统计方法描述统计推断统计参数估计假设检验

1、2021-11-16安徽大学数学科学学院1Mind on statisticsMao Jun JunAnHui University2021-11-16安徽大学数学科学学院21 Measure for teaching ZH31020(专业核心课专业核心课 )双语双语(18 X 4)课件课件(中，英）中，英）教案（英）教案（英）作业（英）作业（英）考试（英）考试（英）(70%+30%)Note:周二周二 、周四、周四 第第1-18周周博南博南B401;2021-11-16安徽大学数学科学学院3Teaching materialMind On Statistics JESSICA M.UTTSH

2、ow is this book different?Two basic premises of learningOne premise is that new material is much easier to learn and remember if it is related to something interesting or previously knownThe second premise is that new material is much easier to learn if you actively ask questions and answer them for

3、 yourself.2021-11-16安徽大学数学科学学院4Mind On Statistics is designed to help you learn statistical ideas by actively thinking about them2021-11-16安徽大学数学科学学院5Teaching reference(1)D. A. Berry and B. W. Lindgren, Statistics Theory and Methods, Duxbury Press,1995, ISBN 0-534-50479-5.(2)G. Casella and J. O. Ber

4、ger, Statistical Inference, 2nd Edition, Brooks Cole, 2001,ISBN 0-534-24312-6.(3)M. H. De Groot, Probability and Statistics, 3rd edition, Addison-Wesley, 2001, ISBN0-201-52488-0.2021-11-16安徽大学数学科学学院6(4)W. Mendenhall, R. L. Scheaffer and D. D. Wackerly, Mathematical Statistics withApplications, Duxbu

5、ry Press, 6th Edition, 2002, ISBN 0-534-37741-6.(5)J. A. Rice, Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Press,1994, ISBN 0-534-20934-3.(6)G. W. Snedecor, W. G. Cochran, Statistical Methods, Iowa State University Press,8th Edition, 1989, ISBN 0-813-81561-4.2021-11-16安徽大学数学科学学院7

6、中文参考 (1) 统计学 高等教育出版社 袁卫等（2）应用数理统计学 人民大学出版社 周复恭 等（3）统计学习题与案例 高等教育出版社 袁卫，庞皓等（4） 统计学原理标准化题型习题集 东北财经大学出版社 栗方忠2021-11-16安徽大学数学科学学院8Teaching Aim2021-11-16安徽大学数学科学学院9Although this course is primarily concerned with mathematical techniques, I will also try, by means of examples and digressions, to introduce

7、 you to some of the non-mathematical aspects of Statistics.In a word: Not only hold mind on statistics but also apply models when we face actual problem2021-11-16安徽大学数学科学学院10Teaching contentsChapter 1 Preface Chapter2 Turning Data Into InformationChapter 3 Gathering Useful Data2021-11-16安徽大学数学科学学院11

8、Ch4 sampling Ch5 relationships between quantitative/categorical variablesCh61415 Regression Analysis and Correlation analysisCh7 Probability2021-11-16安徽大学数学科学学院12Chapter8 random variablesChapter 91012 Sampling Distribution and Estimating( confidence intervals)2021-11-16安徽大学数学科学学院13Chapter 1113 Hypot

9、hesis TestingChapter 16 Variance AnalysisChapter Time Series AnalysisChapter Index Analysis2021-11-16安徽大学数学科学学院14E-mailQQ:401784750(院外交流) (学生交作业) (答疑)2021-11-16安徽大学数学科学学院15几点要求2021-11-16安徽大学数学科学学院16谢谢合作！2021-11-16安徽大学数学科学学院17Chapter 1 Preface2021-11-16安徽大学数学科学学院18Learning objectsBackground of the st

10、atistics What is the statistics ( Emphases and Difficulties)Branches of the statistics2021-11-16安徽大学数学科学学院19【1.1】Background 1662（John Graunt)1676 (William Patty)1710 (Adolphe Quetlet)2021-11-16安徽大学数学科学学院202021-11-16安徽大学数学科学学院21statistics2021-11-16安徽大学数学科学学院22NOWInformation developed through the use

11、of statistics has enhanced our understanding of how life works, helped us learn about each other, allowed control over some societal issues, and helped individuals make informed decisions. There is almost no area of knowledge that has not been advanced by statistical studies.2021-11-16安徽大学数学科学学院23St

12、atistical methods have contributed to our understanding of health, physics, medicine, engineering, psychology, ecology, politics, music, business, lifestyle choices, and dozens of other topics.2021-11-16安徽大学数学科学学院24Although statistics courses are often offered through mathematics departments, statis

13、tics is not a branch of mathematics. Mathematics is to statistics as wood, hammer, and nails are to a house: a partial set of materials and tools. In addition to mathematics, statistics also draws materials and tools from philosophy, graphics, computing, psychology, and language.2021-11-16安徽大学数学科学学院

14、25(1.2) What is statistics ( p1) When you hear the word “statistics”, you probably think of lifeless or gruesome numbers. The word “statistics” however, actually is used to mean different things. The better-known definition is that:Define 1( p1) : numbers measured for some purpose (data)2021-11-16安徽

15、大学数学科学学院26A more complete definition is the following:Define 2 (p1) : a collection of procedures and principles for gathering data and analyzing information in order to help people make decisions when faced with uncertaintythat is to say:2021-11-16安徽大学数学科学学院27Five basic stepsP7Asking the right quest

16、ionsCollecting useful dataSummarizing and analyzing dataMaking decisions and generalizationsTurning the data and subsequent decisions into new knowledgeOR:2021-11-16安徽大学数学科学学院28Step1: Define problem: The definition of the problem is very crucial at the start of an investigation as it acts as the obj

17、ective of investigationStep2: Design experiment: This includes deciding whether to use the population or select a sample. One should also define what it is that is being observed and how it should be measured.2021-11-16安徽大学数学科学学院29 Step3:Collect data: The researcher must decide how to collect the pr

18、imary data and secondary data.Step4:Analysis data: the focus of the course. The aim is turning data into information.Step5: Report results: The reporting of results is one of the most important exercise in the experiment. Results should be presented in such a way that the receiver can understand the

19、m with as little difficulty as possible.2021-11-16安徽大学数学科学学院30Define 3: Statistics is a body of principle and methods concerned with extracting useful information from a set of numerical data. Such as Bayes Theorem.2021-11-16安徽大学数学科学学院31 Statistics: the science of collecting, analyzing, presenting,

20、and interpreting data. 2021-11-16安徽大学数学科学学院32什么是统计学?1. 数据搜集：取得数据2. 数据分析：分析数据 3. 数据表述：图表展示数据4.数据解释：结果的说明2021-11-16安徽大学数学科学学院33收集数据收集数据整理数据整理数据解释数据解释数据分析数据分析数据2021-11-16安徽大学数学科学学院34(1.3) Branches of statistics (a) Descriptive Statistics(b) Inductive (inferential) Statisticsor( A) Theoretical Statistic

21、s(B) Applied statistics2021-11-16安徽大学数学科学学院35Descriptive Statistics:Descriptive Statistics deals with the various measures that can be used to describe a set of data. These measures look at where the focal point of the data is, for example: mean, and how the data fluctuates, for example :median, low

22、er quartile, upper quartile, standard deviation. Story 1.1 1.22021-11-16安徽大学数学科学学院36Case Study 1.1 Who Are Those Speedy Drivers? Question: Whats the fastest you have ever driven a car? mph.Data: 87 male and 102 female students from large statistics class at University.Which gender has driven faster?

23、 How to summarize data? 2021-11-16安徽大学数学科学学院37Males: 110 109 90 140 105 150 120 110 110 90 115 95 145 140 110 105 85 95 100 115 124 95 100 125 140 85 120 115 105 125 102 85 120 110 120 115 94 125 80 85 140 120 92 130 125 110 90 110 110 95 95 110 105 80 100 110 130 105 105 120 90 100 105 100 120 100

24、100 80 100 120 105 60 125 120 100 115 95 110 101 80 112 120 110 115 125 55 90Females: 80 75 83 80 100 100 90 75 95 85 90 85 90 90 120 85 100 120 75 85 80 70 85 110 85 75 105 95 75 70 90 70 82 85 100 90 75 90 110 80 80 110 110 95 75 130 95 110 110 80 90 105 90 110 75 100 90 110 85 90 80 80 85 50 80 1

25、00 80 80 80 95 100 90 100 95 80 80 50 88 90 90 85 70 90 30 85 85 87 85 90 85 75 90 102 80 100 95 110 80 95 90 80 902021-11-16安徽大学数学科学学院38Case Study 1.1 Who Are Those Speedy Drivers? Dotplot: each dot represents the response of an individual student.2021-11-16安徽大学数学科学学院392021-11-16安徽大学数学科学学院40Case St

26、udy 1.1 Who Are Those Speedy Drivers? Five-number summary: the lowest value, the cutoff points for , , and of the data, and the highest value.Note: of men have driven 95 mph or more, only of women have done so.2021-11-16安徽大学数学科学学院412021-11-16安徽大学数学科学学院42Moral: Simple summaries of data can tell an in

27、teresting story and are easier to digest than long lists.Definitions: data median lower quarter upper quarter2021-11-16安徽大学数学科学学院43Case Study 1.2 Safety in the Skies?“Planes get closer in midair as traffic control errors rise”“Errors by air traffic controllers climbed from 746 in fiscal 1997 to 878

28、in fiscal 1998, an 18% increase.” USA Today, Levin, 19992021-11-16安徽大学数学科学学院44Sounds ominous, but “The errors per million flights handled by controllers climbed from 4.8 to 5.5.”So the original rate of errors in 1997, from which the 18% increase was calculated, was less than 5 errors per million fli

29、ghts. 2021-11-16安徽大学数学科学学院45Moral: When discussing the change in the rate or risk of occurrence of something, make sure you also include the base rate or baseline risk.Definitions: rate risk base rate base risk2021-11-16安徽大学数学科学学院46Inductive or inferential statisticsInferential statistics refers to

30、a body of principles that enables one to infer or reach some conclusions about a population by simply observing a sample from that population. It may be time consuming, expensive or inappropriate to look at the population at large. Story 1.3 1.4 1.5 1.6 1.7 2021-11-16安徽大学数学科学学院47“According to a new

31、USA Today/Gallup Poll of teenagers across the country, 57 percent of teens who go out on dates say theyve been out with someone of another race or ethnic group.” (Peterson, 1997).Sacramento Bee headline read: “Interracial dates common among todays teenagers.”Case Study 1.3 Did Anyone Ask Whom Youve

32、Been Dating? 2021-11-16安徽大学数学科学学院48Millions of teenagers in U.S. - Did polltakers ask all of them? No.The article states “the results of the new poll of 602 teens, conducted Oct. 1320, reflect the ubiquity(普遍性） of interracial dating today” Asked only 602 teens. Could such a small sample tell us anyt

33、hing about the millions of teenagers in the U.S.? Yes if those teens constituted a random sample from the population.2021-11-16安徽大学数学科学学院49Case Study 1.3 Did Anyone Ask Whom Youve Been Dating? The percent of all teenagers in the US who date that would say they have dated interracially is likely to b

34、e in the range 57% 5%, or between 52% and 62%.How accurate could this sample be?Margin of error is about 5%.2021-11-16安徽大学数学科学学院50Moral Study 1.3 : A representative sample of only a few thousand, or perhaps even a few hundred, can give reasonably accurate information about a population of millions.2

35、021-11-16安徽大学数学科学学院51Case Study 1.4 Who Are Those Angry Women? Shere Hite sent questionnaires to 100,000 women asking about love, sex, and relationships. Only 4.5% of the women responded, and Hite used those responses to write her book, Women and Love. “The women who responded were fed up with men a

36、nd eager to fight them. For example, 91% of those who were divorced said that they had initiated the divorce. The anger of women toward men became the theme of the book.” Moore (1997, p. 11). 2021-11-16安徽大学数学科学学院52Extensive nonresponse from a random sample, or the use of a self-selected (i.e., all-v

37、olunteer) sample, will probably produce biased results.Moral Study 1.4 : An unrepresentative sample, even a large one, tells you almost nothing about the population.2021-11-16安徽大学数学科学学院53Case Study 1.5 Does Prayer Lower Blood Pressure? “Attending religious services lowers blood pressure more than tu

38、ning into religious TV or radio, a new study says”USA Today headline read: “Prayer can lower blood pressure.” (Davis, 1998)2021-11-16安徽大学数学科学学院54Based on observational study, followed 2391 people 6 years.“People who attended a religious service once a week and prayed or studied the Bible once a day

39、were 40% less likely to have high blood pressure than those who dont go to church every week and prayed and studied the Bible less.” Researchers did observe a relationship, but its a mistake to conclude prayer actually causes lower blood pressure.2021-11-16安徽大学数学科学学院55Case Study 1.5 Does Prayer Lowe

40、r Blood Pressure? In observational studies, groups can differ in important ways that may contribute to the observed relationship.People who attended church regularly may have been less likely to smoke or drink alcohol;had a better social network;been somewhat healthier and able to go to church.These

41、 other factors are possible confounding variables.2021-11-16安徽大学数学科学学院56Moral: Cause-and-effect conclusions cannot generally be made based on an observational study.Definitions :observational study (merely observed ); variable ; confounding variable2021-11-16安徽大学数学科学学院57Case Study 1.6 Does Aspirin R

42、educe Heart Attack Rates? Physicians Health Study (1988)5-year randomized experiment 22,071 male physicians of age 40 - 84; randomly assigned to one of two treatment groups; Group 1 = aspirin every other day; Group 2 = placebo; Physicians blinded as to which group they were in.2021-11-16安徽大学数学科学学院58

43、Case Study 1.6 Does Aspirin Reduce Heart Attack Rates? 2021-11-16安徽大学数学科学学院59Aspirin group: 9.42 heart attacks per 1000 participantsPlacebo group: 17.13 heart attacks per 1000 participantsRandomization = other important factors (age, diet, etc) should have been similar for both groups. Only importan

44、t difference should be whether they took aspirin or placebo.2021-11-16安徽大学数学科学学院60Moral: Unlike with observational studies, cause-and-effect conclusions can generally be made on the basis of randomized experiments.2021-11-16安徽大学数学科学学院61Case Study 1.7 Does the Internet Increase Loneliness and Depress

45、ion? “greater use of the Internet was associated with declines in participants communication with family members in the household, declines in size of their social circle, and increases in their depression and loneliness” (Kraut et al., 1998, p. 1017)New York Times headline read: “Sad, Lonely World

46、discovered in Cyberspace.” (Harmon, 1998)2021-11-16安徽大学数学科学学院62Study included 169 individuals from 73 households in Pittsburgh given free computers and internet service. Participants answered questions at beginning and either 1-2 years later on social contacts, stress, loneliness, depression.2021-11

47、-16安徽大学数学科学学院63Changes were quite small, although significant # of people in “local social network” decreased from average of 23.94 to 22.90 people; on scale 1 to 5, self-reported loneliness decreased from average of 1.99 to 1.89 (lower = greater loneliness). on a scale 0 to 3, self-reported depress

48、ion dropped from average of 0.73 to 0.62 (lower = higher depression).Case Study 1.7 Does the Internet Increase Loneliness and Depression? 2021-11-16安徽大学数学科学学院64Follow-up study: “Using the Internet at home doesnt make people more depressed and lonely after all.” (Elias, 2001)Whether the link ever exi

49、sted will never be known, but it is not surprising, given the small magnitude of the original finding, that it subsequently disappeared.2021-11-16安徽大学数学科学学院65Moral: A “statistically significant” finding does not necessarily have practical importance. When a study reports a statistically significant

50、finding, find out the magnitude of the relationship or difference. Case Study 1.7 Does the Internet Increase Loneliness and Depression? 2021-11-16安徽大学数学科学学院66A secondary moral to this story is that the implied direction of cause and effect may be wrong. In this case, it could be that people who were

51、 more lonely and depressed were more prone to use the Internet. And, as the follow-up research makes clear, remember that “truth” doesnt necessarily remain fixed across time.2021-11-16安徽大学数学科学学院67Therefore ,inferential statistics plays an Important role in enabling the researcher to use only a small

52、 randomly selected sample to predict the characteristics of population. For example: random sample, sample survey, margin of error, Hypothesis Testing, confidence interval2021-11-16安徽大学数学科学学院68统计方法统计方法统计方法描述统计描述统计推断统计推断统计参数估计参数估计假设检验假设检验2021-11-16安徽大学数学科学学院69描述统计(descriptive statistics)1.研究数据收集、整理和描

53、述的统计学分支 2.内容l搜集数据l整理数据l展示数据l描述性分析3. 目的l描述数据特征l找出数据的基本规律2021-11-16安徽大学数学科学学院70推断统计 (inferential statistics)1.研究如何利用样本数据来推断总体特征的统计学分支2.内容l参数估计l假设检验3.目的l对总体特征作出推断2021-11-16安徽大学数学科学学院71描述统计与推断统计的关系2021-11-16安徽大学数学科学学院72(3) Branches of statistics ( A) Theoretical Statistics(B) Applied statistics2021-11-16安徽大学数学科学学院73In Summary: Morals1 Simple summaries of data can tell an interesting story and are easier to digest than long lists.2 When d