ECMT 5001 Year Semester 2REVISIONQUES08.docx

ECMT 5001 Year Semester 2REVISIONQUES08.docx

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ECMT5001YearSemester2REVISIONQUES08

Econometrics&BusinessStatistics

ECMT5001:

PrinciplesofEconometrics

RevisionQuestions.

Pleasenotethatthesequestionsarerevisionquestionsandmaynotbeinthesamestyleastheexaminationquestions.Thesequestionswillallowyoutotestyourknowledgeofthebasicconceptsintheunitofstudy.

PART1:

Q.1Aventurecapitalcompanyfeelsthattherateofreturn（X）onaproposedinvestmentisapproximatelynormallydistributedwithameanof30%andastandarddeviationof10%.（Hint:

Usethepercentagesasawholenumber.Donotuseasaproportion.）

Findtheprobabilitythat

a）thereturnwillexceed55%.

b）thereturnwillbelessthan22%.

c）thereturnwillbebetween21%and29%.

d）theaveragereturnwillbelessthan34%ifasampleof20istaken.

Q.4LetXrepresentthenumberofmachinefailuresforanygivendaybytwoautomaticcanningmachinesinafood-processingplant.

X

0

1

2

3

P（X）

0.35

0.35

0.25

0.05

Whatistheprobabilitythatthenumberofmachinefailuresis

a）exactlythree?

b）morethanone?

c）lessthantwo?

d）Calculatethemeanandvarianceofthenumberofmachinefailures.

Q.5Themaintenancedepartmentofacity'selectricpowercompanyfindsthatitiscostefficienttoreplaceallstreetlightbulbsatonce,ratherthantoreplacethebulbsindividuallyastheyburnout.Assumethatthelifetimeofabulbisnormallydistributedwithameanof3000hoursandastandarddeviationof200hours.Findtheprobabilitythatthelifetimeofabulbis

a）lessthan2750hours.b）between2900hoursand3350hrs.c）between2790hoursand2950hours.

b）Ifthedepartmentwantsnomorethan10%ofthebulbstoburnoutbeforetheyarereplaced,afterhowmanyhoursshouldallthebulbsbereplaced?

c）Ifthedepartmenttakesasampleof40streetlights,whatwouldbetheprobabilitythattheaveragelifetimeofthebulbswasmorethan3050hours?

Q.6Thefollowingtableshowsaconsumer’ssatisfactionwithfoodstores（X）inaparticulartownandY,thenumberofyearstheconsumerhasresidedinthattown.

X

Y

1

2

3

4

1

.04

.14

.23

.07

2

.07

.17

.23

.05

Whatistheprobabilitythat

（a）thenumberofyearsisequaltoone?

（b）Y=2andX=4?

（c）Y=1orX=3?

（d）thelevelofsatisfactionis3ifthenumberofyearsis2?

（e）FindthemarginaldistributionofX.

（f）WhatarethemeanandvarianceofX?

（g）FindtheconditionaldistributionofXgivenY=1.

（h）WhatarethemeanandvarianceofXgivenY=1.

（i）FindthecovarianceofXandY.

Q.8Thelifetimeofacertainbrandoftyresisapproximatelynormallydistributedwithameanof40000kmandastandarddeviationof2500km.Thetyrescarryawarrantyfor33000km.

a）Whatproportionofthetyreswillfailbeforethewarrantyexpires?

b）Whatproportionofthetyreswillfailafterthewarrantyexpiresbutbeforetheyhavelastedfor39000km?

c）Themanufacturerwishestosetthewarrantysuchthatthereisonlya2%failurerate.Whatnumberofkilometresmustheguarantee?

d）Whatistheprobabilitythattheaveragelifetimeof36tyresislessthan39500km?

e）Whatistheprobabilitythattheaveragelifetimeof23tyresismorethan40500km?

Q.9KeepKoolIncmanufactureswindowairconditionersinbothadeluxemodelandastandardmodel.AnauditorengagedinacomplianceauditofthefirmisvalidatingthesalesaccountforApril.Shehascollected200invoicesforthemonth,someofwhichweresenttowholesalersandtheremaindertoretailers.Theinformationisgiveninthetablebelow.

Wholesale

Retail

Deluxe

0.18

0.56

Standard

0.12

0.14

Iftheauditorselectsoneinvoiceatrandom,whatistheprobabilitythattheinvoiceselectedis

a）forthedeluxemodel.

b）forthedeluxemodelandaretailinvoice.

c）awholesaleinvoiceforthedeluxemodel.

d）aninvoiceforthestandardmodelorawholesaleinvoice

e）Aretheevents'retailinvoice'and'invoiceforastandardmodel'mutuallyexclusive?

f）Aretheevents'retailinvoice'and'invoiceforastandardmodel'independent

Q.10Thetablebelowshowsthetotalnumberofaccountingcoursesastudenttakes（X）andY,thesexofthestudent,where1denotesamalestudentand0afemalestudent.

X

Y

2

3

4

5

0

.14

.17

.08

.12

1

.16

.20

.12

.01

Whatistheprobabilitythatastudentselectedatrandom

a）ismale?

b）takes4accountingcoursesandisfemale?

c）takes5accountingcoursesorismale?

d）ismaleifhetakes2accountingcourses?

e）FindthemarginaldistributionofX.

f）WhatarethemeanandstandarddeviationofX?

g）WhatisthedistributionofXifthestudentisamale?

h）Whatarethemeanandvarianceofthedistributioninpart（g）?

i）ArethetwovariablesXandYindependent?

Givereasonsforyouranswer.

Q.11LetYrepresentthenumberoftimesahousewifevisitsasupermarketinaone-weekperiod.

Y

0

1

2

3

P（Y）

0.1

0.2

0.3

0.4

a）Findtheexpectedvalueandthestandarddeviationofthenumberofweeklyvisitstothesupermarket.

b）Whatistheprobabilitythatthehousewifevisitsthestore

i）atleastonce?

ii）nomorethantwice?

iii）exactlyonce?

Q.12LetXrepresentthenumberoftimesastudentvisitsanearbypizzaparlour.

X

0

1

2

3

P（X）

0.1

0.3

0.4

0.2

a）Calculatethemeanandstandarddeviationofthenumberofvisitstothepizzaparlour.

b）Whatistheprobabilitythatastudentvisitsthepizzaparlour

i）atmosttwice?

ii）atleasttwice?

iii）exactlyonce?

Q.13TherandomvariablesXandYhavemeansof10and16respectively,variancesof9and25respectivelyandthecovariancebetweenXandYequalto-4.

i）Findtheexpectedvalueandvarianceof2X-7.

ii）Findtheexpectedvalueandstandarddeviationof3X-4Y

iii）FindE（4-3X）andvar（4-3X）

iv）FindE（4-3X+2Y）andvar（4-3x+2Y）

Q.14Theheightsofboysofage15isnormallydistributedwithameanof150cmandastandarddeviationof20cm.Findtheprobabilitythattheheightofa15yearoldboyis

i）morethan155cms?

ii）between165cmand198cm?

iii）Whatheightmustthehighschoolbasketballteamsetsothatonlythetop10%ofboyswillbeeligibleforteamselection?

iv）IfIrandomlyselect18boys,whatistheprobabilitythattheiraverageheightsislessthan137cm?

Q.15Thedistributionofheartrateforpatientssufferingfromheartdiseaseisapproximatelynormalwithameanof97beatsperminuteandastandarddeviationof18.

a）Whatproportionofthesepeoplehaveheartbeats

i）below120?

ii）between100and115?

b）Ifarandomsampleof30peopleistaken,whatistheprobabilitythatthemeannumberofheartbeatsismorethan105beatsperminute?

Q.16Amanufacturerofcomputerchipsbelievesthatabout90%ofthecomputerchipsconformtospecifications.Ifhesamples100chips,whatistheprobabilitythatbetween82%and87%ofthecomputerchipsconformtospecifications?

Q.17ForthebivariateprobabilitydistributionofX,Ygivenbelow:

Y

X

1

2

0

0.1

0.25

1

0.2

0.15

2

0.25

0.05

a）CalculatethemarginaldistributionofX.

b）CalculatethemeanandstandarddeviationofX.

Whatistheprobabilitythat

c）Y=2andX=1?

d）X=1orY=2?

e）X=2ifY=1?

PART2

Q.1OverthepastfiveyearsthemeanageatdeathintheUnitedStateshasbeen70.5years.Inarandomsampleofthebiographiesof50‘famouspersons’whodiedinthisperiod,themeanageatdeathwas72.0yearswithastandarddeviationof6.3years.Dothesedatasuggestthatfamouspeopleontheaveragelivelongerthantherestofus?

Useasignificancelevelof0.05.

Q.2Inalargecity,22%ofthehouseholdshadtheafternoonpaperdeliveredtotheirdoors.Afterthenewspaperconductedanaggressivemarketingcampaigntoincreasethatfigure,arandomsampleof200householdswastaken.Ofthesamplegroup,61householdsnowhavethepaperdelivered.Canweconcludeatthe5%significancelevelthatthecampaignwasasuccess?

Findthep-valueofthetest.

Q.3Amanufactureroftelevisionpicturetubeshasaproductionlinethatproducesanaverageof100tubesperday.Becauseofnewgovernmentregulations,anewsafetydevicehasbeeninstalled,whichthemanufacturerbelieveswillreducetheaveragedailyoutput.Arandomsampleof15days'outputaftertheinstallationofthesafetydeviceisgiven

93103951019110596941018898941019295

Istheresufficientevidencetoconcludethattheaveragedailyoutputhasdecreasedfollowingtheinstallationofthesafetydevice.（Useα=0.05.）

Q.4Inanefforttoexplaintocustomerswhytheirelectricitybillshavebeensohighlatelyandhow,specifically,theycouldsavemoneybyreducingthermostatsettingsonbothspaceheatersandwaterheaters,anelectricitycompanyhascollectedtotalkilowattconsumption（inkilowatthours/100）figuresforlastyear’swintermonths,aswellasaveragethermostatsettingsonspaceheaters（indegreesCelsius）andaveragethermostatsettingsonwaterheaters（indegreesCelsius）.

Modelfittingresultsfor:

Consumption

Independentvariablecoefficientstd.errort-valuesig.level

CONSTANT-51.5158.618-5.9780.0010

SpaceHeaters-0.7101.635-0.4340.6792

WaterHeaters1.6870.5732.9440.0258

R2=0.9586FStatistic69.5030

SumofSquaredResidual13.1397p-value0.0001

N=9

（i）Writetheestimatedequation,includingthestandarderrorsandR2.

（ii）Interpretthevalueofthecoefficientforspaceheaters.

（iii）Testthehypothesisthatthemodelisusefulforprediction.Useα=0.05.

（iv）Itisbelievedthattheslopeofthevariable,Waterheaters,is