第5章假設(shè)檢驗(yàn)與統(tǒng)計(jì)推斷

1、第5章假設(shè)檢驗(yàn)與統(tǒng)計(jì)推斷一、假設(shè)檢驗(yàn)的概念與思想什么是假設(shè)(hypothesis)? 對(duì)總體參數(shù)的的數(shù)值所作的一種陳述總體參數(shù)包括總體均值、比例、方差等分析之前必需陳述其動(dòng)機(jī)主要是企圖利用人們掌握的反映現(xiàn)實(shí)的數(shù)據(jù)來(lái)找出假設(shè)與現(xiàn)實(shí)之間的矛盾，從而否定這個(gè)假設(shè)我認(rèn)為該地區(qū)新生嬰兒的平均體重為3190克!什么是假設(shè)檢驗(yàn)(hypothesis testing)?事先對(duì)總體參數(shù)或分布形式作出某種假設(shè)，然后利用樣本信息來(lái)判斷原假設(shè)是否成立有參數(shù)假設(shè)檢驗(yàn)和非參數(shù)假設(shè)檢驗(yàn)采用邏輯上的反證法，依據(jù)統(tǒng)計(jì)上的小概率原理假設(shè)檢驗(yàn)的基本思想. 因此我們拒絕假設(shè) = 50. 如果這是總體的真實(shí)均值樣本均值m = 50抽樣

2、分布H0這個(gè)值不像我們應(yīng)該得到的樣本均值 .20總體假設(shè)檢驗(yàn)的過(guò)程抽取隨機(jī)樣本均值 X = 20我認(rèn)為人口的平均年齡是50歲 提出假設(shè) 拒絕假設(shè)! 別無(wú)選擇.作出決策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.顯著性水平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假設(shè)檢驗(yàn)中的兩類錯(cuò)誤1.第一類錯(cuò)誤（棄真錯(cuò)誤）原假設(shè)為真時(shí)拒絕原假設(shè)會(huì)產(chǎn)生一系列后果第一類錯(cuò)誤的概率為被稱為顯著性水平2.第二類錯(cuò)誤（取偽錯(cuò)誤）原假設(shè)為假時(shí)接受原假設(shè)第二類錯(cuò)誤的概率為(Beta)H0: 無(wú)罪假設(shè)檢驗(yàn)中的兩類錯(cuò)誤（決策結(jié)果）陪審

11、團(tuán)審判裁決實(shí)際情況無(wú)罪有罪無(wú)罪正確錯(cuò)誤有罪錯(cuò)誤正確H0 檢驗(yàn)決策實(shí)際情況H0為真H0為假接受H0正確決策(1 a)第二類錯(cuò)誤(b)拒絕H0第一類錯(cuò)誤(a)正確決策(1-b)假設(shè)檢驗(yàn)就好像一場(chǎng)審判過(guò)程統(tǒng)計(jì)檢驗(yàn)過(guò)程 錯(cuò)誤和 錯(cuò)誤的關(guān)系你不能同時(shí)減少兩類錯(cuò)誤!和的關(guān)系就像翹翹板，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、單樣本假設(shè)檢驗(yàn)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、兩樣本假設(shè)檢驗(yàn)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