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POINT ESTIMATION AND INTERVAL ESTIMATION

POINT ESTIMATION AND INTERVAL ESTIMATION. DEFINITIONS.

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POINT ESTIMATION AND INTERVAL ESTIMATION

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  1. POINT ESTIMATIONANDINTERVAL ESTIMATION

  2. DEFINITIONS An estimator of a populationparameter is a randomvariablethatdepends on thesampleinformationandwhoserealizationsprovideapproximationstothisunknownparameter. A Spescificrealization of thatrandomvariable is called an estimate. A pointestimatorof a populationparameter is a function of thesampleinformationthatyields a singlenumber. Thecorrespondingrealization is calledthepointestimateof theparameter.

  3. DEFINITIONS

  4. PROPERTIES OF GOOD POINT ESTIMATORS • A good estimator must satisfy three conditions: • Unbiased: Theestimator is saidto be an unbiasedestimator of theparameterifthemean of thesamplingdistribution of is . Intheotherwords the expected value of the estimator must be equal to the mean of the parameter

  5. UNBIASEDNESS OF SOME ESTIMATORS • Thesamplemean, varianceandproportionareunbiasedestimators of thecorrespondingpopulationquantities. • In general, thesample standart deviation is not an unbiasedestimator of thepopulation standart deviation. Let be an estimator of . Thebias in is defined as thedifferencebetweenitsmeanand ; that is Itfollowsthatthebias of an unbiasedestimator is 0.

  6. EFFICIENCY Letand be twounbiasedestimators of ,based on thesamenumber of sampleobservations. Then is saidto be moreefficientthanif Therelativeefficiency of oneestimatorwithrespecttotheother is theratio of theirvariances; that is Relativeefficiency=

  7. EFFICIENCY is themoreefficientestimator. If is an unbiasedestimator of , andnootherunbiasedestimator has smallervariance, then is saidto be mostefficientor minimum varianceunbiasedestimator of .

  8. CHOICE OF POINT ESTIMATOR • Thereareestimationproblemsforwhichnounbiasedestimator is verysatisfactoryandforwhichtheremay be muchto be gainedfromthesacrifice of acceptinglittlebias. Onemeasure of theexpectedcloseness of an estimatorto a parameter is itsmeansquarederror – theexpectation of thesquareddifferencebetweentheestimatorandtheparameter, that is • It can be shownthat,

  9. CONSISTENCY • Consistencyalsodesirable is that an estimatetendtolienearerthepopulationcharacteristic as thesample size becomeslarger. This is thebasis of theproperty of consistency. • An estimator is a consistentestimator of a populationcharacteristicifthelargerthesample size, themorelikely it is thattheestimatewill be closeto .

  10. INTERVAL ESTIMATION • An intervalestimatorforapopulationparameter is a rulefordetermining (based on sampleinformation) a range, orinterval, in whichtheparameter is likelytofall. Thecorrespondingestimate is called an intervalestimate. • Let be an unknownparameter. Supposethat on thebasis of sampleinformation, we can findrandomvariables A and B suchthat • Ifthespecificsamplerealizations of AandBaredenotedbyaandb ,thentheintervalfromatob is called a 100(1-α)% confidenceintervalfor . Thequantity is calledtheprobabilitycontentorlevel of confidence, of theinterval. • Ifthepopulationwasrepeatedlysampled a verylargenumber of times, theparameterwould be contained in 100(1-α)% of intervalscalculatedthisway.

  11. ELEMENTS OF CONFIDENCE INTERVAL

  12. CONFIDENCE LIMITS FOR POPULATION MEAN

  13. FACTORS EFFECTING INTERVAL WIDTH

  14. CONFIDENCE INTERVALS KNOWN

  15. CONFIDENCE INTERVALS KNOWN

  16. CONFIDENCE INTERVALS UNKNOWN

  17. STUDENT’S t DISTRIBUTION

  18. STUDENT’S t TABLE

  19. ESTIMATION FOR FINITE POPULATIONS • Whensample is largerelativetopopulation, • n/N>0,05 • Usefinitepopulationcorrectionfactor;

  20. CONFIDENCE INTERVALS FOR THE POPULATION PROPORTION • Assumptions; • TwoCategoricalOutcomes (faulty/not faulty – complex/easy), • PopulationFollowsBinomial Distribution Normal Approximation Can Be Usedif: • n·p≥ 5 n·(1 - p) ≥ 5 • ConfidenceIntervalEstimate;

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