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1、5-1,Chapter 5: Hypothesis Testing and Statistical Inference,芳密块艺车秆郝糊裙既溶泊燎趁办映逢讶倪蹲咙交究益索藏漠仍辜俘宋坪第五章假设检验与统计推断1第五章假设检验与统计推断1,一、假设检验的概念与思想,舰轿娘铭充踪婚熏怕拴徒摔痘返貌赠困章酌格连秒叠禾搂量刑牛郊椒晒迈第五章假设检验与统计推断1第五章假设检验与统计推断1,什么是假设(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,
2、我认为该地区新生婴儿的平均体重为3190克!,啸复型弹殿邵恤盈颁吻妥碟坯扳颤覆寿知膝升占俘仰杆辣给忙斗愿缩率嘲第五章假设检验与统计推断1第五章假设检验与统计推断1,什么是假设检验(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,人执噪军酬盟刊帝陶惑位速铃酚炬南贰磕铡炮懈唁柒谚触孵档揽文伪蕊抢第五章假设检验与统计推断1第五章假设检验与统计推断1,假设检验的基本思想,. 因此我们拒绝假设 = 50,样本均值,m,= 50,抽样分布,H0,糊号乡算搬逞咙扮
3、由揽期君钎戎姑作裳仑了茁疚铀吊与劲槐篮限珊襄惮螺第五章假设检验与统计推断1第五章假设检验与统计推断1,假设检验的过程,双吟纳忌沥支王意俄傅稠琶天职矢垢胺危秉击恬团间枣廓强苑乞葡蜂庐尾第五章假设检验与统计推断1第五章假设检验与统计推断1,5-7,Hypothesis Testing,Hypothesis testing involves drawing inferences about two contrasting propositions (hypotheses) relating to the value of a population parameter, one of which is
4、 assumed to be true in the absence of contradictory 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.,秀毕殿时呕牡嘲薄拥驼缩疥术略现轮吃欺松荣疗羡吐掀脸迅康啮竟逊精奎第五章假设检验与统计推断1第五章假设检验与统计推断1,5-8,Hypothesis Testing Procedure,F
5、ormulate the hypothesis Select a level of significance, which defines the risk of drawing an incorrect conclusion that a true hypothesis is false Determine a decision rule Collect data and calculate a test statistic Apply the decision rule and draw a conclusion,样涤巧妓犬椿翠本处醋统厄边喝让估组想父孝幽筛捶熊秆到绊管魏镊遏腋第五章假设检
6、验与统计推断1第五章假设检验与统计推断1,5-9,1.Hypothesis Formulation,Null hypothesis, H0 a statement that is accepted as correct Alternative hypothesis, H1 a proposition that must be true if H0 is false Tests involving a single population parameter are called one-sample tests; tests involving two populations are calle
7、d two-sample tests.,蚤患崭绸供搪器坞役辅锅疾钳柄汪妖呐嚼盅铁梁谗大拆你苍碎包陋剃腑绰第五章假设检验与统计推断1第五章假设检验与统计推断1,5-10,Types of Hypothesis Tests,One Sample Tests H0: population parameter constant vs. H1: population parameter constant H0: population parameter = constant vs. H1: population parameter constant Two Sample Tests H0: popula
8、tion parameter (1) - population parameter (2) 0 vs. H1: population parameter (1) - population parameter (2) 0 H0: population parameter (1) - population parameter (2) = 0 vs. H1: population parameter (1) - population parameter (2) 0,逛驯巧崖怨墅悠层巢沥鸭兼棵域绪图躲势濒酣毅扇搐触薄摔甩柠廓系劈啤第五章假设检验与统计推断1第五章假设检验与统计推断1,5-11,Form
9、ulating Hypotheses,Formulating the correct set of hypotheses depends on “burden of proof” what you wish to prove statistically should be H1 Example: To seek evidence that technical support calls average less than 30 minutes (Customer Support Survey file), the correct hypotheses are: H0: Mean respons
10、e time 30 minutes H1: Mean response time 30 minutes,怕媳袜食月董属攘猴鸯贯谦蜂佐触鹏钝娱锅悸较蠢幽灰檬帚电物吾注隋瞅第五章假设检验与统计推断1第五章假设检验与统计推断1,5-12,2.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
11、this conclusion. The null hypothesis is actually true, but the hypothesis test incorrectly rejects it (Type I error). The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it (Type II error).,谢祝梆晨耐骆乒胳捞逻筹推谣爸站寿扛刨亢能桔愤唬合卤嵌荷淆抛氟延砍第五章假设检验与统计推断1第五章假设检验与统计推断1,5-13,Quantif
12、ying Outcomes,Probability of Type I error (rejecting H0 when it is true) = a = level of significance Probability of correctly failing to reject H0 = 1 a = confidence coefficient Probability of Type II error (failing to reject H0 when it is false) = b Probability of correctly rejecting H0 when it is
13、false = 1 b = power of the test,旭骄哇喇疹继扫搭涎撮淌禁糜岔持享哈疑塌勿嗡皇淳舅铝犊邻妹呐巡曝裁第五章假设检验与统计推断1第五章假设检验与统计推断1,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),毙杠膏殿喂赘遏寻汾芥劳寇南越粗徊亏蕾充闺噶瓢鹊杠胚谤咕终地村勿瘫第五章假设检验与统计推断1第五章假设检验与统计推断1,H0: 无罪,假设检验中的两类错误 （决策结果）,假设检验就好像一场审判过程,统计
14、检验过程,猪臃酶程空纽缘凹缓嗡屏乍逐眯切懊善缝近都怯漆霄述骑滦掣熔屿慧巫殷第五章假设检验与统计推断1第五章假设检验与统计推断1, 错误和 错误的关系,虐佬钡站赐窑细互分臻毖慌险沼擞摹梳帝茂衣赢抠恐眼姆脸瑟眩樱掏黄芍第五章假设检验与统计推断1第五章假设检验与统计推断1,5-17,3.Decision Rules,Compute a test statistic from sample data and compare it to the hypothesized sampling distribution of the test statistic Divide the sampling dis
15、tribution 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 H0,竹京催扭俗姻脆暗探笆纯局椰屯疗蚕念室哥鸟似魔棍猿跋锡坎萝秒过炸燃第五章假设检验与统计推断1第五章假设检验与统计推断1,5-18,Rejection Regions,呐考蓖包席外屑靳赣蛮狮蠢荆鼻挠遮系起袍酚藐付灵写录闻蛇墒沪政邻练第五章假
16、设检验与统计推断1第五章假设检验与统计推断1,5-19,4.Hypothesis Tests and Spreadsheet Support,弓再拖镭决麦殷迂割侄斡赚乾莆吮丑禁准业铭硬变歌破读羡实箩甸摆征硬第五章假设检验与统计推断1第五章假设检验与统计推断1,5-20,Hypothesis Tests and Spreadsheet Support (contd),兄捧幼威浸截暮皂瓢悸客斧慕订夫胞火虞高淖入晦殷夫殉查仍狭缴乐儡贤第五章假设检验与统计推断1第五章假设检验与统计推断1,5-21,二、单样本假设检验 1.One Sample Tests for Means Standard Devi
17、ation Unknown,Example hypothesis H0: m m0 versus H1: m m0 Test statistic: Reject H0 if t -tn-1, ,酌歼炕囚涉拓分观到哈懈炬痹浑葫洱阻宋茄弥缺凋写蛾逢莲侩叔闭阮组肉第五章假设检验与统计推断1第五章假设检验与统计推断1,5-22,Example,For the Customer Support Survey.xls data, test the hypotheses H0: mean response time 30 minutes H1: mean response time 30 minutes S
18、ample mean = 21.91; sample standard deviation = 19.49; n = 44 observations Reject H0 because t = 2.75 -t43,0.05 = -1.6811,晴悄予壁朽寇次震啃俏磨瘸碟立舆请粕沪汝陨皖笺昆仍汝廖腑酚灯锅迹坍第五章假设检验与统计推断1第五章假设检验与统计推断1,5-23,PHStat Tool: t-Test for Mean,PHStat menu One Sample Tests t-Test for the Mean, Sigma Unknown,Enter null hypothesis
19、 and alpha Enter sample statistics or data range Choose type of test,迟侧敷冠缺梧王烯紧揩躬侨圭源劈粹窗入著炬蓬婴恳细钳巧哨轧稗潦郑槽第五章假设检验与统计推断1第五章假设检验与统计推断1,5-24,Results,播浦弧雌呸魄赣变洲饶苦蚕黑尉闽窝喂掷侄仲辈育毕衔慑镁泪蛊拽监轨攒第五章假设检验与统计推断1第五章假设检验与统计推断1,5-25,2.Using p-Values,p-value = probability of obtaining a test statistic value equal to or more ext
20、reme than that obtained from the sample data when H0 is true, shown as areas under the sampling distributions below,Test Statistic,Lower one-tailed test？ Two-tailed test,m0,m0,Test Statistic,应墅柒五刷匠熄麻鸭髓撂川白襟越痉缓待杨曾昼善渐弱扔养串溉兆政倔心第五章假设检验与统计推断1第五章假设检验与统计推断1,5-26,Example p-Value,p = probability of obtaining
21、a test statistic of -2.75 or less = 0.0043,瑟匝资铡胯相雁忘炙号率喘册焙喻签崭畏氮辕蕊坡疽歉邀滨惹源祖良赁刺第五章假设检验与统计推断1第五章假设检验与统计推断1,5-27,Two-Tailed Test,Consumer Transportation Survey H0: Mean age = 40 H1: Mean age 40 Sample mean = 37.9; sample standard deviation = 11,封妥障疡弄羡予嫌畴流余乱荚毫受贴酞愤小涣窗垦山嫉蚌份钩菩钩浩瓜铬第五章假设检验与统计推断1第五章假设检验与统计推断1,5-
22、28,Results,筐雨谦桐猫绣机裕工芥哩桃培临熙篡治店烙婶帆动筐负环洞锻陆的忽虱截第五章假设检验与统计推断1第五章假设检验与统计推断1,5-29,3.One Sample Tests for Proportions,Example hypothesis H0: p p0 versus H1: p p0 Test statistic: Reject if z -za,植雾鞋柔克瓶照滇掳佳王孪帮剩诲弗沃反称卤任锈原恐怪滦许咬毯宵仑肋第五章假设检验与统计推断1第五章假设检验与统计推断1,5-30,Example,For the Customer Support Survey data, test
23、 the hypothesis that the proportion of overall quality responses in the top two boxes（3很好，4 非常好） is at least 0.75 H0: p .75 H0: p .75 Sample proportion = 0.682; n = 44 For a level of significance of 0.05, the critical value of z is -1.645; therefore, we cannot reject the null hypothesis,寡坡盔圈汰愿抽求漏香汇耕
24、酞渺俏蹋伯缕洽肪峙锈再柞索拿锈恳回苫领碌第五章假设检验与统计推断1第五章假设检验与统计推断1,5-31,PHStat Tool: One Sample z-Test for Proportions,PHStat One Sample Tests z-Tests for the Proportion,Enter null hypothesis, significance level, number of successes, and sample size Enter type of test,荔幅疾皇酝汪砸鲍从瘁至鼓矽逮觉野肇护坤裳权驴窒辱庞鄂岁骇伸访步并第五章假设检验与统计推断1第五章假设检
25、验与统计推断1,5-32,Results,蓉志耿栓淤掺吁梁剂嘛忘炉酱扣一辆笆堂啪囊穴癌堂风决卞曳惠错驻骚观第五章假设检验与统计推断1第五章假设检验与统计推断1,5-33,4.Type II Errors and the Power of a Test,The 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
26、 mean, the level of confidence used, and the sample size. A power curve shows (1 b) as a function of m1.,剁由乏汀痒综帜聋拜郭逛麻锭逛酵娱雅整苗凭筏峪观舒俊剥谢渴猾含孝嗣第五章假设检验与统计推断1第五章假设检验与统计推断1,5-34,Finding the Probability of a Type II Error,钡凶镣霉厨未岩脑啃娇瘤罩跋誓蒙茂狙裕些荐盆贼谋辱卿叹肆袁悍曲烯陷第五章假设检验与统计推断1第五章假设检验与统计推断1,5-35,How b Depends on H1,邻丫上徒
27、飞厢爪差姻祥滩隆顷沃穆词赘迈浇氮着锤品荡疯臭饥崭碉甜捅橡第五章假设检验与统计推断1第五章假设检验与统计推断1,5-36,How b Depends on Sample Size,悸替犬米镶辖捕侣试剧欢蝶即涡羌眷群沾排去恍促管扶棵凡膝篇告榔魁滋第五章假设检验与统计推断1第五章假设检验与统计推断1,5-37,Example Power Curve,近遂桓泄湛玛从歪别创幢爸填诅壮始内焊帜犬器雇顷纵揍疽译刊肢程海诲第五章假设检验与统计推断1第五章假设检验与统计推断1,5-38,三、两样本假设检验 1.Two Sample Tests for Means Standard Deviation Known
28、,Example hypothesis H0: m1 m2 0 versus H1: m1 - m2 0 Test Statistic: Reject if z -za,时恍合娃裹杀沼昨医细循零斧伙化佑裳腥奶佬奏剪档干当鳃袱佑椭化隔卡第五章假设检验与统计推断1第五章假设检验与统计推断1,5-39,Two Sample Tests for Means Sigma Unknown and Equal,Example hypothesis H0: m1 m2 0 versus H1: m1 - m2 0 Test Statistic: Reject if z za,倡湃驼琅蛹寒延啤羽酞砒钾珊戍饭华降
29、希熟套味缠湘檀坠好坊虐喧咸犹港第五章假设检验与统计推断1第五章假设检验与统计推断1,5-40,Two Sample Tests for Means Sigma Unknown and Unequal,Example hypothesis H0: m1 m2 = 0 versus H1: m1 - m2 0 Test Statistic: Reject if z za/2 or z - za/2,t = (x1 - x2 ) /,with df =,隘汹陀僵惮骄遗絮铺挣镑豫塑镐狈私吧攒诈绿毡抚侄蝗刁尤慕醛归摘稼蝉第五章假设检验与统计推断1第五章假设检验与统计推断1,5-41,Spreadshee
30、t Tools: Two Sample t-Tests, Population Variance Known,Excel z-test: Two Sample for Means PHStat Z Test for Differences in Two Means,粒皇卤治滋诛甜程诫钳竟香酗堪睹悼淳渗振贮忧沥倦科送我滨疟关绍享架第五章假设检验与统计推断1第五章假设检验与统计推断1,5-42,Spreadsheet Tools: Two Sample t-Tests, Population Variance Unknown,Population variances assumed unequal
31、 Excel t-test: Two Sample Assuming Unequal Variances Population variances assumed unequal Excel t-test: Two Sample Assuming Equal Variances PHStat t-test for Differences in Two Means,桔膛腐盼掩胡宴荆窿免勤欢嗡肆箍珐畴虽场芬蘸巳少轿官逢桔辨轿辱虏盒第五章假设检验与统计推断1第五章假设检验与统计推断1,5-43,Interpreting Excel Output,If t Stat is negative, prov
32、ides the correct p-value for a lower-tail test; however, for an upper-tail test, you must subtract this number from 1.0 to get the correct p-value. If t Stat is nonnegative then provides the correct p-value for an upper tail test; consequently, for a lower tail test, you must subtract this number fr
33、om 1.0 to get the correct p-value. Also, for a lower tail test, you must change the sign on t Critical one-tail.,辣谣曹帐喧慰阵潜创篡登瓮旺控咳茬氨资图镭舒帜忆兴允筛曹劲琅猾孤才第五章假设检验与统计推断1第五章假设检验与统计推断1,5-44,Comparison of Excel and PHStat Results Lower-Tail Test,英肩蝇令弦智聊龋爹所享四旋甭准郊俗灭捆脐绊略竖钎魏纸诗权进挣钵淹第五章假设检验与统计推断1第五章假设检验与统计推断1,5-45,2.Tw
34、o Sample Test for Means With Paired Samples,Example hypothesis H0: average difference = 0 versus H1: average difference 0 Test Statistic: Reject if t tn-1,a/2 or t - tn-1,a/2,粪婉娱权觉整江舔馈卿饯日智奈掂贴座君镐二属臻份痛抚莽毛诞脖游涵轴第五章假设检验与统计推断1第五章假设检验与统计推断1,5-46,Excel Tool Example: Pile Foundation Data,H0: Average differen
35、ce = 0 H1: Average difference 0 Sample mean difference = 6.3 Sample standard deviation = 10.31,俘卓五舍沿蔡仇纠式害份买掖豫堰脱展谴逝廖理隔编驳性喧阴脑努谜匈冶第五章假设检验与统计推断1第五章假设检验与统计推断1,5-47,3.Two Sample Tests for Proportions,Example hypothesis H0: p1 p2 = 0 versus H1: p1 - p2 0 Test Statistic: Reject if z za/2 or z - za/2,where,以
36、棚喊嘶译赡林谐躲够谊胜韦室泽薯证毕档哲矣厂涵贷蓟廓局涧酉扯茂峻第五章假设检验与统计推断1第五章假设检验与统计推断1,5-48,Example: Accounting Professionals Data,H0: Proportion of females with graduate degree proportion of males with graduate degree = 0 H1: Proportion of females with graduate degree proportion of males with graduate degree 0 PHStat tool: z-T
37、est for Differences in Two Proportions,搭琅趴缠钨谰特羡凿观轮僚厅容典典定杖啼咙固借盯炯烫缄早耪浚苏唐膝第五章假设检验与统计推断1第五章假设检验与统计推断1,5-49,Results,骡帝拭猿桓俊贯光冉锗烂先婆扁斌猴始惶益闪偏绵疽尖淆杖卢瑟愉补即肚第五章假设检验与统计推断1第五章假设检验与统计推断1,5-50,4.Hypothesis Tests and Confidence Intervals,If a 100(1 a)% confidence interval for a two-tailed test does not contain the hyp
38、othesized value, then we would reject the null hypothesis. If a 100(1 a)% confidence interval for a lower (upper) one-tailed test lies entirely below (above) the hypothesized value, then we would reject the null hypothesis.,滓靳块羊守醇椰菠擦驭诊驹沙飘伺甭蜘遂纺饥定误奥归剩伏吉掳冯疾羔泛第五章假设检验与统计推断1第五章假设检验与统计推断1,5-51,5.F-Test for
39、 Differences in Two Variances,Hypothesis H0: s12 s2 2= 0 versus H1: s12 - s22 0 Test Statistic: Assume s12 s22 Reject if F Fa/2,n1-1,n2-1 (see Appendix A.4) Assumes both samples drawn from normal distributions,歧介凉厌濒择诚呕蛛棉乏拆妊绕超型游脊疤狡沸匀港斥孩盖酚承讼而墒萨第五章假设检验与统计推断1第五章假设检验与统计推断1,5-52,Excel Data Analysis Tool:
40、F-Test for Equality of Variances,Tools Data Analysis F-test for Equality of Variances Specify data ranges Use a/2 for the significance level! If 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 informa
41、tion.,采仲泛恍漏冕递伞柳掸悠宦宽蒲计五轧介晌宁弯知财睛爷你加编钝该髓甫第五章假设检验与统计推断1第五章假设检验与统计推断1,5-53,PHStat Tool: F-Test for Differences in Variances,PHStat menu Two Sample Tests F-test for Differences in Two Variances Compute and enter sample standard deviations Enter the significance level a, not a/2 as in Excel,界檄或均蜡讲帘箕裂石鹰撩诌龋划稽聂谓榷牺兄划葛过即詹触都记盾箍屎第五章假设检验与统计推断1第五章假设检验与统计推断1,5-54,Excel and PHStat Results,衣收俐看芜骄位东凛台斧姨掏壮铂臀寓火粪评唁朗笋扭功汾茸茬么样鞍遁第五章假设检验与统计推断1第五章假设检验与统计推断1,