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数理统计上机 报 告上机实验题目：用R软件进行一元线性回归上机实验目的：1、 进一步理解假设实验的基本思想，学会使用实验检验和进行统计推断。2、 学会利用R软件进行假设实验的方法。一元线性回归基本理论、方法：基本理论：假设预测目标因变量为Y，影响它变化的一个自变量为X，因变量随自变量的增（减）方向的变化。一元线性回归分析就是要依据一定数量的观察样本（Xi, Yi），i=1，2，n，找出回归直线方程Y=a+b*X方法：对应于每一个Xi，根据回归直线方程可以计算出一个因变量估计值Yi。回归方程估计值Yi 与实际观察值Yj之间的误差记作e-i=Yi-Yi。显然，n个误差的总和越小，说明回归拟合的直线越能反映两变量间的平均变化线性关系。据此，回归分析要使拟合所得直线的平均平方离差达到最小，据此，回归分析要使拟合所得直线的平均平方离差达到最小，简称最小二乘法将求出的a和b代入式（1）就得到回归直线Yi=a+bXi 。那么，只要给定Xi值，就可以用作因变量Yi的预测值。（一）实验实例和数据资料：有甲、乙两个实验员，对同一实验的同一指标进行测定，两人测定的结果如下：实验号12345678甲4.33.23.83.53.54.83.33.9乙3.74.13.83.84.63.92.84.4试问：甲、乙两人的测定有无显著差异？取显著水平=0.05.上机实验步骤：（1） 设置假设：H0：u1-u-2=0：H1：u1-u-20（2） 确定自由度为n1+n2-2=14；显著性水平a=0.05（3） 计算样本均值样本标准差和合并方差统计量的观测值alpha-0.05;n1-8;n2-8;x-c(4.3,3.2,3.8,3.5,3.5,4.8,3.3,3.9);y-c(3.7,4.1,3.8,3.8,4.6,3.9,2.8,4.4);var1-var(x); xbar-mean(x); var2-var(y); ybar-mean(y); Sw2-(n1-1)*var1+(n2-1)*var2)/(n1+n2-2) t-(xbar-ybar)/(sqrt(Sw2)*sqrt(1/n1+1/n2);tvalue-qt(alpha,n1+n2-2)；（4） 计算临界值：tvalue-qt(alpha,n1+n2-2)（5） 比较临界值和统计量的观测值，并作出统计推断实例计算结果及分析：alpha n1 n2 x y var1 xbar var2 ybar Sw2 t var11 0.2926786 xbar1 3.7875 var21 0.2926786 ybar1 3.8875Sw21 0.2926786 t1 -0.3696873 tvalue1 -1.76131分析：t=-0.3696873tvalue=-1.76131，所以接受假设H1即甲乙两人的测定无显著性差异。(二)实验实例和数据资料：2.某型号玻璃纸的横向延伸率要求不低于65%，且其服从正态分布，现对一批该批号的玻璃纸测得100个数据如下： （x%横向延伸率）35.537.539.541.543.545.547.549.551.553.555.557.559.561.563.5频数7811991217145320201上机实验步骤：(1)设置假设：H0：u=65, H1：u65.(2)确定自由度为n=100-1=99；显著性水平a=0.05(3) 输入数据x-c(35.5,35.5,35.5,35.5,35.5,35.5,35.5,37.5,37.5,37.5,37.5,37.5,37.5,37.5,37.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,51.5,51.5,51.5,51.5,51.5,53.5,53.5,53.5,55.5,55.5,59.5,59.5,63.5)（4）用R软件计算临界值（5）比较临界值和统计量的观测值，并作出推断实例计算结果及分析：计算过程如下：alpha-0.05;n-100; x-c(35.5,35.5,35.5,35.5,35.5,35.5,35.5,37.5,37.5,37.5,37.5,37.5,37.5,37.5,37.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,39.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,41.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,43.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,45.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,47.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,49.5,51.5,51.5,51.5,51.5,51.5,53.5,53.5,53.5,55.5,55.5,59.5,59.5,63.5)sd1-sd(x); xbar-mean(x);t-(xbar-65.0)/(sd1/sqrt(n); tvalue-qt(alpha,n-1); sd11 5.815896xbar1 45.06 t1 -34.28534tvalue1 -1.660391分析推断：因为t=-34.28534alpha x y n1 n2 xbar=mean(x); ybar=mean(y); sw q left right n11 16 n21 16 left1 7.819162 right1 8.680838所以置信区间【7.819162，,8.680838】
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