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1、5-1Chapter 5: Hypothesis Testing and Statistical Inference一、假设检验的概念与思想什么是假设(hypothesis)?n 对总体参数的的数值所作的一种陈述n总体参数包括总总体体均均值值、比比例例、方差方差等n分析之前之前必需陈述n其动机主要是企图利用人们掌握的反映现实的数据来找出假设与现实之间的矛盾，从而否定这个假设我认为该地区新生婴儿我认为该地区新生婴儿的平均体重为的平均体重为31903190克克! !什么是假设检验(hypothesis testing)?1.事先对总体参数或分布形式作出某种假设，然后利用样本信息来判断原假设是否成立
2、2.有参数假设检验和非参数假设检验3.采用逻逻辑辑上上的的反反证证法法，依据统计上的小小概概率原理率原理假设检验的基本思想. . . 因此我们拒因此我们拒因此我们拒因此我们拒因此我们拒因此我们拒绝假设绝假设绝假设绝假设绝假设绝假设 = 50= 50= 50. . . 如果这是总如果这是总如果这是总如果这是总如果这是总如果这是总体的真实均值体的真实均值体的真实均值体的真实均值体的真实均值体的真实均值样本均值样本均值样本均值 = 50 = 50抽样分布抽样分布抽样分布抽样分布抽样分布抽样分布H HH0 00这个值不像我这个值不像我这个值不像我这个值不像我这个值不像我这个值不像我们应该得到的们应该得
3、到的们应该得到的们应该得到的们应该得到的们应该得到的样本均值样本均值样本均值样本均值样本均值样本均值 .202020总体总体总体总体假设检验的过程抽取随机样本抽取随机样本抽取随机样本抽取随机样本均值均值均值均值 X X = 20= 20我认为人口的平我认为人口的平均年龄是均年龄是5050岁岁 提出假设提出假设提出假设提出假设 拒绝假设拒绝假设! 别无选择别无选择.作出决策作出决策作出决策作出决策5-7Hypothesis TestingnHypothesis testing involves drawing inferences about two contrasting propositio
4、ns (hypotheses) relating to the value of a population parameter, one of which is assumed to be true in the absence of contradictory data.nWe seek evidence to determine if the hypothesis can be rejected; if not, we can only assume it to be true but have not statistically proven it true.5-8Hypothesis
5、Testing Procedure1.Formulate the hypothesis2.Select a level of significance, which defines the risk of drawing an incorrect conclusion that a true hypothesis is false 3.Determine a decision rule4.Collect data and calculate a test statistic5.Apply the decision rule and draw a conclusion5-91.Hypothesi
6、s FormulationnNull hypothesis, H0 a statement that is accepted as correctnAlternative hypothesis, H1 a proposition that must be true if H0 is falsenTests involving a single population parameter are called one-sample tests; tests involving two populations are called two-sample tests.5-10Types of Hypo
7、thesis TestsnOne Sample TestsnH0: population parameter constant vs. H1: population parameter constantnH0: population parameter = constant vs. H1: population parameter constantnTwo Sample TestsnH0: population parameter (1) - population parameter (2) 0 vs. H1: population parameter (1) - population par
8、ameter (2) 0nH0: population parameter (1) - population parameter (2) = 0 vs. H1: population parameter (1) - population parameter (2) 05-11Formulating HypothesesnFormulating the correct set of hypotheses depends on “burden of proof” what you wish to prove statistically should be H1nExample: To seek e
9、vidence that technical support calls average less than 30 minutes (Customer Support Survey file), the correct hypotheses are:nH0: Mean response time 30 minutesnH1: Mean response time 30 minutes5-122.显著性水平Four Outcomes 1.The null hypothesis is actually true, and the test correctly fails to reject it.
10、 2.The null hypothesis is actually false, and the hypothesis test correctly reaches this conclusion. 3.The null hypothesis is actually true, but the hypothesis test incorrectly rejects it (Type I error). 4.The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it
11、(Type II error). 5-13Quantifying OutcomesnProbability of Type I error (rejecting H0 when it is true) = = level of significancenProbability of correctly failing to reject H0 = 1 = confidence coefficient nProbability of Type II error (failing to reject H0 when it is false) = nProbability of correctly
12、rejecting H0 when it is false = 1 = power of the test8 - 14精品教材精品教材统计学统计学假设检验中的两类错误假设检验中的两类错误1.1.第一类错误（弃真错误）第一类错误（弃真错误）第一类错误（弃真错误）第一类错误（弃真错误）n n原假设为真时拒绝原假设原假设为真时拒绝原假设n n会产生一系列后果会产生一系列后果n n第一类错误的概率为第一类错误的概率为 l l被称为显著性水平被称为显著性水平2.2.第二类错误（取伪错误）第二类错误（取伪错误）第二类错误（取伪错误）第二类错误（取伪错误）n n原假设为假时接受原假设原假设为假时接受原假设n
13、 n第二类错误的概率为第二类错误的概率为(Beta)(Beta)8 - 15精品教材精品教材统计学统计学H H0 0: : 无罪无罪无罪无罪假设检验中的两类错误假设检验中的两类错误（决策结果）（决策结果）陪审团审判陪审团审判陪审团审判陪审团审判裁决裁决裁决裁决实际情况实际情况实际情况实际情况无罪无罪无罪无罪有罪有罪有罪有罪无罪无罪无罪无罪正确正确正确正确错误错误错误错误有罪有罪有罪有罪错误错误错误错误正确正确正确正确H H0 0 检验检验检验检验决策决策决策决策实际情况实际情况实际情况实际情况H H0 0为真为真为真为真H H0 0为假为假为假为假接受接受接受接受H H0 0正确决策正确决策正
14、确决策正确决策(1 (1 )第二类错第二类错第二类错第二类错误误误误( ( ( ()拒绝拒绝拒绝拒绝H H0 0第一类错第一类错第一类错第一类错误误误误( ( ( ()正确决策正确决策正确决策正确决策(1-(1-(1-(1-)假设检验就好像一场审判过程假设检验就好像一场审判过程假设检验就好像一场审判过程统计检验过程统计检验过程统计检验过程8 - 16精品教材精品教材统计学统计学 错误和错误和 错误的关系错误的关系 你不能同时减你不能同时减少两类错误少两类错误! 和和和和 的关系就像的关系就像的关系就像的关系就像翘翘板，翘翘板，翘翘板，翘翘板， 小小小小 就就就就大，大，大，大， 大大大大 就小
15、就小就小就小5-173.Decision RulesnCompute a test statistic from sample data and compare it to the hypothesized sampling distribution of the test statisticnDivide the sampling distribution into a rejection region and non-rejection region.nIf the test statistic falls in the rejection region, reject H0 (concl
16、uding that H1 is true); otherwise, fail to reject H05-18Rejection Regions5-194.Hypothesis Tests and Spreadsheet SupportType of TestExcel/PHStat ProcedureOne sample test for mean, s knownPHStat: One Sample Test Z-test for the Mean, Sigma KnownOne sample test for mean, s unknown PHStat: One Sample Tes
17、t t-test for the Mean, Sigma UnknownOne sample test for proportion PHStat: One Sample Test Z-test for the ProportionTwo sample test for means, s known Excel z-test: Two-Sample for MeansPHStat: Two Sample Tests Z-Test for Differences in Two MeansTwo sample test for means, s unknown, unequal Excel t-t
18、est: Two-Sample Assuming Unequal Variances5-20Hypothesis Tests and Spreadsheet Support (contd)Type of TestExcel/PHStat ProcedureTwo sample test for means, s unknown, assumed equal Excel t-test: Two-Sample Assuming Equal VariancesPHStat: Two Sample Tests t-Test for Differences in Two Means Paired two
19、 sample test for means Excel t-test: Paired Two-Sample for MeansTwo sample test for proportions PHStat: Two Sample Tests Z-Test for Differences in Two Proportions Equality of variances Excel F-test Two-Sample for VariancesPHStat: Two Sample Tests F-Test for Differences in Two Variances 5-21二、二、单样本假本
20、假设检验1.One Sample Tests for Means Standard Deviation UnknownnExample hypothesisnH0: m m0 versus H1: m m0 nTest statistic: nReject H0 if t -tn-1, 5-22Example For the Customer Support Survey.xls data, test the hypotheses H0: mean response time 30 minutesH1: mean response time 30 minutes Sample mean = 2
21、1.91; sample standard deviation = 19.49; n = 44 observations Reject H0 because t = 2.75 One Sample Tests t-Test for the Mean, Sigma UnknownEnter null hypothesis and alphaEnter sample statistics or data rangeChoose type of test5-24Results5-252.Using p-Valuesnp-value = probability of obtaining a test
22、statistic value equal to or more extreme than that obtained from the sample data when H0 is true, shown as areas under the sampling distributions belowTest StatisticLower one-tailed test？ Two-tailed testm0m0Test Statistic5-26Example p-Valuep = probability of obtaining a test statistic of -2.75 or le
23、ss = 0.00435-27Two-Tailed Test nConsumer Transportation Survey nH0: Mean age = 40nH1: Mean age 40nSample mean = 37.9; sample standard deviation = 115-28Results5-293.One Sample Tests for ProportionsnExample hypothesisnH0: p p0 versus H1: p p0nTest statistic:nReject if z -z 5-30ExamplenFor the Custome
24、r Support Survey data, test the hypothesis that the proportion of overall quality responses in the top two boxes（3很好，很好，4 非常好）非常好） is at least 0.75nH0: p .75nH0: p One Sample Tests z-Tests for the ProportionEnter null hypothesis, significance level, number of successes, and sample sizeEnter type of
25、test5-32Results5-334.Type II Errors and the Power of a TestnThe probability of a Type II error, b, and the power of the test (1 b) cannot be chosen by the experimenter.nThe power of the test depends on the true value of the population mean, the level of confidence used, and the sample size. nA power
26、 curve shows (1 b) as a function of m1. 5-34Finding the Probability of a Type II Error5-35How Depends on H15-36How Depends on Sample Size5-37Example Power Curve5-38三、两三、两样本假本假设检验1.Two Sample Tests for Means Standard Deviation KnownnExample hypothesisnH0: m1 m2 0 versus H1: m1 - m2 0nTest Statistic:n
27、Reject if z 0nTest Statistic:nReject if z z 5-40Two Sample Tests for Means Sigma Unknown and UnequalnExample hypothesis nH0: m1 m2 = 0 versus H1: m1 - m2 0nTest Statistic:nReject if z z/2 or z tn-1,/2 or t z/2 or z s22 nReject if F F/2,n1-1,n2-1 (see Appendix A.4)nAssumes both samples drawn from nor
28、mal distributions5-52Excel Data Analysis Tool: F-Test for Equality of VariancesnTools Data Analysis F-test for Equality of VariancesnSpecify data rangesnUse /2 for the significance level!nIf the variance of Variable 1 is greater than the variance of variable 2, the output will specify the upper tail; otherwise, you obtain the lower tail information. 5-53PHStat Tool: F-Test for Differences in VariancesnPHStat menu Two Sample Tests F-test for Differences in Two Variances nCompute and enter sample standard deviationsnEnter the significance level , not /2 as in Excel5-54Excel and PHStat Results
