英文商务统计学ppt-第七章Ch07课件.ppt

1、Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-1Chapter 7Sampling and Sampling DistributionsBusiness Statistics:A First Course5th EditionBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-2Learning ObjectivesIn this chapter, you learn: nTo distinguish between different samp

2、ling methodsnThe concept of the sampling distributionnTo compute probabilities related to the sample mean and the sample proportionnThe importance of the Central Limit TheoremBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-3Why Sample?nSelecting a sample is less time-consuming than sel

3、ecting every item in the population (census).nSelecting a sample is less costly than selecting every item in the population.nAn analysis of a sample is less cumbersome and more practical than an analysis of the entire population.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-4A Sampli

4、ng Process Begins With A Sampling FramenThe sampling frame is a listing of items that make up the populationnFrames are data sources such as population lists, directories, or mapsnInaccurate or biased results can result if a frame excludes certain portions of the populationnUsing different frames to

5、 generate data can lead to dissimilar conclusionsBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-5Types of SamplesSamplesNon-Probability SamplesJudgmentProbability SamplesSimple RandomSystematicStratifiedClusterConvenienceBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-6T

6、ypes of Samples:Nonprobability SamplenIn a nonprobability sample, items included are chosen without regard to their probability of occurrence.nIn convenience sampling, items are selected based only on the fact that they are easy, inexpensive, or convenient to sample.nIn a judgment sample, you get th

7、e opinions of pre-selected experts in the subject matter. Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-7Types of Samples:Probability SamplenIn a probability sample, items in the sample are chosen on the basis of known probabilities.Probability SamplesSimple RandomSystematicStratifie

8、dClusterBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-8Probability Sample:Simple Random SamplenEvery individual or item from the frame has an equal chance of being selectednSelection may be with replacement (selected individual is returned to frame for possible reselection) or withou

9、t replacement (selected individual isnt returned to the frame).nSamples obtained from table of random numbers or computer random number generators.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-9Selecting a Simple Random Sample Using A Random Number TableSampling Frame For Population

10、With 850 ItemsItem Name Item #Bev R. 001Ulan X. 002. . . . .Joann P. 849Paul F. 850Portion Of A Random Number Table49280 88924 35779 00283 81163 0727511100 02340 12860 74697 96644 8943909893 23997 20048 49420 88872 08401The First 5 Items in a simple random sampleItem # 492Item # 808Item # 892 - does

11、 not exist so ignoreItem # 435Item # 779Item # 002Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-10nDecide on sample size: nnDivide frame of N individuals into groups of k individuals: k=N/nnRandomly select one individual from the 1st group nSelect every kth individual thereafterProba

12、bility Sample:Systematic SampleN = 40n = 4k = 10First GroupBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-11Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-12Probability Sample:Stratified SamplenDivide population into two or more subgroups (called strata) according to so

13、me common characteristicnA simple random sample is selected from each subgroup, with sample sizes proportional to strata sizesnSamples from subgroups are combined into onenThis is a common technique when sampling population of voters, stratifying across racial or socio-economic lines.PopulationDivid

14、edinto 4strataBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-13Probability SampleCluster SamplenPopulation is divided into several “clusters,” each representative of the populationnA simple random sample of clusters is selectednAll items in the selected clusters can be used, or items

15、can be chosen from a cluster using another probability sampling techniquenA common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled.Population divided into 16 clusters.Randomly selected clusters for sampleBasic Business Statistic

16、s, 11e 2009 Prentice-Hall, Inc.Chap 7-14Probability Sample:Comparing Sampling MethodsnSimple random sample and Systematic samplenSimple to usenMay not be a good representation of the populations underlying characteristicsnStratified samplenEnsures representation of individuals across the entire popu

17、lationnCluster samplenMore cost effectivenLess efficient (need larger sample to acquire the same level of precision)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-15Evaluating Survey WorthinessnWhat is the purpose of the survey?nIs the survey based on a probability sample?nCoverage er

18、ror appropriate frame?nNonresponse error follow upnMeasurement error good questions elicit good responsesnSampling error always existsBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-16Types of Survey ErrorsnCoverage error or selection biasnExists if some groups are excluded from the fr

19、ame and have no chance of being selectednNon response error or biasnPeople who do not respond may be different from those who do respondnSampling errornVariation from sample to sample will always existnMeasurement errornDue to weaknesses in question design, respondent error, and interviewers effects

20、 on the respondent (“Hawthorne effect”)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-17Types of Survey ErrorsnCoverage errornNon response errornSampling errornMeasurement errorExcluded from frameFollow up on nonresponsesRandom differences from sample to sampleBad or leading question(

21、continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-18Sampling DistributionsnA sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population.nFor example, suppose you sample 50 students from your col

22、lege regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students.Basic Business Statistics, 11e 2009 Prentice-Hall, In

23、c.Chap 7-19Developing a Sampling DistributionnAssume there is a population nPopulation size N=4nRandom variable, X,is age of individualsnValues of X: 18, 20,22, 24 (years)ABCDBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-20.3.2.1 0 18 20 22 24 A B C DUniform DistributionP(x)x(continu

24、ed)Summary Measures for the Population Distribution:Developing a Sampling Distribution21424222018NXi2.236N)(X2iBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-2116 possible samples (sampling with replacement)Now consider all possible samples of size n=21st 2nd Observation Obs 18 20 22

25、24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 (continued)Developing a Sampling Distribution16 Sample Means1stObs2nd Observation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2224,24Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap

26、7-221st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 Sampling Distribution of All Sample Means18 19 20 21 22 23 240 .1 .2 .3 P(X) XSample Means Distribution16 Sample Means_Developing a Sampling Distribution(continued)(no longer uniform)_Basic Business S

27、tatistics, 11e 2009 Prentice-Hall, Inc.Chap 7-23Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)211624191918NXiX1.581621)-(2421)-(1921)-(18N)X(2222XiXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-24Comparing the Population Distributionto the

28、 Sample Means Distribution18 19 20 21 22 23 240 .1 .2 .3 P(X) X 18 20 22 24 A B C D0 .1 .2 .3 PopulationN = 4P(X) X_1.58 21XX2.236 21Sample Means Distributionn = 2_Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-25Sample Mean Sampling Distribution:Standard Error of the MeannDifferent s

29、amples of the same size from the same population will yield different sample meansnA measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

30、nNote that the standard error of the mean decreases as the sample size increasesnXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-26Sample Mean Sampling Distribution:If the Population is NormalnIf a population is normally distributed with mean and standard deviation , the sampling dist

31、ribution of is also normally distributed with andXXnXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-27Z-value for Sampling Distributionof the MeannZ-value for the sampling distribution of :where:= sample mean= population mean= population standard deviation n = sample sizeXn)X()X(ZXXXB

32、asic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-28Normal Population DistributionNormal Sampling Distribution (has the same mean)Sampling Distribution Propertiesn (i.e. is unbiased )xxxxxBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-29Sampling Distribution Properties As n

33、 increases, decreasesLarger sample sizeSmaller sample sizex(continued)xBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-30Determining An Interval Including A Fixed Proportion of the Sample MeansFind a symmetrically distributed interval around that will include 95% of the sample means wh

34、en = 368, = 15, and n = 25.nSince the interval contains 95% of the sample means 5% of the sample means will be outside the intervalnSince the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit.nFrom the standardized normal table, the Z score with 2.5% (0.

35、0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-31Determining An Interval Including A Fixed Proportion of the Sample MeansnCalculating the lower limit of the intervalnCalculating the upper limit of the interval

36、n95% of all sample means of sample size 25 are between 362.12 and 373.8812.3622515)96. 1(368nZXL(continued)88.3732515)96. 1 (368nZXUBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-32Sample Mean Sampling Distribution:If the Population is not NormalnWe can apply the Central Limit Theorem

37、:nEven if the population is not normal,nsample means from the population will be approximately normal as long as the sample size is large enough.Properties of the sampling distribution: andxnxBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-33nCentral Limit TheoremAs the sample size get

38、s large enough the sampling distribution becomes almost normal regardless of shape of populationxBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-34Population DistributionSampling Distribution (becomes normal as n increases)Central TendencyVariationxxLarger sample sizeSmaller sample siz

39、eSample Mean Sampling Distribution:If the Population is not Normal(continued)Sampling distribution properties:xnxxBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-35How Large is Large Enough?nFor most distributions, n 30 will give a sampling distribution that is nearly normalnFor fairly

40、 symmetric distributions, n 15 will usually give a sampling distribution is almost normalnFor normal population distributions, the sampling distribution of the mean is always normally distributedBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-36ExamplenSuppose a population has mean = 8

41、 and standard deviation = 3. Suppose a random sample of size n = 36 is selected. nWhat is the probability that the sample mean is between 7.8 and 8.2?Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-37ExampleSolution:nEven if the population is not normally distributed, the central limit

42、 theorem can be used (n 30)n so the sampling distribution of is approximately normaln with mean = 8 nand standard deviation (continued)xx0.5363nxBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-38Example Solution (continued):(continued)0.31080.4)ZP(-0.43638-8.2n- X3638-7.8P 8.2) X P(7.8

43、Z7.8 8.2-0.4 0.4Sampling DistributionStandard Normal Distribution.1554 +.1554Population Distribution?SampleStandardize8 8X0zxXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-39Population Proportions = the proportion of the population having some characteristicnSample proportion ( p ) p

44、rovides an estimate of :n0 p 1np is approximately distributed as a normal distribution when n is large(assuming sampling with replacement from a finite population or without replacement from an infinite population)size sample interest ofstic characteri the having sample the in itemsofnumbernXpBasic

45、Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-40Sampling Distribution of pnApproximated by anormal distribution if:n where and(where = population proportion)Sampling DistributionP( ps).3.2.1 0 0 . 2 .4 .6 8 1ppn)(1p5)n(15nandBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-41Z

46、-Value for Proportionsn)(1ppZpStandardize p to a Z value with the formula:Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-42ExamplenIf the true proportion of voters who support Proposition A is = 0.4, what is the probability that a sample of size 200 yields a sample proportion between

47、0.40 and 0.45?ni.e.: if = 0.4 and n = 200, what is P(0.40 p 0.45) ?Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-43Examplen if = 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)0.034642000.4)0.4(1n)(1p1.44)ZP(00.034640.400.45Z0.034640.400.40P0.45)pP(0.40Find : Convert to standard

48、ized normal: pBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 7-44ExampleZ0.451.440.4251StandardizeSampling DistributionStandardized Normal Distributionn if = 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)Use standardized normal table: P(0 Z 1.44) = 0.42510.400pBasic Business Stati

49、stics, 11e 2009 Prentice-Hall, Inc.Chap 7-45Chapter SummarynDiscussed probability and nonprobability samplesnDescribed four common probability samplesnExamined survey worthiness and types of survey errorsnIntroduced sampling distributionsnDescribed the sampling distribution of the meannFor normal populationsnUsing the Central Limit TheoremnDescribed the sampling distribution of a proportionnCalculated probabilities using sampling distributions