大三上统计学and distributions

1、 (Statistics in Practice) Mei Di home electric machine company in ShunDe, GuangDong province is one of the largest air- condition manufacture factories in china, it sells 7000000 stages of air-condition per year, and saleroom is twelve thousand million yuan. This low-pitched, some mysterious looked

2、by outside corporation managed by civilian in ShunDe, though it cry “dont want to be the biggest in its industry”, in these years increase strongly, increment rate is up 40percents, profitability is sky-high. The backside causation is Mei Di advance corporation competition ability from early years.

3、For avoiding the low price and profit complexion, realizing to manage in varies of ways, holding advantage in new round competition, the president Mr.He in Mei Di home electric machine company bring out the stratagem target that taking part in car industry. He requested sell-apartment make diagnoses

4、 about throughput, saleroom, management and profit ability, market possess rate, and so on in other big car manufacture factories in world. Fang Hongpo, the sell apartment manager, said he must think about how to collect the economic secret data in other car manufactures, and if got, in which way to

5、 analyze the data and conclude. All of those shall use stat. conclude knowledge. STAT We start to study conclude statistics from this chapter, it save peoples time and property to understand objects best bounds.In real world there is a great of material aggregate, its difficult to pick up the inform

6、ation we need from that. For example, Electorate number: what is the sustain rate of every candidate? Product: whats the unqualified rate?Environment: how about the pollution degree?Market: know the conditions of variety, price, quality, purchasing power and so on . In this chapter, you will learn h

7、ow to draw out stylebook, how to do the stylebook statistics distribution, how to estimate aggregate parameter according to the stylebook statistics?STATSTATEmphasis1,simple random sampling;2,sampling distribution of ;3,sampling distribution of ;4,other sampling methods.Difficultysampling distributi

8、on principle.Bibliography1,LiXinyu:The application economic statistics , University of Peking publisher;2,David S.Moore:The statistics world , ZhongXin publisher;3,YuanWei:The statistics course , economic science publisher;4,statistics website:The UNSD,OECD,Chinese nation statistics bureau.xpSTATExa

9、mple There are 10000 pigs in hogpen, now we want to get the information of the average gross weight of these pigs (establish for , ) It isnt cost effective to weigh up each pig to get datas. Weigh up 100 pigs which draw out from all pigs according to “equal opportunity principle”, compute these 100

10、pigs average gross weight to attain the purpose of our expectation.STATIn this example draw out 100 pigs , called sample population, it means in statistics sampling, according to“equal opportunity principle draw out parts of units from population N(10000) (each unit call the sample unit), whole part

11、s of units constitute sample, also called . Generally the amount of units of sample population means with the n(100), called sample capacity.The sample population do not have only one, its possible number is concern with N,n and the sampling methods.Usually n30 is called big sample, takes the big or

12、 small sample in sampling will influence the characteristic of the sampling distributions directly.I.basic conceptions of statistics samplingII.1. population and sampleIII. IV. Population:The set of all elements of interest in a particular study,also called . The capacity means with N.Have only one.

13、 V. VI. Sample:All of partial units in term of random principle draw out from totally, abbreviation is sample, each unit which was draw out called sample unit.The capacity means with the n.The sample has not only one.STAT2. population parameter and sample statisticsChange the comprehensive index sig

14、n of population and some quantity characteristics of calculating reflect the quantity value according to population each unit, because of population assurance and only one, so call total parameter.As above example Change the comprehensive index sign of sample population and some quantity characteris

15、tics of calculating reflect the quantity value according to sample population each unit, because of the sample do not have only one, so called the sample statistics, it is a random variable.As above example 100 pigs average gross weight(95.5 kg)STAT3. sampling without replacement and sampling with r

16、eplacementSampling with replacement, once an element has been included in the sample, it is returned to the population. A previously selected element can be selected again and therefore may appear in the sample more than once.The probability of sample unit which is draw out is equal every time.We ca

17、ll such sampling is independent mutual trial.Sampling without replacement, once an element has been included in the sample, it is removed from the population and cannot be selected a second time. Compared to sampling with replacement, the conditions of sampling without replacement are different,the

18、previous result will influence the followed result, we call such sampling is not independent mutual trial.Attention:Two methods all followed“equal opportunity principle”STAT Simple random sampling is also called pure random sampling.Sample random sampling is a research method that isnt necessary to

19、categorize or line up the population unit,but samples the sample unit from population directly in term of random principle. STAT For easy to sample the sample unit, generally in the condition of understand the sampling frame,code every unit of population , then use sortition or userandom numerical f

20、orm to carry on sampling. For example:N=500 n=10 Code from 1st-500thOrder two random numbers, get 54-50=4 line, 34 rows.The number then selected by examinations is beginning from this number, because 500 is a three numbers, less than 500 continuously three numbers are elected numbers.Show in the for

21、m below.STATRANDOM NUMBERS 97452389429745238942 12764659091276465909 98747636429874763642 26593059842659305984 1 1676587676587006006 34899624353489962435 98663328909866332890 80365223648036522364 70654363877065436387 1321327697690870879 9 12870877651287087765 21362177212136217721 9878764346987876434

22、6 48908327694890832769 21648965892164896589 70774344317077434431 14228900121422890012 08743211230874321123 0430437 7575967575967 2 2132132577995577995 94242523869424252386 48799034434879903443 2177609552177609554 4 21214874879759754444 75376979977537697997 03777976840377797684 98778084239877808423 2

23、7780068692778006869 21337687902133768790 82621308928262130892 95359535443208443208 21489900852148990085 7067065 5432549432549 06564332230656433223 24379098542437909854 64767964767932433243 43870053454387005345 21648784542164878454 21765908792176590879 21676089652167608965 43436578979665789796 435865

24、08414358650841 93432525349343252534 43876707694387670769 46375674884637567488 12548769871254876987 67432198456743219845 32489060343248906034 07654332450765433245 87078676988707867698 32865489003286548900 80846342128084634212 43326577904332657790 79636453247963645324 90874343299087434329 237698766723

25、76987667 21378607692137860769 88005232678800523267 43797343434379734343 38748560493874856049 32547769073254776907 3243243 3700435700435 21877999902187799990 13587870081358787008 21257497682125749768 23658790482365879048 87659802348765980234 12688032351268803235 93233147669323314766 23236689743166897

26、431 76944327677694432767 90942321559094232155 02323379320232337932 03622123790362212379 34787942353478794235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STATSTAT3. Point estimationPoint estimate is a single numerical value used as an estimate of a population parameter.As above exampl

27、e: 100 pigs average gross weight(95.5 kg) can be used as the point estimator of 10000 pigs average gross weightPoint estimator:(1)refer to sample mean as the point estimator of the population mean ;(2)refer to sample standard deviation as the point estimator of the population standard deviation ;(3)

28、refer to sample proportion As the point estimator of the population proportion P. x22SPSTAT For the population characteristic, not all of the point estimators are good, this bring in the standard of evaluation that point estimator is good or not.Being a good estimator should match as follows three s

29、tandards:STATA, unbiasednessA property of a point estimator when the expected value of the point estimator is equal to the population parameter it estimates, such as the expected value of sample mean is population mean,so sample mean is an unbiased estimator of the population mean.This unbiased esti

30、mator is the unbiasedness of system(not random biasedness)average estimator here, namely if an estimator is unbiaseness, we will not assure that there is no random error in one estimate, just there is no unbiasedness of system.This is the important condition that a good estimator has. is the populat

31、ion parameter, is an unbiased estimator of So: ESTATB, consistency If estimator with the sample capacity n but close to population parameter more and more, then called estimator is the consistency estimator of estimate parameter. The consistency of the estimator is meaning of the limit,it is fit for

32、 big sample.If an estimator is a consistency estimator, big sample is more dependable.Certainly, When the sample capacity n enlarge, the consistency estimator will strengthen, but the human,wealth,material resources of research are also increasing.For example, let sample mean estimate population mea

33、n, meet the request of the consistency, then exist relation as follows:1limxPnSTATis a freewill positive numberC.efficiency efficiency is the estimator of smallest deviation in unbiased estimators.Being not partial to estimate the quantity consideration to estimate whether the average result of the

34、value is equal to a true value that need to be the estimate of parameter, but takes no account of estimated each one and may be worth and count to distributions and need the estimate of parameter to really be worth secondly or not the long-lost degree that leave to differ size.We not only hopes the

35、estimate of value was partial to, but also hopes leave of these estimate of be worth is bad possibly while resolve actual problem small, then have each request to didnt be partial to estimate quantity in with is estimate parameter to leave bad smaller of measure for the valid estimate of.If the samp

36、le equally counts is all total with medians to all be worth of be not partial to estimate quantity, but under the same sample capacity, sample mean is efficiency estimator.nMDnxDe222)()(STAT6.2 sampling distributions From a population draw out various sample with the same capacity random, compute th

37、e possibility distribution of all statistics possible result by these samples, it is called the distributions of the statistics.In conclude sampling, not only population but also sample can describe the characteristics by mean, proportion(or percentage), deviation. When they used to describe the cha

38、racteristic of sample, are called the sample statistics;When they used to describe the characteristic of population, are called population parameter. The structure of sampling distributions include such steps as follows:(1) Draw out the probable sample with the capacity is n from the finite populati

39、on with the capacity is N;(2)Calculate the statistic number of each sample;(3)Calculate the probability for each sample statistics.STATExample:Draw out 40 units to weigh up tobacco(X) in a box of cigarettes (5 50 200=50000), then analyze the tobacco weight of these cigarettes.1xnxSampleSample indexN

40、=50000n =40)!( !nNnNNMnSampling with replaceSampling without replaceX1,X2, XNNXiSTATChapter 6 sampling and sampling distributionsArrange into distribution rows,get a following form:mxxx21Sample meanProbability(Frequency) p1 p2 pk1x2xkxBecame a sampling distributions formThe example below explained t

41、he formation of sampling distributions againSTATReview a population N=6(a 6 point dice), its original distribution is average distributionX123456P1/61/61/61/61/61/6From this population, sample with replacement n=2(two dices throw point at the same time),the total amount of all probable sample is Nn=

42、36, suppose to estimate the population mean through the sample, all 36 possibility results is:STAT FIRSTSECOND123456111.522.533.521.522.533.54322.533.544.542.533.544.55533.544.555.563.544.555.56STAT11.5 22.5 33.5 44.555.5 6 p1/362/363/364/365/366/365/364/363/362/361/36xSTATSTATxx xExSTATxxxxxxxSTATE

43、xample: The age of the population A,B,C is:1,2,3,N=3 N=2. = 2 bias no single 2bias single5.1bias single 1321xxx SampleSample x x ix M Me e （A A、A A） （A A、B B） （A A、C C） （B B、A A） （B B、B B） （B B、C C） （C C、A A） （C C、B B） （C C、C C） 1,11,1 1,21,2 1,31,3 2,12,1 2,22,2 2,32,3 3,13,1 3,23,2 3,33,3 1 1 1.5

44、1.5 2 2 1.5 1.5 2 2 2.5 2.5 2 2 2.52.5 3 3 1 1 1.51.5 2 2 1.51.5 2 2 2.52.5 2 2 2.52.5 3 3 TotalTotal 1818 1818 2918)(MxxEi2)(MMMEee xEMeESTAT2.Sample standard error ( )Definition: All sample statistics the average of sampling error , (adopt standard calculate form).A:The sampling standard error of

45、sampling statistics0)(MxiMxi2)(xiMx2)(N=3(A,B,C)=(1,2,3)N=3(A,B,C)=(1,2,3)n=2n=2 SampleSample Data Data ix ix （A A、A A） （A A、B B） （A A、C C） （B B、A A） （B B、B B） （B B、C C） （C C、A A） （C C、B B） （C C、C C） 1,11,1 1,21,2 1,31,3 2,12,1 2,22,2 2,32,3 3,13,1 3,23,2 3,33,3 1 1 1.51.5 2 2 1.5 1.5 2 2 2.5 2.5 2

46、2 2.5 2.5 3 3 y y1 1= =- -1 1 y y2 2= =- -0.50.5 y y3 3=0=0 y y4 4= =- -0.50.5 y y5 5=0=0 y y6 6=0.5=0.5 y y7 7=0=0 y y8 8=0.5=0.5 y y9 9=1=1 Total Total 1818 0 0 Probable result of sample M=9， =2 B:The sampling standard error of the sampling percentage PMPpip2)(xxxSTAT SAMPLING ERROR（1）Sampling mea

47、ns mistake of Sampling errorA、Repeat sampling58. 03193)(2Mxix32)(2NX312132nxSTATB、sampling without repeat32)(2NX41. 061)(2MxixNnnNnNnx116113232132STATFinite populationInfinite population1NnNnxnxIn the finite population1NnNIs correction factor，generally simplified writingNn1STATAs the sample size n30

48、，then however weather you know population distributions condition，the distribution of sampling mean tend to normal distribution，and this distribution is more concentrate than other distribution，then nx22Of which is variance of sampling mean， is populationdistribution2x2Theorem： Set X is a totality w

49、hich has expected value and variance deviation ，then sampling distribution of sample mean，will tend to normal distribution follow with the larger of n，the style of distribution (parameter) is N （ ）-Central Limit Theoremn2,2xSTAT140 150 160 170 180 1900.50.40.30.20.1HETIGHT(take know totality for exa

50、mple)STATWhen the group number n infinity ，fold line curve。190. 0)(180150180150dxxfXPHEIGHT140 150 160 170 180 1900.050.040.030.020.01190. 0P),(2NX22)(2121)(xexfSTATJustify：“concentration of frequency”（frequency/the distance of the group）“frequency”； The area under histogram or broken line= 1140 150

51、 160 170 180 190HEIGHTSTAT Note: parameter 、 are different the shape and the position of distribution are different。),(211NX),(222NX),(233NX22)(2121)(xexfSTATxZ1)0(0)(2NZZDNZZE x1 x2-Z 0 Z),(2NX) 1 , 0(:ndistrbutio normal standardNZXZIs easy to prove thatStandardization:STAT162 170 178-z/2 0 z/2)8 ,

52、170(2NX) 1 , 0( NZXZ81701788170162178162xPxP%27.6812/ZZP122STATxZ variableppp PpEpSTAT Finite population Infinite population11NnNnpppnppp1p515pnnpSTATTheorem：Set p has expected value P，variance deviation is the totality of P(1-P) ，then the sampling distribution of sample proportion ， and will tend t

53、o normal distribution follow with larger of n ，the style of distribution (parameter) is N （ ）-Central Limit TheoremnPPP1,ppAttention ：when research the sampling distribution of sample proportion ，it can be seem as mean to analysis 。STATP0308. 005. 0P51901510pnpn220095. 005. 0038. 010308. 0nppzSTAT T

54、he way of sampling organization is mean the style about regulate the totality of sampling。According to the different style of regulate the totality ，there are many different ways of sampling organization in sampling investigate ,except simple random sampling，is also have typical sampling、equal space

55、 sampling、cluster sampling、Multi- stages sampling、convenience sampling and the other ways of how to sampling。1、Typical sampling Typical sampling is also named stratified random sampling or classify sampling。The population is first divided into groups of elements called strata，and it use simple rando

56、m sampling or the other sampling methods to sampling sample unite in all of the groups.STATCase The ages of 10 person。N=10 n=3，judge all the average age.person： A B C D E F G H I JAge ： 5 8 12 40 42 46 48 70 72 76simple random sampling （ B 、 H、 I ），（ C、 D 、 E ），（ F 、 G 、 I ）Conclusion ：when totality

57、 e larger typical sampling 。typical sampling （ B 、 E、 I ），（ C、 D 、 H ），（ A 、 G 、 J）STATThe typical sampling should be used in the obvious different sampling of the each unit in totality ，such as research the products of agricultures ，The farming has the plain, the knoll and the mountainous region an

58、d so on；research the lever of salary about workers，the obvious different of any jobs。Typical sampling is exactly divided the statistics into groups .it is also together the sampling organization way of the sampling principle 。Through divided the groups ，it can make the groups May enable in the group

59、 to have the homogeneity, the group has the difference, then from each group of center simple random sampling。It can be sure that sample has more representative than totality, so the sampling error will be smaller. Typical sampling should grasp the main principle is ：the differences in groups must a

60、s small as possible when divided the groups ，and the differences between the groups should be as larger as possible。STAT Set there are N units in totality，now we need to sampling a sample which capacity is n, can follow the mark of totality units N，then divided N into n equal parts ，which each parts