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【统计学第八章】假设检验(全)

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参数估计与假设检验的区别

参数估计与假设检验是统计推断的两个组成部分，都是利用样本信息对总体进行推断，但是角度不同。
参数估计是样本统计量估计总体参数的方法，总体参数在估计前是未知的。
假设检验是先对总体参数提出一个假设，后用样本信息去验证这个假设是否正确。
参数估计不知道总体的信息，也不进行假定，假设不知道总体，需要验证信息。

1.假设检验的基本思想和原理
2.假设检验的步骤
3.一个总体参数的检验
4.两个总体参数的检验
5.P值的计算与应用
6.用EXCEL进行检验

假设检验的基本问题

1.假设问题的提出
2.假设的表达式
3.两类错误
4.假设检验的流程
5.利用P值进行决策

什么是假设？
对总体参数的具体数值所做的陈述
-总体参数包括总体均值、比例、方差等
-分析之前必需陈述

什么是假设检验
1.先对总体的参数(或分布形式)提出某种假设，然后利用样本信息判断假设是否成立的过程
2.有参数检验和非参数检验
3.逻辑上运用反证法，统计上依据小概率原理

原假设
1.研究者想收集证据予以反对的假设
2.又称为"0假设"
3.总是有符号=,≤或≥
4.表示为H0​
H0​:\mu=某一数值
指定为符号=,≤或≥
例如 H0​:μ=10cm
一般情况下，记原假设为H0​所谓为零假设
备择假设
1.研究者想收集证据予以支持的假设
2.也称为研究假设
3.总是有符号 !=,≤或≥
4.表示为H1​
H1​:μ<某一数值，或μ>某一数值
例如：H1​:μ<10cm,或mu>10cm
原假设和备择假设是属于非此即彼的关系，一般来说，原假设是包含等于的
提出假设
1.原假设和备择假设是一个完备事件组，而且相互独立
在一项假设检验中，原假设和备择假设必有一个条件成立，而且只有一个成立
2.先确定备择假设，再确定原假设
3.等号=总是放在原假设上
4.因研究目的不同，对同一问题可能提出不同的假设(也可能得出不同的结论)
双侧检验和单侧检验
1.备择假设没有特定的方向性，并含有符号≠的假设检验，称为双侧检验或双尾检验
2.备择假设具有特定的方向性，并含有符号≤和\req的假设检验，称为单侧检验或单尾检验

  • 备择假设的方向为<,称为左侧检验
  • 备择假设的方向为>,称为右侧检验

双侧检验和单侧检验的假设形式

假设双侧检验单侧检验(左侧)单侧检验(右侧)
原假设 H0​:μ=μ0​ H0​:μ≥μ0​ H0​:μ≤μ0​
备择假设 H1​:μ≠μ0​ H1​:μ<μ0​ H1​:μ>μ0​

假设检验中的两类错误

  • 1.第I类错误(弃真错误)
  • 原假设为真，拒绝原假设
    第I类错误的概率记为α,被称为显著性水平
    2.第II类错误(取伪错误)
  • 原假设为假设时未拒绝原假设
  • 第II类错误的概率记为β
    两类错误时此消彼长的关系，不可能同时达到平衡，需要有所取舍

影响beta错误的因素

  • 1.总体参数的真值
  • 随着假设的总体参数的减少而增大
    2.显著性水平α
    当α减少时增大
    3.总体标准差
    当α增大时增大
    4.样本容量n
    当n减少时增大

显著性水平α
1.是一个概率值
2.原假设为真时，拒绝原假设的概率被称为抽样分布的拒绝域
表示为α

  • 常用的α有0.01,0.05,0.1
    由研究者提出

假设检验中的小概率原理
什么是小概率？
1.在一次试验中，一个几乎不可能发生的事件发生的概率
2.在一次试验中小概率事件一旦发生，就有理由拒绝原假设
3.小概率由研究者事先确定

检验统计量
1.根据样本观测结果计算得到的，并据以对原假设和备择假设作出决策的某个样本统计量
2.对样本估计量的标准化结果

  • 原假设H0​为真
  • 点估计量的抽样分布

3.标准化的检验统计量
标准化检验统计量 ={点估计量-假设值}/{点估计量的抽样标准差}

决策规则
1.给定显著性水平区间下，查表的出相应的临界值z或z2α​​,tm​或t2α​​
2.将检验统计量的值与显著性水平的临界值进行比较
3.做出决策

  • 双侧检验 统计量>临界值，拒绝H0​
  • 左侧检验 统计量<- 临界值，拒绝H0​
  • 右侧检验 统计量>临界值，拒绝H0​

什么是P值
1.在原假设为真的条件下，检验统计量的观察值大于或等于其计算值的概率

  • 双侧检验为分布中两侧面积的总和

2.反映实际观测到的数据与原假设H0​之间不一致的程度
3.被称为观察到的(或实测的)显著性水平
3.决策规则 若P值<α,拒绝原假设

假设检验的结论表述

1.假设检验的目的在于试图找到拒绝原假设，而不在于证明什么是正确的
2.拒绝原假设时结论是清楚的

  • 例如 H0​:μ=10,拒绝H0​时，可以说μ≠10

3.当不拒绝原假设时，

  • 并未给出明确的结论
  • 不能说原假设是正确的，也不能说它不是正确的
  • 例如，当不拒绝H0​:μ=10,并未说它就是10，但也未说它不是10.只能说样本提供的证据不足以推翻原假设。

假设检验步骤的总结

1.陈述原假设和备择假设
2.从研究的总体中抽出一个随机样本
3.确定一个适当的检验统计量，并利用样本统计依据算出具体数值
4.确定一个适当的显著性水平，并计算出其临界值，指定拒绝域
5.将统计量的值域临界值进行比较，做出决策

  • 统计量的值落在拒绝域，拒绝H0​，否则不拒绝H0​
  • 也可以直接利用P值做出决策

一个总体参数的检验

检验统计量的确定
总体均值的检验
总体比例的检验
总体方差的检验

总体均值的检验---大样本

1.假定条件
正态总体或非正态总体大样本n ≥ 30
2.使用z检验统计量
α2 已知 z=n​σ​x−μ0​​ N(0,1)
α2 未知 z=n​s​x−μ0​​ N(0,1)

总体均值的检验---大样本检验方法的总结

假设双侧检验左侧检验右侧检验
假设形式 H0​:μ=μ0​ H0​:μ≥μ0​ H0​:μ≤μ0​
假设形式 H1​:μ≠μ0​ H1​:μ<μ0​ H1​:μ>μ0​
统计量 α已知 z=n​σ​x−μ0​​  
统计量 α未知 z=n​s​x−μ0​​  
拒绝域 z>z2α​​ z<−zα​ z>zα​
P值决策 P<α 拒绝H0​    

总体均值的检验---小样本

1.假定条件

  • 总体服从正态分布
  • 小样本(n<30)

2.检验统计量
α2已知 z=n​σ​x−μ0​​~N(0,1)
α2未知 z=n​s​x−μ0​​~t(0,1)

总体均值的检验---小样本检验方法的总结

假设双侧检验左侧检验右侧检验
假设形式 H0​:μ=μ0​ H0​:μ≥μ0​ H0​:μ≤μ0​
假设形式 H1​:μ≠μ0​ H1​:μ<μ0​ H1​:μ>μ0​
统计量 α已知 z=n​σ​x−μ0​​  
统计量 α未知 z=n​s​x−μ0​​  
拒绝域 t>t2α​​(n−1) t<−tα​(n−1) t>tα​(n−1)
P值决策 P<α 拒绝H0​    

总体比例检验

假定条件

  • 总体服从二项分布
  • 可用正态分布来近似(大样本)
    检验的z统计量
    z=npi0​(1−π0​)​​p−π0​​
    pi0​为假设的总体比例

总体比例的检验

假设双侧检验左侧检验右侧检验
假设形式 H0​:μ=μ0​ H0​:μ≥μ0​ H0​:μ≤μ0​
假设形式 H1​:μ≠μ0​ H1​:μ<μ0​ H1​:μ>μ0​
统计量 z=nπ0​(1−π0​)​​p−π0​​    
统计量      
拒绝域 z>z2α​​ z<−aα​ z>zα​
P值决策 P<α 拒绝H0​    

总体方差的检验

总体方差是服从卡方分布的，检验一个总体的方差或标准差，假设总体近似服从正态分布，使用χ2
检验统计量
χ2=σ02​(n−1)s2​~χ2(n−1)
其中s为样本方差，σ为假设总体方差

两个总体参数的检验

两个统计量的确定
两个总体均值之差的检验
两个总体比例之差的检验
两个总体方差之比的检验
检验中的匹配样本

两个总体均值之差的检验(独立大样本)

1.假定条件

  • 两个样本是独立的随机样本
  • 正态总体或非正态总体大样本(n_1 \geq 30 n_2 \req 30)检验统计量
    σ12​,σ22​已知z=n1​σ12​​+n2​σ22​​​(x1​​−x2​​)−(μ1​−μ2​)​~N(0,1)
    σ12​,σ22​未知z=n1​σ12​​+n2​σ22​​​(x1​​−x2​​)−(μ1​−μ2​)​~N(0,1)

两个总体均值之差的检验(独立小样本)

1.假定条件

  • 两个独立的小样本
  • 两个总体都是正态分布
    σ12​、σ22​已知
    检验统计量
    z=n1​σ12​​+n2​σ22​​​(x1​​−x2​​−(μ1​−μ2​))​~N(0,1)

两个总体均值之差的检验

\sigma_1^2,\sigma_2^2未知但是\sigma_1^2=\sigma_2^2$$
1.假定条件

  • 两个独立的小样本
  • 两个总体都是正态分布
  • σ12​、σ22​未知但是相等，即σ12​=σ22​
    2.检验统计量
    t=sp​n1​1​+n2​1​​(x1​​−x2​​)−(μ1​−μ2​)​
    其中sp2​=n1​+n2​−2(n1​−1)s12​+(n2​−1)s22​​

两个总体均值之差的检验

σ12​,σ22​未知且不相等σ12​≠σ22​
假定条件

  • 两个总体都是正态分布
  • σ22​,σ22​未知且不相等，即σ12​≠σ22​
  • 样本容量不相等，即n1​≠n2​
    检验统计量
    t=n1​s12​​+n2​s22​​​(x1​​−x2​​)−(μ1​−μ2​)​
    自由度 v=n1​−1n22​s12​​​+n2​−1n2​s22​​​(n1​s12​​+n2​s22​​)​

两个总体均值之差的检验(匹配样本)

假定条件
两个总体配对差值构成的总体服从配对差是由差值总体中随机抽取的数据配对或匹配(重复测量(前/后))
检验统计量
t=nd​​Sd​​d−d0​​~t(n−1)
样本差值均值d=nd​∑i=1n​di​​

两个总体比例之差的检验

假定条件
两个总体都服从二项分布
可以用正态分布来近似检验统计量
检验H0​,π1​−π2​=0,z=p(1−p)(n1​1​+n2​1​)​p1​−p2​​
检验H0​,π1​−π2​=d0​,z=n1​p1​(1−p1​)​+n2​p2​(1−p2​)​​(p1​−p2​)−d0​​

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posted on 2019-06-25 14:51  pandaboy1123  阅读(1675)  评论(0)  编辑  收藏  举报

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