第5章假设检验与统计推断

1、第5章假设检验与统计推断一、假设检验的概念与思想什么是假设(hypothesis)? 对总体参数的的数值所作的一种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的平均体重为3190克!什么是假设检验(hypothesis testing)?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有参数假设检验和非参数假设检验采用逻辑上的反证法，依据统计上的小概率原理假设检验的基本思想. 因此我们拒绝假设 = 50. 如果这是总体的真实均值样本均值m = 50抽样

2、分布H0这个值不像我们应该得到的样本均值 .20总体假设检验的过程抽取随机样本均值 X = 20我认为人口的平均年龄是50岁 提出假设 拒绝假设! 别无选择.作出决策Hypothesis TestingHypothesis testing involves drawing inferences about two contrasting propositions (hypotheses) relating to the value of a population parameter, one of which is assumed to be true in the absence of co

3、ntradictory data.We 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.7Hypothesis Testing ProcedureFormulate the hypothesisSelect a level of significance, which defines the risk of drawing an incorrect con

4、clusion that a true hypothesis is false Determine a decision ruleCollect data and calculate a test statisticApply the decision rule and draw a conclusion81.Hypothesis FormulationNull hypothesis, H0 a statement that is accepted as correctAlternative hypothesis, H1 a proposition that must be true if H

5、0 is falseTests involving a single population parameter are called one-sle tests; tests involving two populations are called two-sle tests.9Types of Hypothesis TestsOne Sle TestsH0: population parameter constant vs. H1: population parameter constantH0: population parameter = constant vs. H1: populat

6、ion parameter constantTwo Sle TestsH0: population parameter (1) - population parameter (2) 0 vs. H1: population parameter (1) - population parameter (2) 0H0: population parameter (1) - population parameter (2) = 0 vs. H1: population parameter (1) - population parameter (2) 010Formulating HypothesesF

7、ormulating the correct set of hypotheses depends on “burden of proof” what you wish to prove statistically should be H1Exle: To seek evidence that technical support calls average less than 30 minutes (Customer Support Survey file), the correct hypotheses are:H0: Mean response time 30 minutesH1: Mean

8、 response time 30 minutes112.显著性水平Four Outcomes The null hypothesis is actually true, and the test correctly fails to reject it. The null hypothesis is actually false, and the hypothesis test correctly reaches this conclusion. The null hypothesis is actually true, but the hypothesis test incorrectly

9、 rejects it (Type I error). The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it (Type II error). 12Quantifying OutcomesProbability of Type I error (rejecting H0 when it is true) = a = level of significanceProbability of correctly failing to reject H0 = 1 a =

10、 confidence coefficient Probability of Type II error (failing to reject H0 when it is false) = bProbability of correctly rejecting H0 when it is false = 1 b = power of the test13假设检验中的两类错误1.第一类错误（弃真错误）原假设为真时拒绝原假设会产生一系列后果第一类错误的概率为被称为显著性水平2.第二类错误（取伪错误）原假设为假时接受原假设第二类错误的概率为(Beta)H0: 无罪假设检验中的两类错误（决策结果）陪审

11、团审判裁决实际情况无罪有罪无罪正确错误有罪错误正确H0 检验决策实际情况H0为真H0为假接受H0正确决策(1 a)第二类错误(b)拒绝H0第一类错误(a)正确决策(1-b)假设检验就好像一场审判过程统计检验过程 错误和 错误的关系你不能同时减少两类错误!和的关系就像翘翘板，小就大， 大就小3.Decision RulesCompute a test statistic from sle data and compare it to the hypothesized sling distribution of the test statisticDivide the sling distribu

12、tion into a rejection region and non-rejection region.If the test statistic falls in the rejection region, reject H0 (concluding that H1 is true); otherwise, fail to reject H017Rejection Regions184.Hypothesis Tests and Spreadsheet SupportType of TestExcel/PHStat ProcedureOne sample test for mean, s

13、knownPHStat: One Sample Test Z-test for the Mean, Sigma KnownOne sample test for mean, s unknown PHStat: One Sample Test 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 M

14、eansPHStat: Two Sample Tests Z-Test for Differences in Two MeansTwo sample test for means, s unknown, unequal Excel t-test: Two-Sample Assuming Unequal Variances19Hypothesis Tests and Spreadsheet Support (contd)Type of TestExcel/PHStat ProcedureTwo sample test for means, s unknown, assumed equal Exc

15、el t-test: Two-Sample Assuming Equal VariancesPHStat: Two Sample Tests t-Test for Differences in Two Means Paired two 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 variance

16、s Excel F-test Two-Sample for VariancesPHStat: Two Sample Tests F-Test for Differences in Two Variances 20二、单样本假设检验1.One Sle Tests for Means Standard Deviation UnknownExle hypothesisH0: m m0 versus H1: m m0 Test statistic: Reject H0 if t -tn-1, 21Exle For the Customer Support Survey.xls data, test t

17、he hypotheses H0: mean response time 30 minutesH1: mean response time 30 minutes Sle mean = 21.91; sle standard deviation = 19.49; n = 44 observations Reject H0 because t = 2.75 One Sle Tests t-Test for the Mean, Sigma UnknownEnter null hypothesis and alphaEnter sle statistics or data rangeChoose ty

18、pe of test23Results242.Using p-Valuesp-value = probability of obtaining a test statistic value equal to or more extreme than that obtained from the sle data when H0 is true, shown as areas under the sling distributions belowTest StatisticLower one-tailed test？ Two-tailed testm0m0Test Statistic25Exle

19、 p-Valuep = probability of obtaining a test statistic of -2.75 or less = 0.004326Two-Tailed Test Consumer Transportation Survey H0: Mean age = 40H1: Mean age 40Sle mean = 37.9; sle standard deviation = 1127Results283.One Sle Tests for ProportionsExle hypothesisH0: p p0 versus H1: p p0Test statistic:

20、Reject if z -za 29ExleFor the Customer Support Survey data, test the hypothesis that the proportion of overall quality responses in the top two boxes（3很好，4 非常好） is at least 0.75H0: p .75H0: p One Sle Tests z-Tests for the ProportionEnter null hypothesis, significance level, number of successes, and

21、sle sizeEnter type of test31Results324.Type II Errors and the Power of a TestThe probability of a Type II error, b, and the power of the test (1 b) cannot be chosen by the experimenter.The power of the test depends on the true value of the population mean, the level of confidence used, and the sle s

22、ize. A power curve shows (1 b) as a function of m1. 33Finding the Probability of a Type II Error34How b Depends on H135How b Depends on Sle Size36Exle Power Curve37三、两样本假设检验1.Two Sle Tests for Means Standard Deviation KnownExle hypothesisH0: m1 m2 0 versus H1: m1 - m2 0Test Statistic:Reject if z 0Test Statistic:Reject if z za 39Two Sle Tests for Means Sigma Unknown and UnequalExle hy