Experiment ResultsComputing the Sampling Distribution of PDFs for n=1,2,3,…,30ObservationsThe Central Limit Theorem for ProportionsWhy Surveys WorkAssignmentSamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentSampling Distribution of a SampleProportionLecture 25Sections 8.1 - 8.2Robb T. KoetherHampden-Sydney CollegeFri, Oct 10, 2008SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentOutline1Experiment Results2Computing the Sampling Distribution ofˆp3PDFs for n = 1, 2, 3, . . . , 304Observations5The Central Limit Theorem for Proportions6Why Surveys Work7AssignmentSamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExperiment ResultsThe Results of the ExperimentIn our experiment, we collected a total of 100 samples,each of size 5.The distribution of sample proportions we observedwasSampleProportion Frequency0.0 00.2 40.4 220.6 320.8 321.0 10SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentHomework ReviewThe Results of the ExperimentTo describe this distribution, we could describe itsshape and give the mean and standard deviation.Keep in mind, we are working with a sample of valuesofˆp.That is, a sample of samples.This is not the entire population of all samples of size 5.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentHomework ReviewThe Results of the ExperimentThe shape is approximately normal.To find the mean and standard deviation, we may usethe TI-83.Enter the distinct values ofˆp in list L1.Enter the frequencies in list L2.Use 1-Var Stats as before, but enter both lists:1-Var Stats L1,L2We get 0.644 for the mean and 0.2061 for the standarddeviation.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentHomework ReviewThe Results of the ExperimentTheory says that the shape is approximately normaland thatµˆp= pandσˆp=rp(1 − p)n.We calculate µˆp= 0.66 andσˆp=rp(1 − p)n=r(0.66)(0.34)5= 0.2118.Our experimental results were very close to what thetheory predicts.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentSampling DistributionsDefinition (Sampling Distribution of a Statistic)The sampling distribution of a statistic is the distribution ofvalues of that statistic over all possible samples of a givensize n from the population.We may sample with or without replacement.For large populations, the difference is insignificant.For our purposes, it will be easier to sample withreplacement (allowing repetitions).SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentThe Sample ProportionThe letter p represents the population proportion.The symbolˆp (“p-hat”) represents the sampleproportion.Soˆp is a random variable.The sampling distribution ofˆp is the probabilitydistribution of all the possible values ofˆp.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleSuppose that 2/3 of all males wash their hands afterusing a public restroom.Suppose that we take a sample of 1 male.Find the sampling distribution ofˆp.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleWN1/32/3WN102/31/3SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleLet x be the number of males (in the sample) who wash.The probability distribution of x isx P(x)0 1/3 = 0.33331 2/3 = 0.6667SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleLetˆp be the sample proportion of males who wash.(ˆp =x1.)The sampling distribution ofˆp isˆp P(ˆp)0 1/3 = 0.33331 2/3 = 0.6667SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleNow we take a sample of 2 males, sampling withreplacement.Find the sampling distribution ofˆp.SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleWN1/32/3NWNN102/91/9WN1/32/3WN1/32/3WN 1 2/9WW 2 4/9SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy SurveysWorkAssignmentExampleLet x be the number of males (in the sample) who wash.The probability distribution of x isx P(x)0 1/9 = 0.11111 4/9 = 0.44442 4/9 = 0.4444SamplingDistribution ofa SampleProportionRobb T.KoetherExperimentResultsComputingthe SamplingDistribution ofˆpPDFs for n =1, 2, 3, . . . , 30ObservationsThe CentralLimit TheoremforProportionsWhy
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