光华管理学院2012—2013第一学期期末考试Upper-QuantilesThatMightUsed0.025α0.050.1N(0,1)1.961.641.28χ4211.149.497.78χ8219.0216.9214.68χ19232.8530.1427.20χ20234.1731.4128.41t192.091.731.32t202.091.721.33𝐹19,192.532.171.82𝐹20,202.462.121.791.(40)Considerthelinearregressionmodel:𝑌𝑖=𝛽0+𝛽1𝑥𝑖+𝜀𝑖…………(M1)Where𝜀𝑖’sarei.i.d.andfollowanormaldistributionwithmean0andvarianceσ2,and𝑥𝑖’sarefixedconstants,fori=1,…,n.(a)DerivetheMLEof𝛽0,𝛽1andσ2.(6points)(b)ShowthatMLEof𝛽0,𝛽1arethesameasleastsquareestimators.(5points)(c)ShowthatMLEof𝛽0and𝛽1areunbiased.(6points)(d)ShowthatMLEofσ2isbiasedandfindthebias.Commentontheeffectofsamplesizeonthemagnitudeofbias.Proposeanunbiasedestimatorforσ2.(5points)(e)Explainhowtoperformatestforthehypothesis𝐻0:𝛽1=0𝐻1:𝛽1≠0Andconstructa95%confidenceintervalfor𝛽0.(6points)Nowconsiderareparameterizedmodelof(M1),𝑌𝑖=𝛽0′+𝛽1′(𝑥𝑖−𝑥̅𝑛)+𝜀𝑖…………(M2)where𝑥̅𝑛isthesampleaverageof𝑥𝑖’s.(f)DerivetheMLEof𝛽0′and𝛽1′.(5points)(g)ComparetheMLE’sinpart(f)andthoseofpart(a).(2points)(h)ShowthattheMLEof𝛽0′andthatof𝛽1′areuncorrelated.(5points)2.(10)Anassemblylineproduceswidgetswithameanweightof10andastandarddeviationof0.200.Anewprocesssupposedlywillproducewidgetswiththesamemeanandasmallerstandarddeviation.Asampleof20widgetsproducedbythenewmethodhasasamplestandarddeviation√∑(𝑥𝑖−10)2𝑛20𝑖=1=0.126.Atasignificancelevelof10%canweconcludethatthenewprocessislessvariablethantheold?(1)Stateappropriatenullandalternativehypotheses.(2points)(2)Derivethetestingprocedureatlevel10%.(6points)(3)Isthenewprocesslessvariablethantheold?(2points)3.(10)Anewdrugisbeingproposedforthetreatmentofmigraineheadaches.Unfortunatelysomeusersinearlytestsofthedrughavereportedmildnauseaasasideeffect.TheFDA(FoodandDrugAdministration)willrejectthedrugifitthinksthatmorethan15%(i.e.0.15)ofthepopulationwouldsufferfromthissideeffect.Inanexperimenttotestthissideeffect,400peoplewhosufferfrommigraineheadachesreceivethenewdrugand80ofthemreportnauseaasa...
