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1、Chap 7-1Chapter 7Sampling And Sampling DistributionsStatistics For Managers Using Microsoft Excel 6th EditionChap 7-2Learning ObjectivesIn this chapter, you learn: The concept of the sampling distributionTo compute probabilities related to the sample mean and the sample proportionThe importance of t
2、he Central Limit TheoremTo distinguish between different survey sampling methodsTo evaluate survey worthiness and survey errorsChap 7-3Sampling DistributionsSampling DistributionsSampling Distribution of the MeanSampling Distribution of the ProportionChap 7-4Sampling DistributionsA sampling distribu
3、tion is a distribution of all of the possible values of a statistic for a given size sample selected from a populationChap 7-5Developing a Sampling DistributionAssume there is a population Population size N=4Random variable, X,is age of individualsValues of X: 18, 20,22, 24 (years)ABCDChap 7-6.3.2.1
4、 0 18 19 20 21 22 23 24 A B C DUniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing a Sampling DistributionChap 7-716 possible samples (sampling with replacement)Now consider all possible samples of size n=2(continued)Developing a Sampling Distribution16 Sa
5、mple Means1stObs2nd Observation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2224,24Chap 7-8Sampling 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)(
6、no longer uniform)_Chap 7-9Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)Chap 7-10Comparing the Population with its Sampling Distribution18 19 20 21 22 23 240 .1 .2 .3 P(X) X 18 19 20 21 22 23 24 A B C D0 .1 .2 .3 PopulationN = 4P(X) X_Sample Means Distri
7、butionn = 2_Chap 7-11Sampling Distribution of the MeanSampling DistributionsSampling Distribution of the MeanSampling Distribution of the ProportionChap 7-12Standard Error of the MeanDifferent samples of the same size from the same population will yield different sample meansA measure of the variabi
8、lity 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)Note that the standard error of the mean decreases as the sample size increasesChap 7-13If the Population i
9、s NormalIf a population is normal with mean and standard deviation , the sampling distribution of is also normally distributed with andChap 7-14Z-value for Sampling Distributionof the MeanZ-value for the sampling distribution of :where:= sample mean= population mean= population standard deviation n
10、= sample sizeChap 7-15Normal Population DistributionSampling Distribution is also normal (and has the same mean)Sampling Distribution Properties (i.e. is unbiased )Chap 7-16Sampling Distribution Properties As n increases, decreasesLarger sample sizeSmaller sample size(continued)Chap 7-17If the Popul
11、ation is not NormalWe can apply the Central Limit Theorem:Even if the population is not normal,sample means from the population will be approximately normal as long as the sample size is large enough.Properties of the sampling distribution: andChap 7-18nCentral Limit TheoremAs the sample size gets l
12、arge enough the sampling distribution becomes almost normal regardless of shape of populationChap 7-19Population DistributionSampling Distribution (becomes normal as n increases)Central TendencyVariationLarger sample sizeSmaller sample sizeIf the Population is not Normal(continued)Sampling distribut
13、ion properties:Chap 7-20How Large is Large Enough?For most distributions, n 30 will give a sampling distribution that is nearly normalFor fairly symmetric distributions, n 15For normal population distributions, the sampling distribution of the mean is always normally distributedChap 7-21ExampleSuppo
14、se a population has mean = 8 and standard deviation = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?Chap 7-22ExampleSolution:Even if the population is not normally distributed, the central limit theorem can be used (n 30) s
15、o the sampling distribution of is approximately normal with mean = 8 and standard deviation (continued)Chap 7-23Example Solution (continued):(continued)Z7.8 8.2-0.4 0.4Sampling DistributionStandard Normal Distribution.1554 +.1554Population Distribution?SampleStandardizeXChap 7-24Sampling Distributio
16、n of the ProportionSampling DistributionsSampling Distribution of the MeanSampling Distribution of the ProportionChap 7-25Population Proportions = the proportion of the population having some characteristicSample proportion ( p ) provides an estimate of :0 p 1p has a binomial distribution(assuming s
17、ampling with replacement from a finite population or without replacement from an infinite population)Chap 7-26Sampling Distribution of pApproximated by anormal distribution if: where and(where = population proportion)Sampling DistributionP( p).3.2.1 0 0 . 2 .4 .6 8 1pChap 7-27Z-Value for Proportions
18、Standardize p to a Z value with the formula:Chap 7-28ExampleIf 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 0.40 and 0.45?i.e.: if = 0.4 and n = 200, what is P(0.40 p 0.45) ?Chap 7-29Example if
19、= 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)Find : Convert to standard normal: Chap 7-30ExampleZ0.451.440.4251StandardizeSampling DistributionStandardized Normal Distribution if = 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)Use cumulative standard normal table: P(0 Z 1.44) = P(Z 1.
20、44) P(Z 0) = 0.9251 0.5000 = 0.42510.400pChap 7-31Reasons for Drawing a SampleLess time consuming than a censusLess costly to administer than a censusLess cumbersome and more practical to administer than a census of the targeted populationChap 7-32Nonprobability SampleItems included are chosen witho
21、ut regard to their probability of occurrenceProbability SampleItems in the sample are chosen on the basis of known probabilitiesTypes of Samples UsedChap 7-33Types of Samples UsedQuotaSamplesNon-Probability SamplesJudgementChunkProbability SamplesSimple RandomSystematicStratifiedClusterConvenience(c
22、ontinued)Chap 7-34Probability SamplingItems in the sample are chosen based on known probabilitiesProbability SamplesSimple RandomSystematicStratifiedClusterChap 7-35Simple Random SamplesEvery individual or item from the frame has an equal chance of being selectedSelection may be with replacement or
23、without replacementSamples obtained from table of random numbers or computer random number generatorsChap 7-36Decide on sample size: nDivide frame of N individuals into groups of k individuals: k=N/nRandomly select one individual from the 1st group Select every kth individual thereafterSystematic Sa
24、mplesN = 64n = 8k = 8First GroupChap 7-37Stratified SamplesDivide population into two or more subgroups (called strata) according to some common characteristicA simple random sample is selected from each subgroup, with sample sizes proportional to strata sizesSamples from subgroups are combined into
25、 onePopulationDividedinto 4strataSampleChap 7-38Cluster SamplesPopulation is divided into several “clusters,” each representative of the populationA simple random sample of clusters is selectedAll items in the selected clusters can be used, or items can be chosen from a cluster using another probabi
26、lity sampling techniquePopulation divided into 16 clusters.Randomly selected clusters for sampleChap 7-39Advantages and DisadvantagesSimple random sample and systematic sampleSimple to useMay not be a good representation of the populations underlying characteristicsStratified sampleEnsures represent
27、ation of individuals across the entire populationCluster sampleMore cost effectiveLess efficient (need larger sample to acquire the same level of precision)Chap 7-40Evaluating Survey WorthinessWhat is the purpose of the survey?Is the survey based on a probability sample?Coverage error appropriate fr
28、ame?Nonresponse error follow upMeasurement error good questions elicit good responsesSampling error always existsChap 7-41Types of Survey ErrorsCoverage error or selection biasExists if some groups are excluded from the frame and have no chance of being selectedNonresponse error or biasPeople who do
29、 not respond may be different from those who do respondSampling errorVariation from sample to sample will always existMeasurement errorDue to weaknesses in question design, respondent error, and interviewers effects on the respondentChap 7-42Types of Survey ErrorsCoverage errorNon response errorSamp
30、ling errorMeasurement errorExcluded from frameFollow up on nonresponsesRandom differences from sample to sampleBad or leading question(continued)Chap 7-43Chapter SummaryIntroduced sampling distributionsDescribed the sampling distribution of the meanFor normal populationsUsing the Central Limit TheoremDescribed the sampling distribution of a proportionCalculated probabilities using sampling distributionsDescribed different types of samples and sampling techniquesExamined survey worthiness and types of survey errors
