第十一章抽样法(统计学原理-南开大学,陆宇建)21078.pptx

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1、第十一章第十一章抽样法抽样法1第一节第一节抽样法的意义和作用抽样法的意义和作用一、抽样法的特点一、抽样法的特点抽抽样样法法在在统统计计调调查查和和统统计计分分析析中中都都有有广广泛泛的的应用。应用。抽样法是按照随机原则从全部研究对象中抽抽样法是按照随机原则从全部研究对象中抽取一部分单位进行观察，并依据所获得的数取一部分单位进行观察，并依据所获得的数据对全部研究对象的数量特征做出具有一定据对全部研究对象的数量特征做出具有一定可靠性的估计判断，从而达到对全部研究对可靠性的估计判断，从而达到对全部研究对象的认识的一种统计方法。象的认识的一种统计方法。2抽样法的基本特点：抽样法的基本特点：（1）根据）

2、根据部分实际资料部分实际资料对全部总体的数量对全部总体的数量特征作出估计。特征作出估计。通过抽样调查，取得部分单位的实际材料，通过抽样调查，取得部分单位的实际材料，据以计算抽样的综合指标，然后对于总体据以计算抽样的综合指标，然后对于总体的规模、水平、结构指标作出估计。的规模、水平、结构指标作出估计。3（2）按）按随机的原则随机的原则从全部总体中抽选样本单从全部总体中抽选样本单位。位。（3）抽样推断的）抽样推断的抽样误差抽样误差可以事先计算并反可以事先计算并反加以控制。加以控制。抽样推断是以部分资料推算全体，虽然存在抽样推断是以部分资料推算全体，虽然存在一定的抽样误差，但它可以事先通过一定资一定

3、的抽样误差，但它可以事先通过一定资料加以计算，并且能够采取一定的组织措施料加以计算，并且能够采取一定的组织措施来控制这个误差范围，保证抽样准断的结果来控制这个误差范围，保证抽样准断的结果达到一定的可靠程度。达到一定的可靠程度。4二、抽样法的作用二、抽样法的作用第一，对某些不可能进行全面调查而又要了解第一，对某些不可能进行全面调查而又要了解其全面情况的社会经济现象，必须应用抽样法。其全面情况的社会经济现象，必须应用抽样法。如，工业生产中检验某些产品的质量时，常常如，工业生产中检验某些产品的质量时，常常具有破坏性。如轮胎的里程检验、灯泡的寿命具有破坏性。如轮胎的里程检验、灯泡的寿命检验，纱布的强力

4、检验、炮弹的杀伤力检验等。检验，纱布的强力检验、炮弹的杀伤力检验等。有些现象的总体过大，单位过于分散，进行全有些现象的总体过大，单位过于分散，进行全面调查实际上是不可能的，例如要检验水库的面调查实际上是不可能的，例如要检验水库的鱼苗数，森林的木材积蓄丝等。鱼苗数，森林的木材积蓄丝等。5第二，对某些社会经济现象虽然可以进行全面第二，对某些社会经济现象虽然可以进行全面调查但抽样法仍然有其独到的作用，例如：调查但抽样法仍然有其独到的作用，例如：抽样调查可以节省人力、费用，提高调查的经抽样调查可以节省人力、费用，提高调查的经济效果。济效果。抽样调查可以节省时间，提高调查的时效性。抽样调查可以节省时间，

5、提高调查的时效性。抽样调查由于调查单位少，调查队伍经过专门抽样调查由于调查单位少，调查队伍经过专门训练，可以增加调查项目，取得比较详细的资训练，可以增加调查项目，取得比较详细的资料，并且提高资料的准确性。料，并且提高资料的准确性。6第三，抽样调查和全面调查同时进行，第三，抽样调查和全面调查同时进行，可以发挥相互补充和检查质量的作用。可以发挥相互补充和检查质量的作用。第四，抽样法可以用于工业生产过程的第四，抽样法可以用于工业生产过程的质量控制。质量控制。第五，利用抽样法原理，还可以对于第五，利用抽样法原理，还可以对于某种总体的假设进行检验，来判断这种某种总体的假设进行检验，来判断这种假设的真伪，

6、决定行动的取舍。假设的真伪，决定行动的取舍。7三、抽样法的理论基础三、抽样法的理论基础（一）大数法则（一）大数法则就数量关系来说，抽样推断是建立在概就数量关系来说，抽样推断是建立在概率论的大数法则基础上，大数法则的一率论的大数法则基础上，大数法则的一系列定理为抽样推断提供了数学依据。系列定理为抽样推断提供了数学依据。8大数法则即关于大量的随机现象具有稳定大数法则即关于大量的随机现象具有稳定性质的法则。它说明如果被研究的总体是性质的法则。它说明如果被研究的总体是由由大量的相互独立的大量的相互独立的随机因素所构成，而随机因素所构成，而且每个因素对总体的影响都且每个因素对总体的影响都相对地小相对地小

7、，那，那么对这些大量因素加以综合平均的结果，么对这些大量因素加以综合平均的结果，因素的个别影响将相互抵消，而显现出它因素的个别影响将相互抵消，而显现出它们们共同作用的倾向共同作用的倾向，使总体具有稳定的性，使总体具有稳定的性质。质。9联系到抽样推断来看，大数法则证明：如果随联系到抽样推断来看，大数法则证明：如果随机变量总体存在着有限的平均数和方差，则对机变量总体存在着有限的平均数和方差，则对于充分大的抽样单位数于充分大的抽样单位数n，可以几乎趋近于，可以几乎趋近于l的的概率，来期望抽样平均数与总体平均数的绝对概率，来期望抽样平均数与总体平均数的绝对离差为任意小，即对于任意的正数。有：离差为任意

8、小，即对于任意的正数。有：10（二）中心极限定理（二）中心极限定理大数法则论证了大数法则论证了抽样平均数抽样平均数趋近于总体平均数的趋趋近于总体平均数的趋势，这为抽样推断提供了重要的依据。但是，抽样势，这为抽样推断提供了重要的依据。但是，抽样平均数和总体平均数的离差究竟有多大平均数和总体平均数的离差究竟有多大?离差不超过离差不超过一定范围的概率究竞有多少一定范围的概率究竞有多少?这个离差的分市怎样？这个离差的分市怎样？大数法则并没有在这方面给出什么信息。大数法则并没有在这方面给出什么信息。这个问题要利用另一重要的定理，即这个问题要利用另一重要的定理，即中心极限定理中心极限定理来研究。中心极限定

9、理证明：如果总体变量存在有来研究。中心极限定理证明：如果总体变量存在有限的平均数和方差，那么不论这个总体变量的分布限的平均数和方差，那么不论这个总体变量的分布如何，随着抽样单位数如何，随着抽样单位数n的增加，抽样平均数的分布的增加，抽样平均数的分布便趋近于便趋近于正态分布正态分布。11INTRODUCTION TO INFERENTIAL STATISTICSINTRODUCTION TO INFERENTIAL STATISTICS Statistical inferenceStatistical inference is the process of making generalizati

10、on about a is the process of making generalization about a population from a sample.population from a sample.Since most of the characteristics of a population can be described by Since most of the characteristics of a population can be described by parameters,inferential statistics primarily dealspa

11、rameters,inferential statistics primarily dealswith the estimation of an unknown with the estimation of an unknown population parameter from the population parameter from the corresponding sample statistic.corresponding sample statistic.with the verification whether a belief with the verification wh

12、ether a belief or hypothesis about a parameter is or hypothesis about a parameter is supported by the sample evidence.supported by the sample evidence.EstimationEstimationHypothesis testingHypothesis testing（E.g.E.g.：We estimate We estimate probability measures probability measures from relative fro

13、m relative frequencies.frequencies.）（E.g.E.g.：We believe that We believe that the probability of an event is the probability of an event is 0.2 and using just a sample 0.2 and using just a sample we want to find out whether we want to find out whether this is a reasonable this is a reasonable assump

14、tion.assumption.）12Ex 1Ex 1：Suppose we are interested in the following populationSuppose we are interested in the following population：X X=1,2,3,4,5=1,2,3,4,5）.a)a)Since this is a very small population Since this is a very small population（N Nx x=5 5）,it is easy to observe the,it is easy to observe

15、the whole population,to illustrate it with a relative frequency histogram and to whole population,to illustrate it with a relative frequency histogram and to find the parameters,like the find the parameters,like the population meanpopulation mean and the and the population variancepopulation varianc

16、e.andand（Check the Check the details.details.）The key concept behind these statistical procedures is The key concept behind these statistical procedures is the probability distribution,called sampling distribution,the probability distribution,called sampling distribution,of a sample statistic.of a s

17、ample statistic.A summary of all possible values of a statistic A summary of all possible values of a statistic along with the corresponding probabilities.along with the corresponding probabilities.131234512345b)b)Though these calculations were really simple,assume that,for some Though these calcula

18、tions were really simple,assume that,for some reason,we do not observe the whole population,but draw all possible reason,we do not observe the whole population,but draw all possible samples of size two samples of size two（n n =2=2）with replacement.with replacement.There are 25 possible samples.They

19、are shown in the first row and There are 25 possible samples.They are shown in the first row and first column of the table below.first column of the table below.1 1stst draw draw（x x1 1）2 2ndnd draw draw（x x2 2）1.01.0Compute the sample mean from each of these samples.Compute the sample mean from eac

20、h of these samples.（E.g.E.g.：If If x x1 1=1 and=1 and x x2 2=4,=4,x x-bar is 2.5.-bar is 2.5.）2.52.53.53.52.02.02.52.53.03.01.51.52.02.02.52.53.03.03.53.52.02.02.52.53.03.03.53.54.04.03.03.03.53.54.04.04.54.53.03.01.51.54.04.04.54.55.05.0These sample mean values These sample mean values form a form

21、a secondsecond population,population,X X-bar-bar1 1.（Check the Check the details.details.）14c)c)Repeat part Repeat part b b assuming this time that sampling is assuming this time that sampling is without without replacement.replacement.123451-1.52.02.53.021.5-2.53.03.532.02.5-3.54.042.53.03.5-4.553.

22、03.54.04.5-1 1stst draw draw（x x1 1）2 2ndnd draw draw（x x2 2）These sample mean values These sample mean values form a form a thirdthird population,population,X X-bar-bar2 2.（Since samples are drawn Since samples are drawn without replacement,the without replacement,the same number cannot turn up sam

23、e number cannot turn up twice.twice.）15d)d)Compare the Compare the X X,X X-bar-bar1 1 and and X X-bar-bar2 2 populations to each other.populations to each other.XX-bar1X-bar2Size（N）52520Mean（）333Variance（2）210.75i.i.X X-bar-bar1 1 and and X X-bar-bar2 2 are larger populations than are larger populat

24、ions than X X.ii.ii.X X,X X-bar-bar1 1 and and X X-bar-bar2 2 have the same mean.have the same mean.iii.iii.X X has the biggest variance and has the biggest variance and X X-bar-bar2 2 has the smallest.has the smallest.iv.iv.These results suggest thatThese results suggest thatIt is easier to guess I

25、t is easier to guess X X-bar-bar2 2 than than X X-bar-bar1 1 or or X X.16Apart from their means and variances,the Apart from their means and variances,the X X-bar-bar1 1 and and X X-bar-bar2 2 populations populations can also be characterized by their shapes.can also be characterized by their shapes

26、.Note however,Note however,thatthatandand（These relationships between the variances of These relationships between the variances of X X,X X-bar-bar1 1 and and X X-bar-bar2 2 are valid in are valid in general.general.）These relative frequency histograms are graphical representations of the These rela

27、tive frequency histograms are graphical representations of the X X-bar-bar1 1 and and X X-bar-bar2 2 sampling distributions.sampling distributions.Apparently,both sampling distributions are symmetrical around Apparently,both sampling distributions are symmetrical around =3.3.Nevertheless,they are di

28、fferent from each other,and both of them are Nevertheless,they are different from each other,and both of them are different from the distribution of different from the distribution of X X.17 How to take a sample?How to take a sample?In this example the original population,In this example the origina

29、l population,X X,is very small,so we could,is very small,so we could easily calculate the population mean,easily calculate the population mean,x x.There was no real need to draw.There was no real need to draw samples.samples.In practice,however,the In practice,however,the target populationtarget pop

30、ulation is usually much larger.is usually much larger.The population about which The population about which we want to draw inferences.we want to draw inferences.Since it might be impossible or impractical to observe the whole Since it might be impossible or impractical to observe the whole populati

31、on,we draw a sample.population,we draw a sample.It is not necessarily the same than the It is not necessarily the same than the sampled sampled populationpopulation,i.e.the population from which we actually,i.e.the population from which we actually take the sample.take the sample.The sample must be

32、representative,i.e.it must have The sample must be representative,i.e.it must have similar attributes than the population itself.similar attributes than the population itself.In order to obtain reliable information from a sampleIn order to obtain reliable information from a sample The target and sam

33、pled populations should be very The target and sampled populations should be very similar,or similar,or the same if possible.the same if possible.18Even Even a a small samplesmall sample is is likely to give us fairly accurate likely to give us fairly accurate information about the populationinforma

34、tion about the population.（E.g.the sample mean can be expected to be close E.g.the sample mean can be expected to be close to the population mean.to the population mean.）If the sample items are selected If the sample items are selected randomlyrandomly,the sample the sample is is like a like a scale

35、d-scaled-downdown version of the population,unless we are very unlucky version of the population,unless we are very unlucky.If the sample is not randomly selected,it is likely to produce If the sample is not randomly selected,it is likely to produce misleading,misleading,biased results,biased result

36、s,even if even if the sample is relatively the sample is relatively large.large.（E.g.If we were attempting to estimate mean earnings,but we E.g.If we were attempting to estimate mean earnings,but we failed to failed to sample people in more affluent suburbs,the sample mean would sample people in mor

37、e affluent suburbs,the sample mean would almost certainly underestimate the true population mean.almost certainly underestimate the true population mean.）19This example suggests thatThis example suggests thatIf we intend to use a sample statistic for statistical If we intend to use a sample statisti

38、c for statistical inference,first we have to study its sampling inference,first we have to study its sampling distribution.distribution.（Ex 1Ex 1）We studied all possible samples of size 2 drawn with replacement.There We studied all possible samples of size 2 drawn with replacement.There were 25 diff

39、erent samples.If we select only one of them,but make sure were 25 different samples.If we select only one of them,but make sure that all these samples have the same chance of being selected,then that all these samples have the same chance of being selected,then sampling is assured to be simple rando

40、m sampling.sampling is assured to be simple random sampling.If we draw just one sample and it happens to be If we draw just one sample and it happens to be x x1 1=1 and=1 and x x2 2=3,=3,x x-bar is-bar is 2.Since the true mean of 2.Since the true mean of X X is 3,the error is 1.This is a sampling er

41、ror.is 3,the error is 1.This is a sampling error.Sample statistics,like e.g.Sample statistics,like e.g.X X-bar,are random variables since their-bar,are random variables since their actual values vary depending on which particular sample is actual values vary depending on which particular sample is s

42、elected.selected.The 25 possible samples of size 2 had 9 different sample mean valuesThe 25 possible samples of size 2 had 9 different sample mean values：1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5 and 5.0.1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5 and 5.0.SAMPLING DISTRIBUTIONSSAMPLING DISTRIBUTIONS20The results from Ex

43、1 can be generalized as follows.The results from Ex 1 can be generalized as follows.The sampling distribution of the sample means calculated from random The sampling distribution of the sample means calculated from random samples of the same size have the following characteristicssamples of the same

44、 size have the following characteristics：1 1）i.e.the expected value of the sample mean is the mean i.e.the expected value of the sample mean is the mean of the sampled population.of the sampled population.The sample mean from a random sample is our The sample mean from a random sample is our best be

45、st guess guess of the true population mean.of the true population mean.2 2）granted that sampling is with replacement,or from a granted that sampling is with replacement,or from a relatively large,maybe infinite,population.relatively large,maybe infinite,population.（The population is considered to be

46、 large compared to the The population is considered to be large compared to the sample if sample if n n/N N0.10,i.e.0.10,i.e.n n is less than 10%of is less than 10%of N N.）OtherwiseOtherwiseFinite population Finite population correction factor.correction factor.Like probability distributions in gene

47、ral,a sampling distribution can be Like probability distributions in general,a sampling distribution can be described by its three important propertiesdescribed by its three important properties：meanmean,standard deviationstandard deviation and and shapeshape.21As for the shape of the sampling distr

48、ibution of the sample mean,it depends on As for the shape of the sampling distribution of the sample mean,it depends on the distribution of the sampled population and on the sample size.the distribution of the sampled population and on the sample size.The standard deviation of the sample mean The st

49、andard deviation of the sample mean（or any other statisticor any other statistic）is is called called standard errorstandard error.Since the mean of the sample mean is Since the mean of the sample mean is ,the standard deviation,the standard deviation of the sample mean measures the average distance

50、between of the sample mean measures the average distance between the sample mean and the population mean.the sample mean and the population mean.NamelyNamely：3 3）If the population If the population（X X）is normally distributed,is normally distributed,X X-bar is also-bar is also normally distributed,r