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UW-Madison ECON 310 - 310_spring2012_chapter10

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Chapter TenIntroduction to EstimationWhere we have been…Chapter 7 and 8:Binomial, Poisson, normal, and exponential distributions allow us to make probability statements about X (a member of the population). To do so we need the population parameters.Binomial: pPoisson: µNormal: µ and σExponential: λ or µWhere we have beenChapter 9: Sampling distributions allow us to make probability statements about statistics.We need the population parameters.Sample mean: µ and σSample proportion: pDifference between sample means: µ1,σ1 ,µ2, and σ2Where we are going• However, in almost all realistic situations parameters are unknown.• We will use the sampling distribution to draw inferences about the unknown population parameters.Statistical InferenceStatistical inference is the process by which we acquire information and draw conclusions about populations from samples.In order to do inference, we require the skills and knowledge of descriptive statistics, probability distributions, and sampling distributions.ParameterPopulationSampleStatisticInferenceDataStatisticsInformationEstimation• There are two types of inference: estimation and hypothesis testing; estimation is introduced first.• The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.• For example, the sample mean ( ) is employed to estimate the population mean ( ).EstimationThe objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.There are two types of estimators:Point Estimato r:Interval Estimator:Point Estimato rA point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point.Point estimators don’t reflect the random nature of sampling distributions. To incorporate this randomness, we employ inte rval estimators to estimate population parameters…Interval EstimatorAn interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.That is we say (with some ___% certainty) (for example, with 95% certainty) that the population parameter of inte rest is between some lower and upper bounds.Point vs. Interval EstimationFor example, suppose we want to estimate the mean summer income of a class of business students. For n = 25 s tudents, is calculated to be 400 $/week.point estimate interval estimateAn alte rnative statement is:The mean income is between 380 and 420 $/week.Qualities of EstimatorsQualities desirable in estimators include unbiasedness, consistency, and relative efficiency:An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.Unbiased EstimatorsAn unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.For example, we concluded in previous lectures that the sample mean is an unbiasedestimator of the population mean µ , since:ConsistencyAn unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.For example, is a consistent estimator of µ because, as we showed in previous lectures, Therefore, as n grows larger, the variance of X grows smaller. Since is unbiased, this necessarily means that, as the sample size grows larger, must become more and more likely to be closer to µ.Relative EfficiencyIf there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.For example, instead of using , we could estimate µ as That is, we could simply tak e the first observation X1 in our random sample and use it as an estimator of µ. This estimator is clearly unbiased, since .However, is more efficient than , since, while Therefore, as long as n ≥ 2 (two or more observations in the sample), we will have < . Therefore, is more efficient than Estimating µ when  is known• A point‐estimate of µ is simply given by  (the sample mean).• How do we construct an interval‐estimate for µ? We do it by invoking the Central Limit Theorem.• Fix a probability . Using the notation in Chapter 8, we define /as the value that satisfies //2 where Z is a Standard Normal• Symmetry around zero of the Standard Normal distribution (we discussed this previously) implies that //2• Therefore, //󰇛/) 󰇛/)• Therefore, if Z is a Standard Normal variable, and if we pre‐specify a cover


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UW-Madison ECON 310 - 310_spring2012_chapter10

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