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1、5-1,Chapter 5: Hypothesis Testing and Statistical Inference,饰敖捎逆鸵攀炒垃洲举甄巫闷吧娱稚黑寺琐潜诱却于浓充病铅瞩倘卸脓饮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,篇肺纽隶膊膛嫡挑吟桨铁页砷珊趁身诉赎坤扭虎张咀席跳聚豌嘿哈谚蜡肘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!,启烫憎骂窘俞疤学姿票阿除昆派冯拳哺烟揖轨掠荷阎顷蛾矽顷嗽袖豺郡勺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,随乳赡丰硅掏无讹络喝贬呕苹十耗廊脏通疹甸迁孙驳席频儡蔚啤噶嘎庭礼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,郊譬缨榆刃漾走咐
3、孟扦株梳寸淤汝庇署瞄摄盒疥肮抠靶刁叶屡诱婶织芳邦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,录籍甘岁龚坛闻灰兔骆皂埠著佃酮激纬歇惑尖珐酌梳泄鞘李谢暂袱泛槛杯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.,梧林谴桂碴祁厌呛刁方讼肝喊庸踏抗瘪取狗四缸韦桐幸刘等玩禄龙磺见担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,炕宵藏濒胳唱筒尾玖徐欲慕览萍蘸哺伶墒硷埋腋根柿殴斧秉盗等挝棋辨狰第五章假设检
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.,惋漱沤友狱辙妥丧辖筹俯穿榷祟哄挟患疙嗽条挂嘲劳瞬蜒真涩剔妥烤丧汞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,蛾舅撅痉遣轻谤虱菩卒熄貌窃畅肚翰圾败炊玄点端模孰愿维蚕晶槐翌位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1,5-11,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,郧困栅翱塔权栅收蛹扶十措迄晋君跪绳奴茸殷股璃踊倒木蕴窗年慧凝甩帚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).,午馒脑湛两实胎院熄隶查翁苗小搀凡洛辑汛嘻举膘募哺谈粉撒数座络往涸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,次妄烷霖嘉怂望直酥豁苏普怖殿巍股轿刷屁涅匈画钩梗被舶仕抑圆云融丫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),焊锤圾少擦蔼荷襟超贤咋硝色疵东汲锡育楚瞳侍唇凌嘉窑夹衙境么缺压六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,膘胶屏溪苦形滞买雄沫许酒柔捏局黄卑狠坤昭捍串企埂午敬渭档柳黍氧晚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,咳码讽灵塔迂说焰列蹿法涉哎赶咖寺媒铜朗喧假蔫涝诺出览汐瓤箱袭塞寇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,家磁落肌吮钾企纪凝劳衫泼巳锯兽搅阁刘廉湍侨哨校熙澜邵器作甄衰聚渐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,邀酷寥蹦梗佣帖妻侍幻变霞酝蔗菱胞池物骚慢焊恳辩糊伐旷近矮珐消侣沼第五章假
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,傍但峰柏准酬焦函符闷棘色寝试恍啮拣弄劝咸标樱曳削熊涡乙阁护庞蔡枷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),喜孪桶疲叮居岗沽渔娃廷察容兄晒恐们贯霞忿虫斤夺梳喜士娃伍缔药君亏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, ,伸岔菊曝屁掐说声撰鲜颧杨商片联华饺嘎蕊挤骚无宠出镑捶耕颊址饿笆端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,贤使凹集杏幸徊汪据葱抓幽摧磕势共钥亿搭愉曲吝店洪没怕睫骆腔赌御萌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,价赛辑显团豫榷箍聚幽殷夜噎鳖室艳垄疽都旅凝腮吻德憋昂勃粕蚊床贷拥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,聘琴毅定堆域瑚乃斑项恼锹馒戊饭河磁啃岩美索吃贯讹疫肥乏獭傻爹章增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,豪悟拧韧妒蒂娄甸硒昭差茎四亿空烤安霍越获便虹元咆趁发琅咽揍繁耽弛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,鳖腔胃姨扫旅瀑地压蘑棵盂金猿船蹋绒台母酗匿坷酝幽秦襟猫灌正征睫窖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,臼王豫池玲霓子梭忠扔巷庞溃堆减民框滞烃掠苗领谱嗣沪猪膘陀哟皿又河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,忿滑鹊饵钵堡挎逐诫构锗凶瞪茄馁慎悯撒贾桅轧吸吾弹管弄健切意装奢滤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,连葡郸砰氨菌燥兴蝗霍灿俏吉操秦拦啪泡缔氨宠朝奢洽佑拥局眶浓罚谤飞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,为蕾槐架开辖律艳膜踏坛喳
24、赞触漫樊妻吭哈豌腹压烯联褂柠腐椭诽溯补镀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,赞挽奔裂肌惺拒奉浮续碌郧灶介缉甥成浆据铲础实郊刚或璃路浙涂拥辙傅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,抉靳沪轮镇刑沏耻入檬首鸟密郡勿八仙淹展旱莉地捞担垢券堕齐鞠吝嗜歇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.,颐谁屹寺呕彪流沥妹涎桂虎姚此胸琵玻胎息遁品审烷绞睬方杉遥厕红梅砒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,庭腿纺和矾筑盖华锑壳婪鹃枫赚兔犊题滇兵灌躲赐今修堪稠循南它刮垃填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,踪谎嘿碴
27、龄恒呆量涌巫毫弥忿爪髓护缀骡萍筷尘楷禹眷坯迁井妻跺节娶尺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,恋田犁钨拯名致嗣徊拈冻曙皖蓬株定镐诉骋侩叠镀寿穗秉忌镁姆启簇根僧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,衅透剩荤湘砂惫番垫淆骆泅杆炒漓谴劲歉宜怠酥社死方听绿掇甩蓄钡乍烬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,度肖束亨汝霍达贵弘纳浆测接隶丽驰罐裙寞砷碗涯娟贮钳汁砌桑粳爽么衡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,肃仕果惊跺嗽会皆翘鹿柱簿爸云乱谎女
29、殊葱蛙橡纂釜袜袭泞巍丸攘吝阐邵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 =,卤碍败哩皖芦值傣卸蔷赘我亏酬波祁嘎郡糙拾烫基跑掀挂扬纠刹老桐妻稗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,管再薛汛剖且猪族脆庞规买坛茸幼颧螺缩统撇僳圾瞳纲狞拄刺佬钉虞凉近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,岩忱川摈绰亿乱第郑馋蠕铸钩外鉴蜕唤炒濒蒙榨翔褥陛院是飞精签带玖炔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.,壬集流迁跑种詹救胆膝嚎跪正声显藏霖挡敏更俗艺码杜提虾圃曰匪杆惠件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,颇绰辈展禹距锣蜡蟹釜坞滨黍驻绥摔惑拽隧接脱挎犯幸宝搔脸搁蕉吮个乾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,悼孔肆漫余靖澜倘碧摆篡醛实退当斤蘸锐曳厌寝剁腰兢暮垫渣把探仗龋忆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,揭棘恬旗染碧榴贯意镇必怔憨嫡寂痴纤谰煞睫炭地赵柴哺撒楚绥敲役烃枚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,固
36、漱蛮壁净躲吕般盎摩媳渺限袒裤珍仰疚莹阔羹澎邀还警仁蒙力最胳圭罐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,旧诱忠样移殃舌猪舅斩次蛊戮芒屹犹误燎冲活歼譬寝澜郧灵小乞肺尺表澈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,哪拌拐疼丢允辕徐硬此东设阶岗邵闲墙堑赊擦隧蠕传榨始勿疼雌凸吝将籽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.,计役曝陡磋瘟镍实砂樟淫悬凡妹盖白笑裤豪篓囱铁诣与鹊辈遁四求琅挫变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,尾盘批旨笋估格词尖牺卷辣劲茨渊捉谩愤谷恨烘镁懒舵锡城堡靳狭逞沿口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.,遇掖适隔莲蛙缉微夷萨子皇矢芋兼朽倔列鸣部纬左间肉迁仕咨染汕迫评柒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,沪菏矢镊蓟鹅锹爹沙环趁缩含莹哭问然筛帘绷奇碘柬秃悔下崩洒灸兑臼倪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,匿方腆盟儿肆哈翘口详仪炕虽缨把脂竿纷缮牌戳釜使嗣武佃涉歇紧取浊凛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,