Lecture 19 - PKUmwfy.gsm.pku.edu.cn/miao_files/ProbStat/lecture19.pdf19 The Critical Region!Let...
Transcript of Lecture 19 - PKUmwfy.gsm.pku.edu.cn/miao_files/ProbStat/lecture19.pdf19 The Critical Region!Let...
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Lecture 19
! Hypothesis Testing
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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?
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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?
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No title gender student score
1 Associate Professor Female MBA 3.175
2 Associate Professor Female MBA 3.523
3 Associate Professor Female MBA 3.47
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 4.667
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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?
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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)
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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。
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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。
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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?
问题:如何选择?
4.2z x w= - ³拒绝域:
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你会犯什么错误?
实际上 H0为真
实际上 H1为真
你的 决策
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
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两类错误
! 第一类错误的概率
! 第二类错误的概率
! 找一个两类错误都不犯的检验!?
( )0 0 0( ) 4.2 |P H H P X Ha w= = - ³在 为真时,拒绝
( )1 0 1( ) 4.2 |P H H P X Hb w= = - <在 为真时,接受
【问】:要使第一类错误的概率变小,ω应变__?(大or小)。要是第二类错误的概率变小,ω应变__?(大or小)。
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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 :控制犯第一类错误的概率不大于 。a
a
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控制显著性水平
! 通常显著性水平 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) 值?
! 如果犯第一类错误的成本不高,则选择(较大)(较小) 值?
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a
a
a
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Basic concepts in hypothesis testing
! H0: null hypothesis--- tentative assumption about a population parameter
! H1: alternative hypothesis
!
! Hypothesis testing: using data from a sample to decide whether to accept H0 or to accept H1
! Begin with the assumption that the null hypothesis is TRUE. (Similar to the notion of innocent until proven guilty)
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Formulating Hypotheses:Testing Research Hypothesis
! A particular automobile model currently attains anaverage fuel efficiency of 24 miles per gallon. Aproduct-research group has developed a newcarburetor (汽化器) specifically designed to increasethe miles-per-gallon rating. To evaluate this newcarburetor, several will be installed in automobiles,and subjected to research-controlled driving tests.
! Research hypothesis:Generally formulated as alternative hypothesis.
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Formulating Hypotheses: Testing in Decision-Making Situations
! On the basis of a sample of parts from a shipmentthat has just arrived, a quality-control inspector mustdecide whether to accept the entire shipment or toreturn the shipment to the supplier because it doesnot meet specifications. The specifications for aparticular part requires a mean length of 2 inches perpart.
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Simple and Composite Hypothesis
! If Wi contains just a single value of q, then it is said that the hypothesis Hi is a simple hypothesis. --- The distribution of the observations is completely specified.
! E.g. 00 : qq =H
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! If the set Wi contains more than one value of q, then it is said that the hypothesis Hi is a composite hypothesis. --- It is only specified that the distribution of the observations belongs to a certain class.
! E.g.
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The Critical Region! Let S denote the sample space of the random vector X=(X1,…,Xn). A test procedure is specified by partitioning the sample space S into two subsets.• One subset contains the values of X for which H0
will 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 of the test.
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Test Statistics! In 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 µ and known variance . We wish to test the hypotheses
It might seem reasonable to reject H0 if is far from µ0. We define , and for each c>0, we create a test procedure dc that rejects H0 if .
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The Power Function
! Let C denote the critical region of the test, then the power function is defined by
! is the probability that the test procedure d will lead to the rejection of H0 .
! is the probability that the test procedure d will lead to the acceptance of H0 .
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! Suppose that a random sample X1,…,Xn is taken from a uniform distribution on the interval [0,q].
! The M.L.E. of q is . Suppose that the critical region of the test d contains all the values of X1,…,Xn for which either or .
Example
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! The power function of the test is
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! The size a(d) of a given test d is defined by:
a(d) is the maximum probability of type I error .
! d is a level a0 test if and only if
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Size of the Test
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! Suppose that a random sample X1,…,Xn is taken from a uniform distribution on the interval [0,q].
! The M.L.E. of q is . Suppose that the critical region of the test d contains all the values of X1,…,Xn for which either or .
Example Revisit
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! The power function of the test is
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! The size of the test is
! When n=68, the size of d is .
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