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Econ 3780: Business and Economics Statistics Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal Email: [email protected] Chapter 8: Interval Estimation Population Mean: Known Population Mean: Unknown MarginMargin ofof ErrorError andand thethe IntervalInterval EstimateEstimate AA pointpoint estimatorestimator cannotcannot bebe expectedexpected toto provideprovide thethe exactexact valuevalue ofof thethe populationpopulation parameter.parameter. AnAn intervalinterval estimateestimatecancan bebe computedcomputed byby addingadding andand subtracting a margin of error to the point estimate. subtractingsubtracting aa marginmargin ofof error errortoto thethe pointpoint estimate.estimate. Point Estimate +/ Margin of Error TheThe purposepurpose ofof anan intervalinterval estimateestimate isis toto provideprovide informationinformation aboutabout howhow closeclose thethe pointpoint estimateestimate isis toto thethe valuevalue ofof thethe parameter.parameter. MarginMargin ofof ErrorError andand thethe IntervalInterval EstimateEstimate TheThe generalgeneral formform ofof anan intervalinterval estimateestimate ofof aa populationpopulation meanmean isis xx MarMargginin ofof ErrorError In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation , or the sample standard deviation s These are also Confidence Interval. Interval Estimate of a Population Mean: Known Interval Estimate of x z/ 2 / 2 n where: x is the sample mean 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the population standard deviation n is the sample size IntervalInterval EstimationEstimation ofof aa PopulationPopulation Mean:Mean: KnownKnown There is a 1 probability that the value of a sample mean will provide a margin of error of z/2/2 x or less. SamplingSampling distributiondistribution ofof x /2/2 1 - ofof allall /2/2 xx valuesvalues x z/2/2 x z/2/2 x SummarySummary ofof PointPoint EstimatesEstimates ObtainedObtained fromfrom aa SimpleSimple RandomRandom SampleSample Population Parameter Point Point Parameter Value Estimator Estimate = Population mean 40.9 x = Sample mean = Population std. 20.5 s = Sample std. ……. deviation deviation p = Population pro- .62 p = Sample pro- portion portion Example: Air Quality Consider our air quality example. Suppose the population is approximately normal with μ = 40.9 and σ = 20.5. This is σ known case. If you guys remember, we picked a sample of size 5 (n =5). Given all this information, What is the margin of error at 95% confidence level? Example: Air Quality What is the margin of error at 95% confidence level. 20 . 5 20 . 5 0z . 0251 . 961 . 96 9 . 2 18 0 . 025 n 5 We can say with 95% confidence that population mean (μ) is between ± 18 of the sample mean. x0 . z 025 0 . 025 n With 95% confidence, μ is between …. and …... Interval Estimation of a Population Mean: Unknown If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation s to estimate . This is the unknown case. In this case, the interval estimate for is based on the t distribution. (We’ll assume for now that the population is normally distributed.) tt DistributionDistribution TheThe tt distributiondistributionisis aa familyfamily ofof similarsimilar probabilityprobability distributions. distributions.distributions. AA specificspecific tt distributiondistribution dependsdepends onon aa parameterparameter known as the degrees of freedom. knownknown asas thethe degreesdegrees ofof freedomfreedom.. DegreesDegrees ofof freedomfreedom referrefer toto thethe numbernumber ofof independentindependent piecespieces ofof informationinformation thatthat gogo intointo thethe computationcomputation ofof ss.. tt DistributionDistribution AA tt distributiondistribution withwith moremore degreesdegrees ofof freedomfreedom hashas less dispersion. lessless dispersion.dispersion. AsAs thethe numbernumber ofof degreesdegrees ofof freedomfreedom increases,increases, thethe differencedifference betweenbetween thethe tt distributiondistribution andand thethe standard normal probability distribution becomes standardstandard normalnormal probabilityprobability distributiondistribution becomesbecomes smallersmaller andand smaller.smaller. t Distribution t distribution Standard (20 degrees normal of freedom) distribution t distribution (10 degrees of freedom) z, t 0 tt DistributionDistribution ForFor moremore thanthan 100100 degreesdegrees ofof freedom,freedom, thethe standardstandard normalnormal zzvaluevalue providesprovides aa goodgood approximationapproximation toto the t value. thethe ttvalue.value. TheThe standardstandard normalnormal zzvaluesvalues cancan bebe foundfound inin thethe infinite degrees ( ) row of the t distribution table. infiniteinfinite degreesdegrees (()) rowrow ofof thethe ttdistributiondistribution table.table. tt DistributionDistribution Degrees Area in Upper Tail of Freedom 0.2 .10 .05 .025 .01 .005 . ...... 50 0.849 1.299 1.676 2.009 2.403 2.678 60 0.848 1.296 1.671 2.000 2.390 2.660 80 0.846 1.292 1.664 1.990 2.374 2.639 100 0.845 1.290 1.660 1.984 2.364 2.626 0.842 1.282 1.645 1.960 2.326 2.576 Standard normal z values IntervalInterval EstimationEstimation ofof aa PopulationPopulation Mean:Mean: UnknownUnknown Interval Estimate s x t/ 2 / 2 n where: 1 - = the confidence coefficient t/2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation Example: Air quality when σ is unknown Now suppose that you did not know what σ is. You can estimate using the sample and then use t-distribution to find the margin of error. What is 95% confidence interval in this case? The sample size n =5. So, the degrees of freedom for the t-distribution is 4. The level of significance ( ) is 0.05. s = …… s x t 2/ 2/ n SummarySummary ofof IntervalInterval EstimationEstimation ProceduresProcedures forfor aa PopulationPopulation MeanMean Can the Yes No population standard deviation be assumed known ? Use the sample standard deviation s to estimate σ Known Case Use Use Unknown s xt xz /2 Case xt /2 n n IntervalInterval EstimationEstimation ofof aa PopulationPopulation ProportionProportion TheThe generalgeneral formform ofof anan intervalinterval estimateestimate ofof aa populationpopulation proportionproportion isis pp MarginMargin ofof ErrorError Interval Estimation of a Population Proportion Interval Estimate pp()1 pz //2 n where: 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution p is the sample proportion.
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