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參數估計與假設檢驗的區別

參數估計與假設檢驗是統計推斷的兩個組成部分，都是利用樣本信息對總體進行推斷，但是角度不同。
參數估計是樣本統計量估計總體參數的方法，總體參數在估計前是未知的。
假設檢驗是先對總體參數提出一個假設，后用樣本信息去驗證這個假設是否正確。
參數估計不知道總體的信息，也不進行假定，假設不知道總體，需要驗證信息。

1.假設檢驗的基本思想和原理
2.假設檢驗的步驟
3.一個總體參數的檢驗
4.兩個總體參數的檢驗
5.P值的計算與應用
6.用EXCEL進行檢驗

假設檢驗的基本問題

1.假設問題的提出
2.假設的表達式
3.兩類錯誤
4.假設檢驗的流程
5.利用P值進行決策

什么是假設？
對總體參數的具體數值所做的陳述
-總體參數包括總體均值、比例、方差等
-分析之前必需陳述

什么是假設檢驗
1.先對總體的參數(或分布形式)提出某種假設，然后利用樣本信息判斷假設是否成立的過程
2.有參數檢驗和非參數檢驗
3.邏輯上運用反證法，統計上依據小概率原理

原假設
1.研究者想收集證據予以反對的假設
2.又稱為"0假設"
3.總是有符號=,≤或≥
4.表示為H0?
H0?:mu=某一數值
指定為符號=,≤或≥
例如 H0?:μ=10cm
一般情況下，記原假設為H0?所謂為零假設
備擇假設
1.研究者想收集證據予以支持的假設
2.也稱為研究假設
3.總是有符號 !=,≤或≥
4.表示為H1?
H1?:μ<某一數值，或μ>某一數值
例如：H1?:μ<10cm,或mu>10cm
原假設和備擇假設是屬于非此即彼的關系，一般來說，原假設是包含等于的
提出假設
1.原假設和備擇假設是一個完備事件組，而且相互獨立
在一項假設檢驗中，原假設和備擇假設必有一個條件成立，而且只有一個成立
2.先確定備擇假設，再確定原假設
3.等號=總是放在原假設上
4.因研究目的不同，對同一問題可能提出不同的假設(也可能得出不同的結論)
雙側檢驗和單側檢驗
1.備擇假設沒有特定的方向性，并含有符號≠的假設檢驗，稱為雙側檢驗或雙尾檢驗
2.備擇假設具有特定的方向性，并含有符號≤和eq的假設檢驗，稱為單側檢驗或單尾檢驗

備擇假設的方向為<,稱為左側檢驗
備擇假設的方向為>,稱為右側檢驗

雙側檢驗和單側檢驗的假設形式

假設	雙側檢驗	單側檢驗(左側)	單側檢驗(右側)
原假設	H0?:μ=μ0?	H0?:μ≥μ0?	H0?:μ≤μ0?
備擇假設	H1?:μ≠μ0?	H1?:μ<μ0?	H1?:μ>μ0?

假設檢驗中的兩類錯誤

1.第I類錯誤(棄真錯誤)
原假設為真，拒絕原假設
第I類錯誤的概率記為α,被稱為顯著性水平
2.第II類錯誤(取偽錯誤)
原假設為假設時未拒絕原假設
第II類錯誤的概率記為β
兩類錯誤時此消彼長的關系，不可能同時達到平衡，需要有所取舍

影響beta錯誤的因素

1.總體參數的真值
隨著假設的總體參數的減少而增大
2.顯著性水平α
當α減少時增大
3.總體標準差
當α增大時增大
4.樣本容量n
當n減少時增大

顯著性水平α
1.是一個概率值
2.原假設為真時，拒絕原假設的概率被稱為抽樣分布的拒絕域
表示為α

常用的α有0.01,0.05,0.1
由研究者提出

假設檢驗中的小概率原理
什么是小概率？
1.在一次試驗中，一個幾乎不可能發生的事件發生的概率
2.在一次試驗中小概率事件一旦發生，就有理由拒絕原假設
3.小概率由研究者事先確定

檢驗統計量
1.根據樣本觀測結果計算得到的，并據以對原假設和備擇假設作出決策的某個樣本統計量
2.對樣本估計量的標準化結果

原假設H0?為真
點估計量的抽樣分布

3.標準化的檢驗統計量
標準化檢驗統計量 ={點估計量-假設值}/{點估計量的抽樣標準差}

決策規則
1.給定顯著性水平區間下，查表的出相應的臨界值z或z2α??,tm?或t2α??
2.將檢驗統計量的值與顯著性水平的臨界值進行比較
3.做出決策

雙側檢驗 統計量>臨界值，拒絕H0?
左側檢驗 統計量<- 臨界值，拒絕H0?
右側檢驗 統計量>臨界值，拒絕H0?

什么是P值
1.在原假設為真的條件下，檢驗統計量的觀察值大于或等于其計算值的概率

雙側檢驗為分布中兩側面積的總和

2.反映實際觀測到的數據與原假設H0?之間不一致的程度
3.被稱為觀察到的(或實測的)顯著性水平
3.決策規則 若P值<α,拒絕原假設

假設檢驗的結論表述

1.假設檢驗的目的在于試圖找到拒絕原假設，而不在于證明什么是正確的
2.拒絕原假設時結論是清楚的

例如 H0?:μ=10,拒絕H0?時，可以說μ≠10

3.當不拒絕原假設時，

并未給出明確的結論
不能說原假設是正確的，也不能說它不是正確的
例如，當不拒絕H0?:μ=10,并未說它就是10，但也未說它不是10.只能說樣本提供的證據不足以推翻原假設。

假設檢驗步驟的總結

1.陳述原假設和備擇假設
2.從研究的總體中抽出一個隨機樣本
3.確定一個適當的檢驗統計量，并利用樣本統計依據算出具體數值
4.確定一個適當的顯著性水平，并計算出其臨界值，指定拒絕域
5.將統計量的值域臨界值進行比較，做出決策

統計量的值落在拒絕域，拒絕H0?，否則不拒絕H0?
也可以直接利用P值做出決策

一個總體參數的檢驗

檢驗統計量的確定
總體均值的檢驗
總體比例的檢驗
總體方差的檢驗

總體均值的檢驗---大樣本

1.假定條件
正態總體或非正態總體大樣本n ≥ 30
2.使用z檢驗統計量
α2 已知 z=n?σ?x?μ0??N(0,1)
α2 未知 z=n?s?x?μ0??N(0,1)

總體均值的檢驗---大樣本檢驗方法的總結

假設	雙側檢驗	左側檢驗	右側檢驗
假設形式	H0?:μ=μ0?	H0?:μ≥μ0?	H0?:μ≤μ0?
假設形式	H1?:μ≠μ0?	H1?:μ<μ0?	H1?:μ>μ0?
統計量	α已知	z=n?σ?x?μ0??
統計量	α未知	z=n?s?x?μ0??
拒絕域	z>z2α??	z<?zα?	z>zα?
P值決策	P<α 拒絕H0?

總體均值的檢驗---小樣本

1.假定條件

總體服從正態分布
小樣本(n<30)

2.檢驗統計量
α2已知 z=n?σ?x?μ0??~N(0,1)
α2未知 z=n?s?x?μ0??~t(0,1)

總體均值的檢驗---小樣本檢驗方法的總結

假設	雙側檢驗	左側檢驗	右側檢驗
假設形式	H0?:μ=μ0?	H0?:μ≥μ0?	H0?:μ≤μ0?
假設形式	H1?:μ≠μ0?	H1?:μ<μ0?	H1?:μ>μ0?
統計量	α已知	z=n?σ?x?μ0??
統計量	α未知	z=n?s?x?μ0??
拒絕域	t>t2α??(n?1)	t<?tα?(n?1)	t>tα?(n?1)
P值決策	P<α 拒絕H0?

總體比例檢驗

假定條件

總體服從二項分布
可用正態分布來近似(大樣本)
檢驗的z統計量
z=npi0?(1?π0?)??p?π0??
pi0?為假設的總體比例

總體比例的檢驗

假設	雙側檢驗	左側檢驗	右側檢驗
假設形式	H0?:μ=μ0?	H0?:μ≥μ0?	H0?:μ≤μ0?
假設形式	H1?:μ≠μ0?	H1?:μ<μ0?	H1?:μ>μ0?
統計量	z=nπ0?(1?π0?)??p?π0??
統計量
拒絕域	z>z2α??	z<?aα?	z>zα?
P值決策	P<α 拒絕H0?

總體方差的檢驗

總體方差是服從卡方分布的，檢驗一個總體的方差或標準差，假設總體近似服從正態分布，使用χ2
檢驗統計量
χ2=σ02?(n?1)s2?~χ2(n?1)
其中s為樣本方差，σ為假設總體方差

兩個總體參數的檢驗

兩個統計量的確定
兩個總體均值之差的檢驗
兩個總體比例之差的檢驗
兩個總體方差之比的檢驗
檢驗中的匹配樣本

兩個總體均值之差的檢驗(獨立大樣本)

1.假定條件

兩個樣本是獨立的隨機樣本
正態總體或非正態總體大樣本(n_1 geq 30 n_2 eq 30)檢驗統計量
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)

兩個總體均值之差的檢驗(獨立小樣本)

1.假定條件

兩個獨立的小樣本
兩個總體都是正態分布
σ12?、σ22?已知
檢驗統計量
z=n1?σ12??+n2?σ22???(x1???x2???(μ1??μ2?))?~N(0,1)

兩個總體均值之差的檢驗

sigma_1^2,sigma_2^2未知但是sigma_1^2=sigma_2^2$$
1.假定條件

兩個獨立的小樣本
兩個總體都是正態分布
σ12?、σ22?未知但是相等，即σ12?=σ22?
2.檢驗統計量
t=sp?n1?1?+n2?1??(x1???x2??)?(μ1??μ2?)?
其中sp2?=n1?+n2??2(n1??1)s12?+(n2??1)s22??

兩個總體均值之差的檢驗

σ12?,σ22?未知且不相等σ12?≠σ22?
假定條件

兩個總體都是正態分布
σ22?,σ22?未知且不相等，即σ12?≠σ22?
樣本容量不相等，即n1?≠n2?
檢驗統計量
t=n1?s12??+n2?s22???(x1???x2??)?(μ1??μ2?)?
自由度 v=n1??1n22?s12???+n2??1n2?s22???(n1?s12??+n2?s22??)?

兩個總體均值之差的檢驗(匹配樣本)

假定條件
兩個總體配對差值構成的總體服從配對差是由差值總體中隨機抽取的數據配對或匹配(重復測量(前/后))
檢驗統計量
t=nd??Sd??d?d0??~t(n?1)
樣本差值均值d=nd?∑i=1n?di??

兩個總體比例之差的檢驗

假定條件
兩個總體都服從二項分布
可以用正態分布來近似檢驗統計量
檢驗H0?,π1??π2?=0,z=p(1?p)(n1?1?+n2?1?)?p1??p2??
檢驗H0?,π1??π2?=d0?,z=n1?p1?(1?p1?)?+n2?p2?(1?p2?)??(p1??p2?)?d0??

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