的区间估计normrnd(，1，1)-产生一个服从正态分布N（，）的随机数1用随机数产生函数rand（）产生，的一系列随机值；2由每一对，的值，用函数normrnd（）产生服从正态分布，的500个样本；3根据，置信度为的置信区间的求法，求出置信区间即可。程序如下:n=500;%样本容量for i=1:120 %模拟120次 j=i %记录模拟的次数 mu=10*rand(1,1);%产生一次模拟的均值的随机值 sigma=50*rand(1,1);%产生一次模拟的标准差的随机值 rd=normrnd(mu,sigma,1,500);%产生服从一次模拟时正态分布N（mu,sigma）的500个随机数 mu,sigma%输出每一次模拟产生的mu随机值和sigma随机值mu = mean (rd);%计算样本均值for alpha=0.05,0.1%显著水平为0.05,0.1 alphau = norminv(1-alpha/2,0,1); %计算置信度为1 - alpha/ 2 的正态分布临界值chi2 = sum( (rd - mu).2); %计算离差的平方和lambda1 = chi2inv(1-alpha/2,n); %计算卡方分布的临界值lambda2 = chi2inv(alpha/2 ,n);muci = mu-u*sqrt(sigma2/n),mu+u*sqrt(sigma2/n) %计算均值的置信区间sigmaci = sqrt(chi2/lambda1) ,sqrt(chi2/lambda2) %计算方差的置信区间endend运行结果（部分数据如下）：j = 1ans = 9.5013 11.5569alpha = 0.0500muci = 7.7609 9.7869sigmaci = 10.2928 11.6526alpha = 0.1000muci = 7.9238 9.6240sigmaci = 10.3921 11.5325j = 2ans = 6.0684 24.2991alpha = 0.0500muci = 3.3742 7.6340sigmaci = 21.5052 24.3462alpha = 0.1000muci = 3.7166 7.2915sigmaci = 21.7127 24.0953j = 3ans = 8.9130 38.1048alpha = 0.0500muci = 6.5808 13.2608sigmaci = 37.5508 42.5116alpha = 0.1000muci = 7.1178 12.7238sigmaci = 37.9131 42.0734j = 4ans = 4.5647 0.9252alpha = 0.0500muci = 4.5434 4.7056sigmaci = 0.8838 1.0005alpha = 0.1000muci = 4.5564 4.6925sigmaci = 0.8923 0.9902j = 5ans = 8.2141 22.2352alpha = 0.0500muci = 7.8733 11.7712sigmaci = 21.6469 24.5066alpha = 0.1000muci = 8.1866 11.4578sigmaci = 21.8557 24.2540j = 6ans = 6.1543 39.5969alpha = 0.0500muci = 1.2232 8.1648sigmaci = 37.7223 42.7057alpha = 0.1000muci = 1.7813 7.6067sigmaci = 38.0863 42.2656j = 7ans = 9.2181 36.9104alpha = 0.0500muci = 7.2870 13.7576sigmaci = 34.8475 39.4511alpha = 0.1000muci = 7.8072 13.2374sigmaci = 35.1837 39.0445j = 8ans = 1.7627 20.2853alpha = 0.0500muci = 0.3363 3.8924sigmaci = 18.6959 21.1658alpha = 0.1000muci = 0.6222 3.6065sigmaci = 18.8763 20.9476j = 9ans = 9.3547 45.8452alpha = 0.0500muci = 3.9887 12.0256sigmaci = 44.6448 50.5427alpha = 0.1000muci = 4.6348 11.3795sigmaci = 45.0755 50.0218j = 10ans = 4.1027 44.6825alpha = 0.0500muci = 1.2412 9.0743sigmaci = 42.7757 48.4267alpha = 0.1000muci = 1.8709 8.4446sigmaci = 43.1884 47.9276j = 11ans = 0.5789 17.6434alpha = 0.0500muci = -1.3928 1.7001sigmaci = 17.1260 19.3884alpha = 0.1000muci = -1.1442 1.4515sigmaci = 17.2912 19.1886j = 12ans = 8.1317 0.4931alpha = 0.0500muci = 8.0846 8.1710sigmaci = 0.4782 0.5413alpha = 0.1000muci = 8.0915 8.1640sigmaci = 0.4828 0.5358j = 13ans = 1.3889 10.1383alpha = 0.0500muci = 0.8506 2.6279sigmaci = 9.3635 10.6005alpha = 0.1000muci = 0.9935 2.4850sigmaci = 9.4539 10.4913j = 14ans = 1.9872 30.1896alpha = 0.0500muci = -2.1245 3.1679sigmaci = 29.1724 33.0263alpha = 0.1000muci = -1.6991 2.7424sigmaci = 29.4539 32.6859j = 15ans = 2.7219 9.9407alpha = 0.0500muci = 1.0677 2.8103sigmaci = 9.0980 10.2999alpha = 0.1000muci = 1.2078 2.6702sigmaci = 9.1858 10.1938j = 16ans = 0.1527 37.3393alpha = 0.0500muci = -0.1283 6.4175sigmaci = 33.3635 37.7711alpha = 0.1000muci = 0.3979 5.8913sigmaci = 33.6854 37.3818j = 17ans = 4.4510 46.5907alpha = 0.0500muci = -1.0942 7.0733sigmaci = 43.3399 49.0654alpha = 0.1000muci = -0.4377 6.4168sigmaci = 43.7580 48.5597j = 18ans = 4.6599 20.9325alpha = 0.0500muci = 2.7472 6.4167sigmaci = 20.3591 23.0487alpha = 0.1000muci = 3.0422 6.1218sigmaci = 20.5556 22.8112j = 19ans = 8.4622 26.2576alpha = 0.0500muci = 7.3293 11.9324sigmaci = 24.0194 27.1925alpha = 0.1000muci = 7.6993 11.5624sigmaci = 24.2511 26.9122j = 20ans = 2.0265 33.6069alpha = 0.0500muci = -1.9581 3.9334sigmaci = 31.2406 35.3677alpha = 0.1000muci = -1.4845 3.4598sigmaci = 31.5420 35.0031- 11 -(三)、算法分析与总结：本程序主要运用了for循环语句实现了100次以上的模拟，它能较好的解决由每一对随机值，对应的产生500个样本的问题。另一种想法是可以由随机数产生函数直接产生关于，的一个数组，但接下来遇到了问题，无法用正态分布产生函数得到与每一对，值相对应的正态样本，这就无法求出置信区间了。本程序在for语句的前提下，可以改进的是直接用Matlab中求置信区间函数求解，那样会简单很多，但为了说明置信区间的求解过程，选用的是本程序。对于求均值和标准差的点估计和区间估计的算法还可以进行改进：muhat ,sigmahat ,muci ,sigmaci = normfit (rd ,0.05);此时可得到改进的更简单的程序：n=500;%样本容量for i=1:120 %模拟100次 j=i %记录模拟的次数 mu=10*rand(1,1);%产生一次模拟的均值的随机值 sigma=50*