课件-徐敏亚lecture 19

1、Lecture 19Hypothesis Testing11假设检验获得估计量、标准误差、以及置信区间有时还不够实际工作往往需要在两个对立的决策中选择一个。例如：新产品上市？还是不上市？ 2案例：教学评估数据：某学院进行了教学改革之后，收集了下一个学期各课程的教学评估。 问题：为什么关心这个数据？能分析什么？ 34No title gender student score 1 Associate Professor Female MBA 3.175 2 Associate Professor Female MBA 3.523 3 Associate Professor Female MBA 3

2、.47 4 Associate Professor Female MBA 3.184 5 Associate Professor Female MBA 3.548 6 Associate Professor Female Graduate Student 4.103 7 Associate Professor Female MBA 4.833 8 Associate Professor Male Undergraduate 4.481 9 Associate Professor Male Undergraduate 4.195 118 Associate Professor Male MBA

3、4.667 具体案例： 教学改革以前，MBA教学评估的长期平均水平为4.2。 当前收集的教学评估涉及118门课程，其中有52门MBA课程，它们教学评估的平均水平为4.34，标准差为0.54。 问题：MBA教学质量比以往提高了吗？这两个均数不等原因？ 5这两个均数不等原因？n 教学改革的影响。（是真提高）n 由于抽样误差所致，如抽到的都是好老师上的课。（是假提高） 6两类假设原假设H0 ：MBA教学平均分等于4.2。备择假设 H1 ：MBA教学平均分大于4.2（ MBA教学质量比以往提高了）。 根据调查得到的样本，对假设做出决策：拒绝还是接受 什么时候拒绝原假设（接受备择假设）？ 4.34是否已

4、经足够好？ 要多好才算真的好？ 要找出拒绝域-即样本均值落在哪个区域时拒绝原假设 。 7假设检验的第一种思路我们希望能把样本(均值)空间分解成两个不相交部分，其中一个称为接受域，另一个称为拒绝域。 当样本落在拒绝域时就拒绝原假设，当样本落在接受域时就接受原假设。 8拒绝域拒绝域：样本MBA教学平均分比4.2大多大，才会拒绝原假设？x4.2 t Z拒绝域：z问题：如何选择？99你会犯什么错误？你的决策110 0实际上 H0 为真 实际上 H1 为真 接受 H0你是正确的 你犯的是 第二类错误拒绝 H0你犯的是 第一类错误你是正确的 两类错误第一类错误的概率 P(在H0为真时，拒绝H0 )D4.2

5、 t Z | H0PX第二类错误的概率 E4.2Z | H1P(在H1为真时，接受H0 )PX找一个两类错误都不犯的检验！？ 11【问】：要使第一类错误的概率变小，应变 ?(大or小)。要是第二类错误的概率变小，应变 ?(大or小)。 Neymann-Pearson原则N-P原则： 控制你犯第一类错误的概率,然后使犯第二类错误的概率尽量的小。 显著性水平D：控制犯第一类错误的概率不大于D。 112 2控制显著性水平通常显著性水平D 取为 0.1, 0.05 或 0.01如果犯第一类错误的成本很高，则选择(较大)(较小) D 值?如果犯第一类错误的成本不高，则选择(较大)(较小) D 值?113

6、 3Basic concepts in hypothesis testingT :0H0: null hypothesis- tentative assumption about a population parameterT :1H1: alternative hypothesis:0 :1I:0 :1:,Hypothesis testing: using data from a sample to decide whether to accept H0 or to accept H1Begin with the assumption that the null hypothesis is

7、TRUE.(Similar to the notion of innocent until proven guilty)14Formulating Hypotheses: Testing Research HypothesisAparticularautomobilemodelcurrentlyattainsanaveragefuelefficiencyof24milespergallon.Aproduct-researchgrouphasdevelopedanew carburetor (汽化器) specifically designed to increasethemiles-per-g

8、allonrating.Toevaluatethisnewcarburetor,severalwillbeinstalledinautomobiles, and subjected to research-controlled driving tests.Research hypothesis: P !24Generally formulated as alternative hypothesis.H0 : P d 24H1 : P ! 2415Formulating Hypotheses: Testing in Decision-Making SituationsOn the basis o

9、f a sample of parts from a shipment that has just arrived, a quality-control inspector must decide whether to accept the entire shipment or to return the shipment to the supplier because it does not meet specifications. The specifications for a particular part requires a mean length of 2 inches per

10、part.H0 : P2H1 : P z 216Simple and Composite HypothesisIf :i contains just a single value of T, then it is said that the hypothesis Hi is a simple hypothesis.- The distribution of the observations is completely specified.H0 :TT0E.g.17If the set :i contains more than one value of T, then it is said t

11、hat the hypothesis Hi is a composite hypothesis.- It is only specified that the distribution of the observations belongs to a certain class.E.g.H0 :T tT0H0 :T dT0H1 :T !T0ororH1 :TT0orH1 :T z T018The Critical RegionLet S denote the sample space of the random vector X=(X1,Xn).A test procedure is spec

12、ified by partitioning the sample space S into two subsets. One subset contains the values of X for which H0will be accepted. The other subset contains the values of X for which H0 will be rejected and H1 will be accepted (critical region).Determining a test procedure = specifying the critical region

13、 of the test.19Test StatisticsIn most hypothesis testing problems, the critical region is defined in terms of a test statistic, T=r(X).Example 8.1.2. Suppose that X=(X1,Xn) is a random sample from a normal distribution with mean P and known variance V 2 . We wish to test the hypothesesH0 : PP0H1 : P

14、 z P0It might seem reasonable to reject H0 ifis far fromXnT| XnP0 |P0.We define, and for each c0, wecreate a test procedure dc that rejects H0 ifT t c.20The Power FunctionLet C denote the critical region of the test,then the power functionS (T | d )is defined byS (T | d )Pr( X C | T )T :S (T | d )is

15、 the probability that the testprocedure d will lead to the rejection of H0 .1S (T | d )is the probability that the testprocedure d will lead to the acceptance of H0 .21ExampleSuppose that a random sample X1,Xn is taken from a uniform distribution on the interval 0,T.H0 : 3 d T d4H1 :TT ! 43orThe M.L

16、.E. of T is.SupposeYnmax( X1 , Xn )that the critical region of the test d contains alld 2.9the values of X1,Xn for which either YnYn t 4 .or22The power function of the test isS(T |d)Pr(Yn d 2.9 |T )1Pr(Yn t 4 |T )T d 2.9(2.9/T )n2.9T d 4T ! 4(2.9/T )n1(4 /T )n23Size of the Testof a given test d is d

17、efined by:The size D dD (d )sup S (T | d )T:0D dis the maximum probability of type Ierror .d is a level D0 test if and only ifD (d ) d D024Example RevisitSuppose that a random sample X1,Xn is taken from a uniform distribution on the interval 0,T.H0 : 3 d T d4H1 :TT ! 43orThe M.L.E. of T ismax( X1 , Xn ) .YnSupposethat the critical region of the test d contains allYn d 2.9the val