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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 sampling m
2、ethodsnThe 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 selecting e
3、very 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 Sampling Process
4、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 generate dat
5、a 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 7-6Types of Samples:N
6、onprobability 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 the opinions of pre-sele
7、cted 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 RandomSystematicStratifiedClusterBasic Business Sta
8、tistics,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 without replacement(selected individ
9、ual 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 With 850 ItemsItem Name Item#Bev
10、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 not exist so ignoreItem#435Item#779Item#002Basic Bus
11、iness 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 thereafterProbability Sample:Systematic SampleN=40n=4k=10First GroupBasic Busi
12、ness 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 some common characteristicnA simple random sample is selected from each subgr
13、oup,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.PopulationDividedinto 4strataBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-13P
14、robability 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 can be chosen from a cluster using another probability sampling techniquenA common
15、 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 Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-14Probability Sample:Comparing Sampling Methods
16、nSimple random sample and Systematic samplenSimple to usenMay not be a good representation of the populations underlying characteristicsnStratified samplenEnsures representation of individuals across the entire populationnCluster samplenMore cost effectivenLess efficient(need larger sample to acquir
17、e 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 error appropriate frame?nNonresponse error follow upnMeasurement error good questions elic
18、it 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 frame and have no chance of being selectednNon response error or biasnPeople who do not resp
19、ond 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 on the respondent(“Hawthorne effect”)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.C
20、hap 7-17Types of Survey ErrorsnCoverage errornNon response errornSampling errornMeasurement errorExcluded from frameFollow up on nonresponsesRandom differences from sample to sampleBad or leading question(continued)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-18Sampling Distributionsn
21、A 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 college regarding their mean GPA.If you obtained many different samples of 50,you will compute a diff
22、erent 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,Inc.Chap 7-19Developing a Sampling DistributionnAssume there is a population nPopulation size N=4nRandom
23、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(continued)Summary Measures for the Population Distribution:Developing a Sampling DistributionBasic Business Statistic
24、s,11e 2009 Prentice-Hall,Inc.Chap 7-2116 possible samples(sampling with replacement)Now consider all possible samples of size n=2(continued)Developing a Sampling Distribution16 Sample Means1stObs2nd Observation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2
25、224,24Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-22Sampling 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 Statistics,11e 2009 Prentice-Hall,Inc
26、.Chap 7-23Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-24Comparing the Population Distributionto the 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 Populat
27、ionN=4P(X)X_Sample Means Distributionn=2_Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-25Sample Mean Sampling Distribution:Standard Error of the MeannDifferent samples of the same size from the same population will yield different sample meansnA measure of the variability in the mean f
28、rom 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)nNote that the standard error of the mean decreases as the sample size increasesBasic Business Statistics,11e 2009 Prentice-H
29、all,Inc.Chap 7-26Sample Mean Sampling Distribution:If the Population is NormalnIf a population is normally distributed with mean and standard deviation,the sampling distribution of is also normally distributed with andBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-27Z-value for Sampling
30、 Distributionof the MeannZ-value for the sampling distribution of :where:=sample mean=population mean=population standard deviation n=sample sizeBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-28Normal Population DistributionNormal Sampling Distribution(has the same mean)Sampling Distrib
31、ution Propertiesn (i.e.is unbiased)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-29Sampling Distribution Properties As n increases,decreasesLarger sample sizeSmaller sample size(continued)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-30Determining An Interval Including A
32、Fixed Proportion of the Sample MeansFind a symmetrically distributed interval around that will include 95%of the sample means when =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 abov
33、e the upper limit and 2.5%will be below the lower limit.nFrom the standardized normal table,the Z score with 2.5%(0.0250)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 Proporti
34、on of the Sample MeansnCalculating the lower limit of the intervalnCalculating the upper limit of the intervaln95%of all sample means of sample size 25 are between 362.12 and 373.88(continued)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-32Sample Mean Sampling Distribution:If the Popul
35、ation is not NormalnWe can apply the Central Limit Theorem: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:andBasic Business Statistics,11e 2009 Prentice-Hall,Inc.C
36、hap 7-33nCentral Limit TheoremAs the sample size gets large enough the sampling distribution becomes almost normal regardless of shape of populationBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-34Population DistributionSampling Distribution(becomes normal as n increases)Central Tendenc
37、yVariationLarger sample sizeSmaller sample sizeSample Mean Sampling Distribution:If the Population is not Normal(continued)Sampling distribution properties:Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-35How Large is Large Enough?nFor most distributions,n 30 will give a sampling distri
38、bution that is nearly normalnFor fairly 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 7-36Exampl
39、enSuppose a population has mean =8 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 dis
40、tributed,the central limit theorem can be used(n 30)n so the sampling distribution of is approximately normaln with mean =8 nand standard deviation(continued)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-38Example Solution(continued):(continued)Z7.8 8.2-0.4 0.4Sampling DistributionStan
41、dard Normal Distribution.1554+.1554Population Distribution?SampleStandardizeXBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-39Population Proportions =the proportion of the population having some characteristicnSample proportion(p)provides an estimate of :n0 p 1np is approximately distri
42、buted as a normal distribution when n is large(assuming sampling with replacement from a finite population or without replacement from an infinite population)Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-40Sampling Distribution of pnApproximated by anormal distribution if:n where and(w
43、here =population proportion)Sampling DistributionP(ps).3.2.1 0 0 .2 .4 .6 8 1pBasic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-41Z-Value for ProportionsStandardize p to a Z value with the formula:Basic Business Statistics,11e 2009 Prentice-Hall,Inc.Chap 7-42ExamplenIf the true proportion
44、of voters who support Proposition A is =0.4,what is the probability that a sample of size 200 yields a sample proportion between 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
45、)?(continued)Find :Convert to standardized normal:Basic 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)=0.4251
46、0.400pBasic Business Statistics,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