ACTMStateExam-Statistics

Forthe25multiple-choicequestions,makeyouranswerchoiceandrecorditontheanswersheetprovided.Onceyouhavecompletedthatsectionofthetest,proceedtothetie-breakeritems.Thesewillbegradedinnumericalorder,soit’sinyourbestinteresttorespondtotheseinthatorder.Besurethatyournameisoneachsheetofthetie-breakeritems.

1. Generally,Greeklettersareusedtorepresent___________,andRomanlettersareusedtorepresent______________.

a. Statistics,parametersb. Parameters,statisticsc. Measuresofvariation,measuresofspreadd. Measuresofspread,measuresofvariation

2. Ifthemodeistotheleftofmedianandthemeanistotherightofthemedian,thenthedistributionis_____________skewed.

a. Positivelyb. Negativelyc. Notd. Highly

3. Asetof45valueshasameanof23andasetof53valueshasameanof47,whatisthemeanofthecombinedsetofvalues?(Roundtonearesttenthsifneeded.)

a. 49.0b. 50.4c. 35.0d. Noneofthese

4. Supposeweusealeastsquareslinearregressionmodelonasetofdatapoints(x,y).Wefindthecoefficientofdeterminationis0.81andtheregressionlineisgivenbyy=-51x+32.Whatisthevalueofthecorrelationcoefficient?

a. 0.6561b. 0.9c. -0.6561d. -0.9

5. Giventhesamescenarioasinquestion#4andtheQQplotofresidualsbelow,doyoubelieve

thelinearmodelisagoodone?

a. Yes,becausetheabsolutevalueofthecorrelationcoefficientisclosetooneandthe

residualsareclosetotheidentityline.b. No,becausetheabsolutevalueofthecorrelationcoefficientisnotclosetoone.c. No,becausetheQQplotshowsthattheresidualsarenotnormallydistributed.d. Yes,becausetheQQplotshowsthattheresidualsarenormallydistributedandthe

absolutevalueofthecorrelationcoefficientisclosetoone.

6. Thefollowinglistgivesnumberofhoursastudentstudiesperweek.Whatisthelargestvalueinthesamplethatisnotconsideredanoutlier?

0,1,2,2,5,5,5,6,6,7,7,7,8,8,8,8,8,8,9,10,10,10,12,20

a. 12b. 20c. 0d. Noneoftheabove

7. Youstepontoabusthatwilltakebetween15and25minutestoarriveoncampus.Assumethatthedistributionofthesetimesisuniform,yourstatisticsclassbeginsin20minutes,anditwilltaketwominutesforyoutowalkfromthebustotheclassroom.Whatistheprobabilitythatyouarrivelatetoclass?

a. 70%b. 30%c. 50%d. Noneoftheabove

8. Adrawercontains10differentpairsofgloves.If6glovesaredrawnrandomlyfromthedrawer,whatistheprobabilitythattherewillbenomatchingpairinthesample?

a. !(!",!)!(!",!)

b. !!(!",!)!(!",!)

c. !!!(!",!)!(!",!)

d. Noneofthese

9. Afamilyhastwochildren.Giventhatatleastoneofthemisagirl,wasittheprobabilitythatthefamilyhastwogirls?

a. 1/2b. 1/3c. 1/4d. ¾

10. Supposeweknowthatatleast88.89%ofthemembersofagymhaveanagebetween32years62years.WhatcanyouconcludeusingChebyshev’stheoremaboutthemeanandstandarddeviationofthedistributionofagesforthemembers?

a. Youcannotdrawaconclusionfromtheinformationgiven.b. Thestandarddeviationis3years,butwecannotfindthemean.c. Themeanis47yearsandthestandarddeviationis5years.d. Themeanis47yearsandthestandarddeviationis1.67years.

11. Redjellybeansaremyfavorite.IfIrandomlyselectajellybeanfrombowl,theprobabilityofdrawingaredoneis0.21.SupposeIgrabahandfulof13jellybeans,whatistheprobabilityofgettingatleast5thatarered?

a. 0.0797b. 0.1173c. 0.8827d. 0.9625

12. TherandomvariableXisthenumberthatshowsupwhenaloaded4-sideddieisrolled.Its

probabilitydistributionisgiveninthetable.DeterminethemeanofthedistributionofX.

x 1 2 3 4P(X=x) 0.30 0.30 0.15 0.25

a. 2.35b. 2c. 0.5875d. 2.5

13. Inadatasetwitharangeof52.2to114.5and500observations,thereare345observations

withvalueslessthan87.1.Findthepercentilefor87.1.a. 69b. 131.458c. 698d. 32

14. Whichscorehasahigherrelativeposition,ascoreof48onatestforwhichthemeanis40andstandarddeviationis4,orascoreof288.6onatestforwhichthemeanis260andstandarddeviationis26?

a. Ascoreof48b. Ascoreof288.6c. Bothscoreshavethesamerelativeposition

15. Thebrandmanagerforabrandoftoothpastemustplananewcampaigndesignedtoincreasebrandrecognition.Aprevioussurveysuggestedthatabout76%ofadultshaveheardofthebrand.Hewantstofirstdeterminethepresentpercentageofadultswhohaveheardofthebrand.Howmanyadultsmusthesurveyinordertobe80%confidentthathisestimateiswithinsixpercentagepointsofthetruepopulationpercentage?

a. Atleast84b. Atleast83c. Atleast115d. Atleast116

16. WhichofthefollowingisNOTtrueoftheconfidencelevelofaconfidenceinterval?

a. Theconfidencelevelgivesusthesuccessrateoftheprocedureusedtoconstructtheconfidenceinterval.

b. Theconfidencelevelisoftenexpressedasanarea1-α,whereαisthecomplementoftheconfidencelevel.

c. Theconfidencelevelisalsocalledthedegreeofconfidence.d. Thereisa1-αchance,whereαisthecomplementoftheconfidencelevel,thatthetrue

valueoftheparameterwillfallintheconfidenceintervalproducedbythesample.

17. Usinga95%confidencelevel,findaconfidenceintervalforestimatingthetruemeanweight,µ,ofallpackagesreceivedbytheparcelservicegiventhatasampleof31packageshadameanweightof13.2lb.andthestandarddeviationofallpackagesisσ=2.4lbs.

a. 12.2lb<µ<14.2lb.b. 12.4lb<µ<14.0lb.c. 12.1lb<µ<14.3lb.d. 12.5lb<µ<13.9lb.

18. Themeanreplacementtimeforarandomsampleof20washingmachinesis10.8yearsandthestandarddeviation2.8years.Constructa99%confidenceintervalforthestandarddeviation,σ,ofthereplacementtimesofallwashingmachinesofthistype.

a. 2.5years<σ<5.9yearsb. 2.0years<σ<4.7yearsc. 0.3years<σ<5.3yearsd. 1.2years<σ<4.4years

19. Afarmerhasdecidedtouseanewadditivetogrowhiscrops.Hedividedhisfarminto10plotsandkeptrecordsofthecornyield(inbushels)beforeandafterusingtheadditive.Theresultsareinthetablebelow.Usinga1%levelofsignificance,findthep-valuefortestingtheclaimthattheadditiveproducesahighermeanyield.

a. 0.1355b. 0.0004c. 0.0070d. 0.2709

20. Whichofthefollowingisnotanadvantageofpoolingsamplevariances?a. Confidenceintervalsarealittlenarrower.b. Thenumberofdegreesoffreedomisalittlehigher.c. Hypothesistestshavemorepower.d. Weoftenknowthatthepopulationvariancesareequal.

21. Astudywasdoneusingatreatmentgroupandaplacebogroup.Wewanttotesttheclaimthatthesamplesarefrompopulationswiththesamemean.Theresultsareshowninthetable.Assumethatthetwosamplesareindependentsimplerandomsamplesfromnormallydistributedpopulations.Usinga0.01significancelevel,statetheconclusionofthetest.

Treatment PlaceboSamplesize 26 38Samplemean 2.33 2.64Samplestandarddeviation 0.51 0.99

Plot Before After1 9 102 9 93 8 94 7 85 6 76 8 107 5 68 9 109 10 1010 11 12

a. Rejectthenullhypothesis.Thereissufficientevidencetowarrantrejectionoftheclaimthatthetwosamplesarefrompopulationswiththesamemean.

b. Rejectthenullhypothesis.Thereisnotsufficientevidencetowarrantrejectionoftheclaimthatthetwosamplesarefrompopulationswiththesamemean.

c. Failtorejectthenullhypothesis.Thereisnotsufficientevidencetowarrantrejectionoftheclaimthatthetwosamplesarefrompopulationswiththesamemean.

d. Failtorejectthenullhypothesis.Thereissufficientevidencetowarrantrejectionoftheclaimthatthetwosamplesarefrompopulationswiththesamemean.

22. Theactivationtimesforanautomaticsprinklersystemareasubjectofstudybythesystem’smanufacturer.Asampleofactivationtimesis27,41,22,27,23,35,30,33,24,27,28,22,and24seconds.Thedesignofthesystemcallsforitsactivationinatmost25seconds.Assuminganormalpopulation,doesthedatacontradictthevalidityofthisdesignspecification?

a. Yesb. Noc. Notenoughdatatodrawaconclusion

23. Callscomeintoatelephoneswitchboardatarateof4perminute.Findtheprobabilityof

getting6callsinanintervalof2minutesroundedtothreeplaces.UseaPoissondistribution.a. 0.122b. 0.104c. 0.313d. Noneofthese

24. SupposeweareinterestedinstudyingthefactorsthataffectGPAofseniorhighschoolstudents.

FiftyschoolsareselectedatrandomandwecollecttheGPAofonemaleandonefemalefromeachschool.Wealsoclassifyeachschoolaspublicorprivate.Whichofthefollowingproceduresismostappropriatetoconductfirst.

a. Anindependent2sampledttestonthemaleandfemaleGPA.b. ApairedttestonthemaleandfemaleGPA.c. ANOVAonthefoursetsofGPAs:malesatprivateschool,femalesatprivateschool,

malesatpublicschool,femalesatpublicschool.d. Two-WayANOVAusingtheschoolstatusasonefactorandgenderasthesecondfactor.

25. Theinterceptinlinearregressionrepresents:

a. Thestrengthoftherelationshipbetweenxandy.b. Theexpectedx-valuewhenyiszero.c. Theexpectedy-valuewhenxiszero.d. Theerrorinthemeasurements.

TieBreakers

Name:___________________________________________

1. Yourteachertellstheclassthatthetestscoresonthemostrecentexamwerenormally

distributedwithmeanof82%andavarianceof4%.Youaredisappointedbecauseyourscorewas76%andusuallyyouhavethetopscoreintheclass!Soyourandomlyselect5ofyour25classmatesandfindouttheirtestscoreswere56%,78%,84%,67%,and81%.Assumingthefivescoresyoucollectedwereasimplerandomsample,answerthefollowingquestions.

a. Whatistheprobabilitythatarandomlyselectedstudentreceivesascoreof76%orlowerontheexam?

b. Assumingtheteacherwastruthful,whatistheprobabilitythattheaverageof5randomlyselectedstudentswouldbehigherthan83%.

c. Basedonthefivescoresyouknow,testtheclaimthattheaveragefortheclasswasactuallylessthan82%.Use0.05significancelevel.

d. Supposenowthattheteacherhadonlytoldyouthatthescoreswerenormallydistributedwithameanof82%,buthe/shedidnotcalculatethevariancefortheclass.Basedonthefivescoresyouknow,testtheclaimthattheaveragefortheclasswasactuallylessthan82%.Use0.05significancelevel.

e. DescribetheinterpretationsofatypeIandtypeIIerrorinthisscenario.

Name:______________________________________

2. Inaboltfactory,machinesA,BandCmanufacture20%,30%and50%(respectively)ofthetotaloutput.Oftheiroutput,3%,2%and1%aredefective.

a. Suppose500boltsaremadeeveryhour,howmanydefectiveboltsdoyouexpecttofindoverthecourseofa40hourworkweek?

b. Ifaboltisselectedatrandom,whatistheprobabilitythatitisdefective?

c. Aboltisselectedatrandomandfoundtobedefective,whatistheprobabilitythatitcamefrommachineA?

3. AnExperimentwasconductedtocomparetheeffectivenessofthreetrainingprograms,A,B,

andC,intrainingassemblersofapieceofelectronicequipment.Fifteenemployeeswererandomlyassigned,fiveeachtothethreeprograms.Aftercompletionofthecourses,eachpersonwasrequiredtoassemblefourpiecesoftheequipment,andtheaveragelengthoftimerequiredtocompletetheassemblywasrecorded.Severaloftheemployeesresignedduringthecourseoftheprogram;theremainderwereevaluated,producingthedatashownintheaccompanyingtable.Usethefollowingoutputtoanswerthequestions.TrainingProgram AverageAssemblyTime(min)A 59 64 57 62 B 52 58 54 C 58 65 71 63 64

Source DF SS MS F P Program 2 170.5 85.2 5.7 .025 Error 9 134.5 14.9 Total 11 304.9

Level N Mean StDev A 4 60.500 3.109 B 3 54.667 3.055 C 5 64.200 4.658

a. Dothedataprovidesufficientevidencetoindicateadifferenceinmeanassemblytimesforpeopletrainedbythethreeprograms?Useasignificancelevelof0.05.

b. Findsimultaneous99%confidenceintervalsforthemeanassemblytimesbetweenpersonstrainedbyprogramsA,B,andC.Whatcanyouconcludebasedontheseconfidenceintervals?

Solutions

1.b

2.a

3.d

4.d

5.c

6.a

7.b

8.c

9.b

10.c

11.b

12.a

13.a

14.a

15.a

16.d

17.b

18.b

19.b

20.d

21.c

22.a

23.a

24.d

25.c

SolutionstoTieBreakers

1. Yourteachertellstheclassthatthetestscoresonthemostrecentexamwerenormallydistributedwithmeanof82%andavarianceof4%.Youaredisappointedbecauseyourscorewas76%andusuallyyouhavethetopscoreintheclass!Soyourandomlyselect5ofyour25classmatesandfindouttheirtestscoreswere56%,78%,84%,67%,and81%.Assumingthe6scoresyoucollectedwereasimplerandomsample,answerthefollowingquestions.

a. Whatistheprobabilitythatarandomlyselectedstudentreceivesascoreof76%orlowerontheexam?

P(X<76)=0.0013bycalculator.Iftheyuseempiricalrulewithz-scoreof-3,theprobabilityisapproximately0.0015.

b. Assumingtheteacherwastruthful,whatistheprobabilitythattheaverageof5

randomlyselectedstudentswouldbehigherthan83%.

Thesamplingdistributionofthemeanhasameanof82andvarianceof4/5.

P(𝑿>83)=0.1318

c. Basedonthe6scoresyouknow,testtheclaimthattheaveragefortheclasswasactuallylessthan82%.Use0.05significancelevel.

Ho:µ=82vs.H1:µ<82Zstat=-9,839andp-valueisvery,veryclosetozero.Rejectthenullhypothesis.Wehaveevidencethattheclassaveragewasprobablylessthan82%.Weshouldconcludethatmaybetheteachermiscalculatedtheaverageorthevarianceoftheclass.

d. Supposenowthattheteacherhadonlytoldyouthatthescoreswerenormally

distributedwithameanof82%,buthe/shedidnotcalculatethevariancefortheclass.Basedonthe6scoresyouknow,testtheclaimthattheaveragefortheclasswasactuallylessthan82%.Use0.05significancelevel.

Ho:µ=82vs.H1:µ<82Tstat=-1.702andp-valueis0.082.Failtorejectthenullhypothesis.Wedonothaveevidencethattheclassaveragewaslessthan82%.

e. DescribetheinterpretationsofatypeIandtypeIIerrorinthisscenario.

AtypeIerrorwouldmeanthatwerejectthenullhypothesiswheninfactthenullhypothesisistrue.HereatypeIerrorwouldmeanthatwewouldconcludetheteacherwaswrong(orlied)aboutthemeanoftheclassbasedonthesamplecollected,butinfacttheaveragereallywas82%.

AtypeIIerrorwouldmeanthatwefailtorejectthenullhypothesiswhenthenullhypothesisisinfactfalse.HereatypeIIerrorwouldmeanthatwewouldconcludetherewasnotenoughevidenceforsayingtheteacherwasincorrect,butactuallytheaverageoftheclasswaslessthan82%.

2. Inaboltfactory,machinesA,BandCmanufacture20%,30%and50%(respectively)ofthetotaloutput.Oftheiroutput,3%,2%and1%aredefective.

a. Suppose500boltsaremadeeveryhour,howmanydefectiveboltsdoyouexpecttofindoverthecourseofa40hourworkweek?

Amakes100boltswithapprox.3defectiveperhour.Bmakes150boltswithapprox..3defectiveperhour.Cmakes250boltswithapprox.2.5defectiveperhour.Totalweexpect8.5defectiveboltsperhouror340per40hours.(TheycouldalsousepartB)

b. Ifaboltisselectedatrandom,whatistheprobabilitythatitisdefective?

P(defective)=P(defective|A)P(A)+P(defective|B)P(B)+P(defective|C)P(C)=0.03*0.20+0.02*0.30+0.01*0.5=0.017

c. Aboltisselectedatrandomandfoundtobedefective,whatistheprobabilitythatitcamefrommachineA?

P(A|defective)=P(defective|A)P(A)/P(defective)=(0.03*0.20)/0.017=0.3529

3. a. For the hypotheses 𝑯𝟎: 𝝁𝑨 = 𝝁𝑩 = 𝝁𝑪; 𝑯𝟏:At least one mean is different.

The cr it ical value is 𝑭𝑪𝑽 = 𝟒.𝟐𝟔, Reject 𝑯𝟎 s ince the test stat ist ic fa l ls ins ide the r ight ta i led reject ion region with p value of 0.025. There is evidence to indicate a difference in the mean assembly t imes for people trained by the three programs.

b . (Using F isher’s Least S ignif icant Difference) The t-value for a l l confidence

intervals is 3.250 and the est imate for s is square root of MSE= 𝟏𝟒.𝟗 ≈ 𝟑.𝟖𝟔𝟎𝟎𝟓

The 99% Confidence intervals for the mean assembly t ime for programs A, B, and C are

𝟓𝟒.𝟐𝟐𝟕 < 𝝁𝑨 < 𝟔𝟔.𝟕𝟕𝟑 𝟒𝟕.𝟒𝟐𝟒 < 𝝁𝑩 < 𝟔𝟏.𝟗𝟏𝟎 𝟓𝟖.𝟓𝟗𝟎 < 𝝁𝑪 < 𝟔𝟗.𝟖𝟏𝟎

With 99% confidence there doesn’t seem to be a s ignif icant difference among the three methods. This is because there is an intersect ion in a l l confidence intervals .

This is not contradictory to our result because the test in (a) had s ignif icance level 0.05 (or confidence 95%). You can verify this by construct ing the 95% simultaneous CIs (B and C do not intersect.)

I t is more l ikely that they wi l l compute confidence intervals for each set separately, g iv ing even wider intervals that st i l l overlap.

𝟓𝟏.𝟒𝟐 < 𝝁𝑨 < 𝟔𝟗.𝟓𝟖 𝟑𝟕.𝟏𝟔𝟏 < 𝝁𝑩 < 𝟕𝟐.𝟏𝟕𝟐 𝟓𝟒.𝟔𝟎𝟖 < 𝝁𝑪 < 𝟕𝟑.𝟕𝟗𝟐