两正态分布均值差检验

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2023年12月18日发(作者：)

两正态总体均值差的假设检验基于WolframMathematica,给出了两正态分布Ν[μ1,σ1]、Ν[μ2,σ2]总体均值差μ1-μ2在两总体方差已知、未知但相等、未知但样本量相等、未知但已知方差比、未知近似、未知精确的假设检验方法。"一.两总体方差-σ12=σ102、σ22=σ202已知,Z检验"-定理1:U=X1-X2-(μ1-μ2)σ12Ｎ[0,1],σ12=σ21n1+σ22n2.

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正态分布正态分布统计分析两正态分布均值差检验.nbNeeds["HypothesisTesting`"]需要σ1=1;σ2=2;X1=RandomVariate[NormalDistribution[3,σ1],1000];伪随机变数伪随机变数正态分布正态分布长度长度X2=RandomVariate[NormalDistribution[2,σ2],1500];μ0=1.02;α=0.01;n1=Length[X1];n2=Length[X2];σ12n1σ22n2m=Mean[X1]-Mean[X2];σ=平均值平均值+;u=m-μ0σ;"1.双侧Z检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[NormalDistribution[0,1],Abs[u]]⋯正态分布绝对值备择假设不等ZTest{X1,X2},σ12,σ22,μ0,"TestDataTable",AlternativeHypothesis→"Unequal""2.右侧Z检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[NormalDistribution[0,1],u]⋯Z检验正态分布备择假设大于ZTest{X1,X2},σ12,σ22,μ0,"TestDataTable",AlternativeHypothesis→"Greater""3.左侧Z检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[NormalDistribution[0,1],u]⋯Z检验正态分布备择假设小于ZTest{X1,X2},σ12,σ22,μ0,"TestDataTable",AlternativeHypothesis→"Less"1.双侧Z检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.640232StatisticP-ValueZ0.4673750.6402322.右侧Z检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.320116StatisticP-ValueZ0.4673750.3201163.左侧Z检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.679884StatisticP-ValueZ0.4673750.679884"二.两总体方差σ1-2-=σ22未知,T检验"SW定理2:T=X1-X2-(μ1-μ2)tn1+n2-2，SW=1n1+1n22(n1-1)S21+(n2-1)S2n1+n2-2.

正态分布正态分布统计分析两正态分布均值差检验.nb

3Needs["HypothesisTesting`"]需要σ1=σ2=2;X1=RandomVariate[NormalDistribution[3,σ1],1000];伪随机变数伪随机变数正态分布正态分布X2=RandomVariate[NormalDistribution[2,σ2],1500];μ0=1.05;α=0.01;n1=Length[X1];长度n2=Length[X2];长度m=Mean[X1]-Mean[X2];平均值方差平均值V1=Variance[X1];V2=Variance[X2];方差Sw=t=1n1+1n2(n1-1)V1+(n2-1)V2n1+n2-2;m-μ0Sw;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n1+n2-2],Abs[t]]⋯学生t分布备择假设绝对值不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n1+n2-2],t]⋯T检验学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n1+n2-2],t]⋯T检验学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.181834StatisticP-ValueT-1.33550.1818342.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.909083StatisticP-ValueT-1.33550.9090833.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.0909169

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正态分布正态分布统计分析两正态分布均值差检验.nbStatisticP-ValueT-1.33550.0909169"三.两总体方差σ1定理3:T=SX2、σ22未知,但样本容量n1=n2=n,T检验"nX-(μ1-μ2)tn-1,X=X1-X2.

正态分布正态分布统计分析两正态分布均值差检验.nb

5Needs["HypothesisTesting`"]需要n=1000;X1=RandomVariate[NormalDistribution[3,2],n];伪随机变数伪随机变数正态分布正态分布X2=RandomVariate[NormalDistribution[2,1],n];μ0=1.0;α=0.01;X=X1-X2;m=Mean[X];平均值S=StandardDeviation[X];标准偏差t=m-μ0Sn;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n-1],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n-1],t]⋯学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n-1],t]⋯T检验学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.316987StatisticP-ValueT-0.9836090.3254682.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.841506StatisticP-ValueT-0.9836090.8372663.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.158494StatisticP-ValueT-0.9836090.162734"四.两总体方差σ12、σ2未知,但已知方差比2σ12σ22=r,T检验"

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正态分布正态分布统计分析两正态分布均值差检验.nb定理4:X1-X2-(μ1-μ2)SX1----tn1-1,X1-X2-(μ1-μ2)SX2rn1--tn2-1,1n1+1n2r+1n2X1-X2-(μ1-μ2)1rn+n1r(n1-1)SX12+n+121tn1+n2-2.(n2-1)SX221n2n1+n2-2Needs["HypothesisTesting`"]需要X1=RandomVariate[NormalDistribution[3,2],1200];伪随机变数伪随机变数正态分布正态分布X2=RandomVariate[NormalDistribution[2,1],2500];μ0=1.1;α=0.01;r=4;n1=Length[X1];长度n2=Length[X2];长度m=Mean[X1]-Mean[X2];平均值标准偏差平均值标准偏差S1=StandardDeviation[X1];S2=StandardDeviation[X2];X1-X2-(μ1-μ2)SX11n1--"(一)t(n1-1)"+11n2rSw=S11n1+n2r;t=m-μ0Sw;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n1-1],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n1-1],t]⋯学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n1-1],t]⋯学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]X1-X2-(μ1-μ2)SX2rn1--"(二)t(n2-1)"+1n21n2Sw=S2rn1+;t=m-μ0Sw;

正态分布正态分布统计分析两正态分布均值差检验.nb

7"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n2-1],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n2-1],t]⋯学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n2-1],t]⋯学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]X1-X2-(μ1-μ2)n11+n2r(n1-1)1--"(三)t(n1+n2-2)"SX12+nr1n1+n2-2+1n2(n2-1)SX22Sw=t=1n1+n2-21n1+1n2r(n2-1)S12+rn1+1n2(n2-1)S22;m-μ0Sw;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n2-1],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n2-1],t]⋯学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n2-1],t]⋯学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]X1-X2-(μ1-μ2)SX11n1--(一)t(n1-1)+1n2r1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.0516561StatisticP-ValueT-1.95070.05128092.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.974172StatisticP-ValueT-1.95070.97436

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正态分布正态分布统计分析两正态分布均值差检验.nb3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.0258281StatisticP-ValueT-1.95070.0256405(二)X1-X2-(μ1-μ2)SX2rn1--t(n2-1)+1n21.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.0484695StatisticP-ValueT-1.95070.05128092.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.975765StatisticP-ValueT-1.95070.974363.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.0242347StatisticP-ValueT-1.95070.0256405(三)n111X1-X2-(μ1-μ2)+n2r(n1-1)SX1+nr12--t(n1+n2-2)2+n1+n2-21n2(n2-1)SX21.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.0917784StatisticP-ValueT-1.95070.05128092.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.954111StatisticP-ValueT-1.95070.974363.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.0458892StatisticP-ValueT-1.95070.0256405"五.两总体方差σ1--2、σ22未知,近似T检验"~t(n),n=Round舍入定理5:X-Y-(μ1-μ2)SX1SX+2n1n222SX12n1+SX222SX122n2n1SX222n1-1+n2n2-1.

正态分布正态分布统计分析两正态分布均值差检验.nb

9Needs["HypothesisTesting`"]需要X1=RandomVariate[NormalDistribution[3,2],1800];伪随机变数伪随机变数正态分布正态分布X2=RandomVariate[NormalDistribution[2,1],1000];μ0=1.0;α=0.01;m=Mean[X1]-Mean[X2];平均值方差平均值V1=Variance[X1];V2=Variance[X2];方差n=Round舍入V1n1+V2n22V1n12n1-1+V2n22n2-1;t=m-μ0V1V2+n1n2;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n],t]⋯T检验学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n],t]⋯T检验学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.617823StatisticP-ValueT-0.5401430.5891422.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.691089StatisticP-ValueT-0.5401430.7054293.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.308911StatisticP-ValueT-0.5401430.294571"六.两总体方差σ12、σ22未知,T检验(n1

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正态分布正态分布统计分析两正态分布均值差检验.nb定理6:T=n1n2X1-X2-(μ1-μ2)SXX3i+n1n2n--tn1-1,X3i=X2ii=1,2,⋯,n1,n1n1Xi=X1i-X3-X2,X=1n1Xi,SX=i=11n1-1Xi-X.i=12Needs["HypothesisTesting`"]需要X1=RandomVariate[NormalDistribution[3,2],1200];伪随机变数伪随机变数正态分布正态分布X2=RandomVariate[NormalDistribution[2,1],1500];μ0=1.0;α=0.01;n1=Length[X1];长度n2=Length[X2];长度m1=Mean[X1];平均值m2=Mean[X2];平均值X3=X2[[1;;n1]];m3=Mean[X3];X=X1-Sqrtn1n2X3+Sqrtn1n2m3-m2;平均值平方根平方根m=Mean[X];S=StandardDeviation[X];t=平均值标准偏差m-μ0Sn1;"1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ0"p=21-CDF[StudentTDistribution[n1-1],Abs[t]]⋯学生t分布绝对值备择假设不等TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Unequal"]"2.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ0"p=1-CDF[StudentTDistribution[n1-1],t]⋯学生t分布备择假设大于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Greater"]"3.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ0"p=CDF[StudentTDistribution[n1-1],t]⋯T检验学生t分布备择假设小于TTest[{X1,X2},μ0,"TestDataTable",AlternativeHypothesis→"Less"]1.双侧T检验H0:μ1-μ2=μ0,H1:μ-μ2≠μ00.618077StatisticP-ValueT0.5029170.615092.右侧T检验H0:μ1-μ2≤μ0,H1:μ1-μ2>μ00.309038

正态分布正态分布统计分析两正态分布均值差检验.nb

11StatisticP-ValueT0.5029170.3075453.左侧T检验H0:μ1-μ2≥μ0,H1:μ1-μ2<μ00.690962StatisticP-ValueT0.5029170.692455

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