Chap 8-1 Statistics for Business and Economics, 6e 2007 Pearson Education, IncChapter 8Estimation: Single PopulationStatistics for Business and Economics 6th EditionStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"2hapter #oalsAfter completing this chapter, you should be able to: $istinguish %et&een a point estimate and a confidence inter'al estimateonstruct and interpret a confidence inter'al estimate for a single population mean using %oth the(andtdistri%utions)orm and interpret a confidence inter'al estimate for a single population proportionStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*onfidence Inter'alsContent of this chapteronfidence Inter'als for the Population +ean, ,&hen Population -ariance .2 is /no&n&hen Population -ariance .2 is 0n1no&nonfidence Inter'als for the Population Proportion, 2large samples3pStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"4$efinitions5n estimator of a population parameter is a random 'aria%le that depends on sample information . . . &hose 'alue pro'ides an appro6imation to this un1no&n parameter5 specific 'alue of that random 'aria%le is called an estimateStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"7Point and Inter'al Estimates5 point estimate is a single num%er, a confidence inter'al pro'ides additional information a%out 'aria%ilit8Point EstimateLower Confidence LimitUpperConfidence LimitWidth of confidence interalStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"69e can estimate a Population Parameter :Point Estimates&ith a SampleStatistic2a Point Estimate3+eanProportionP 6 ,pStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"70n%iasedness5 point estimatoris said to %e an un%iased estimator of the parameterif the e6pected 'alue, or mean, of the sampling distri%ution ofis,E6amples:;he sample mean is an un%iased estimator of ,;he sample 'ariance is an un%iased estimator of .2;he sample proportion is an un%iased estimator of P7 = = .0272J= .0272J=Point EstimateLower Confidence LimitUpperConfidence Limit: units:& units:Point Estimate))ind @.027 F =.>6 from the standard normal distri%ution ta%leStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"2*ommon ?e'els of onfidenceommonl8 used confidence le'els are >0D, >7D, and >>DConfidence LevelConfidence Coefficient, Z/2 value9158910.(91,051;;51(8;1)8;157D confidence inter'al for the true mean resistance of the population.Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"262.406! , =.>>*2.206! 2.203 == 2.*7I =.>6 2.20n.@ 6< < = =E6ample5 sample of == circuits from a large normal population has a mean resistance of 2.20 ohms.9e 1no& from past testing that the population standard de'iation is .*7 ohms.Solution:(continued)Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"27Interpretation9e are >7D confident that the true mean resistance is %et&een =.>>*2and2.406! ohms 5lthough the true mean ma8 or ma8 not %e in this inter'al, >7D of inter'als formed in this manner &ill contain the true meanStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"2!onfidence Inter'alsPopulation %eanConfidence+nteralsPopulationProportion 45 Un#nown45 6nownStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"2>StudentOst$istri%utiononsider a random sample of n o%ser'ations&ith mean6and standard de'iations from a normall8 distri%uted population &ith mean,;hen the 'aria%lefollo&s the StudentOs t distri%ution &ith 2n " =3 degrees of freedomn sI, 6t=Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*0If the population standard de'iation.is un1no&n, &e can su%stitute the sample standard de'iation, s ;his introduces e6tra uncertaint8, sincesis 'aria%le from sample to sampleSo &e use thetdistri%ution instead of the normal distri%utiononfidence Inter'al for ,2.2 0n1no&n3 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*=5ssumptionsPopulation standard de'iation is un1no&nPopulation is normall8 distri%utedIf population is not normal, use large sample0se StudentOs t$istri%utiononfidence Inter'al Estimate:&heretn"=,JI2is the critical 'alue of the t distri%ution &ithn"=d.f. and an area ofJI2in each tail: onfidence Inter'al for ,2. 0n1no&n3 nSt 6 ,nSt 6JI2 =, " n JI2 =, " n+ < < (continued)JI2 3 t P2tJI2 =, n = n= > Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*2StudentOs t $istri%ution;hetis a famil8 of distri%utions;het 'aluedepends on degrees of freedom 2d.f.3Hum%er of o%ser'ations that are free to 'ar8 after sample mean has %een calculatedd.f. F n " =Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"**StudentOs t $istri%utiont0t2df F 73 t2df F =*3t"distri%utions are %ell"shaped and s8mmetric, %ut ha'e PfatterO tails than the normalStandard Hormal2t &ith df F Q3Hote:t (asnincreasesStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*4StudentOs t ;a%leUpper =ail Areadf .=0 .0271)(= =2.7065* *.=!2t)51,5);he %od8 of the ta%le contains t 'alues, not pro%a%ilities?et: n F * df F n " = F 2 F .=0I2 F.07I2 F .07*.07!=.!!6=.6*!6.*=451,5)2.*7*4.*0*Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*7t distri%ution 'alues9ith comparison to the ( 'alueConfidence t tt:Leel!9) d1f1$ !5) d1f1$ !;) d1f1$ >>>> .!0 =.*72=.*27 =.*=0=.2!2 .>0=.!=2=.727 =.6>7=.647 .>72.22!2.0!6 2.042=.>60 .>>*.=6>2.!47 2.7702.776Hote:t (asnincreasesStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"*6E6ample 5 random sample of n F 27 has 6 F 70 and s F !.)orm a >7D confidence inter'al for ,d.f. F n M = F 24,so;he confidence inter'al is 2.06*> t t24,.027 JI2 =, n= =7*.*02 , 46.6>!27!22.06*>3 70 ,27!22.06*>3 70nSt 6 ,nSt 6JI2 =, " n JI2 =, " n< P 0.=67==00.272.773=.>6=0027P=00.272.773=.>6=0027n3 p 2= p@ p Pn3 p 2= p@ pJI2 JI2< D.5lthough the inter'al from 0.=67= to 0.**4>ma8 or ma8 not contain the true proportion, >7D of inter'als formed from samples of si@e =00 in this manner &ill contain the true proportion.Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"44PBStat Inter'al RptionsoptionsStatistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"470sing PBStat 2for ,, . un1no&n3 5 random sample of n F 27 has S F 70 and S F !.)orm a >7D confidence inter'al for ,Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.hap !"46hapter Summar8Introduced the concept of confidence inter'als$iscussed point estimates$e'eloped confidence inter'al estimatesreated confidence inter'al estimates for the mean 2.21no&n3Introduced the StudentOs t distri%ution$etermined confidence inter'al estimates for the mean 2.2 un1no&n3reated confidence inter'al estimates for the proportion