《Statistics 1历年考试真题分类汇编》教程文件.docx

《Statistics 1历年考试真题分类汇编》教程文件.docx

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《Statistics1历年考试真题分类汇编》教程文件

SectionA………………………………………………………………………1

Representationofdata.Measuresoflocation.Measuresofspread

SectionB………………………………………………………………………9

Probability

SectionC………………………………………………………………………14

Permutationsandcombinations

SectionD………………………………………………………………………19

Probabilitydistributions

ExpectationandVarianceofarandomVariable

SectionE………………………………………………………………………22

Thebinomialdistribution

SectionF………………………………………………………………………26

Thenormaldistribution

Answer…………………………………………………………………………30

SectionA………………………………………………………………………30

SectionB………………………………………………………………………36

SectionC………………………………………………………………………39

SectionD………………………………………………………………………40

SectionE………………………………………………………………………42

SectionF………………………………………………………………………44

SectionARepresentationofdata

Measuresoflocation

Measuresofspread

1Theweightsinkilogramsoftwogroupsof17-year-oldmalesfromcountryPandcountryQaredisplayedinthefollowingback-to-backstem-and-leafdiagram.Inthethirdrowofthediagram,...4|7|1...denotesweightsof74kgforamaleincountryPand71kgforamaleincountryQ.（02w）

（i）FindthemedianandquartileweightsforcountryQ.[3]

（ii）Youaregiventhatthelowerquartile,medianandupperquartileforcountryPare84,94and98kgrespectively.Onasinglediagramongraphpaper,drawtwobox-and-whiskerplotsofthedata.[4]

（iii）Maketwocommentsontheweightsofthetwogroups.[2]

2（i）

（03s）

ThediagramrepresentsthesalesofSuperclenetoothpasteoverthelastfewyears.Giveareasonwhyitismisleading.[1]

（ii）Thefollowingdatarepresentthedailyticketsalesatasmalltheatreduringthreeweeks.

52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86.

（a）Constructastem-and-leafdiagramtoillustratethedata.[3]

（b）Useyourdiagramtofindthemedianofthedata.[1]

3Arandomsampleof97peoplewhoownmobilephoneswasusedtocollectdataontheamountoftimetheyspentperdayontheirphones.Theresultsaredisplayedinthetablebelow.（03s）

Timespentper

day（tminutes）

0≤t<5

5≤t<10

10≤t<20

20≤t<30

30≤t<40

40≤t<70

Number

ofpeople

11

20

32

18

10

6

（i）Calculateestimatesofthemeanandstandarddeviationofthetimespentperdayonthesemobilephones.[5]

（ii）Ongraphpaper,drawafullylabelledhistogramtorepresentthedata.[4]

4Acomputercangeneraterandomnumberswhichareeither0or2.Onaparticularoccasion,itgeneratesasetofnumberswhichconsistsof23zerosand17twos.Findthemeanandvarianceofthissetof40numbers.[4]（03w）

5Thefloorareas,xm2,of20factoriesareasfollows.（03w）

150350450578595644722798802904

1000133015331561177819602167233024333231

Representthesedatabyahistogramongraphpaper,usingintervals

0≤x<500,500≤x<1000,1000≤x<2000,2000≤x<3000,

3000≤x<4000.[4]

6Twocricketteamskeptrecordsofthenumberofrunsscoredbytheirteamsin8matches.Thescoresareshowninthefollowingtable.（04s）

TeamA

150220773029811816057

TeamB

1661421709311113014886

（i）FindthemeanandstandarddeviationofthescoresforteamA.[2]

ThemeanandstandarddeviationforteamBare130.75and29.63respectively.

（ii）Statewithareasonwhichteamhasthemoreconsistentscores.[2]

7Inarecentsurvey,640peoplewereaskedaboutthelengthoftimeeachweekthattheyspentwatchingtelevision.Themediantimewasfoundtobe20hours,andthelowerandupperquartileswere15hoursand35hoursrespectively.Theleastamountoftimethatanyonespentwas3hours,andthegreatest

amountwas60hours.（04s）

（i）Ongraphpaper,showtheseresultsusingafullylabelledcumulativefrequencygraph.[3]

（ii）Useyourgraphtoestimatehowmanypeoplewatchedmorethan50hoursoftelevisioneachweek.[2]

8Thelengthsofcarstravellingonacarferryarenoted.Thedataaresummarisedinthefollowingtable.（04w）

Lengthofcar（xmetres）

Frequency

Frequencydensity

2.80≤x<3.00

17

85

3.00≤x<3.10

24

240

3.10≤x<3.20

19

190

3.20≤x<3.40

8

a

9Theages,xyears,of18peopleattendinganeveningclassaresummarisedbythefollowingtotals:

Σx=745,Σx2=33951.（04w）

（i）Calculatethemeanandstandarddeviationoftheagesofthisgroupofpeople.[3]

（ii）Onepersonleavesthegroupandthemeanageoftheremaining17peopleisexactly41years.Findtheageofthepersonwholeftandthestandarddeviationoftheagesoftheremaining17people.[4]

10Thefollowingtableshowstheresultsofasurveytofindtheaveragedailytime,inminutes,thatagroupofschoolchildrenspentininternetchatrooms.

（05s）

Timeperday

（tminutes）

Frequency

0≤t<10

2

10≤t<20

f

20≤t<40

11

40≤t<80

4

Themeantimewascalculatedtobe27.5minutes.

（i）Formanequationinvolvingfandhenceshowthatthetotalnumberofchildreninthesurveywas26.[4]

（ii）Findthestandarddeviationofthesetimes.[2]

11Thefollowingback-to-backstem-and-leafdiagramshowsthecholesterolcountforagroupof45peoplewhoexercisedailyandforanothergroupof63whodonotexercise.Thefiguresinbracketsshowthenumberofpeoplecorrespondingtoeachsetofleaves.（05s）

（i）Giveoneusefulfeatureofastem-and-leafdiagram.[1]

（ii）Findthemedianandthequartilesofthecholesterolcountforthegroupwhodonotexercise.[3]

Youaregiventhatthelowerquartile,medianandupperquartileofthecholesterolcountforthegroupwhoexerciseare4.25,5.3and6.6respectively.

（iii）Onasinglediagramongraphpaper,drawtwobox-and-whiskerplotstoillustratethedata.[4]

12Agroupof10marriedcouplesand3singlemenfoundthatthemeanage

ofthe10womenwas41.2yearsandthestandarddeviationofthewomen’sageswas15.1years.Forthe13men,themeanage

was46.3yearsandthestandarddeviationwas12.7years.（05w）

（i）Findthemeanageofthewholegroupof23people.[2]

（ii）Theindividualwomen’sagesaredenotedby

andtheindividualmen’sagesby

.ByfirstfindingΣ

andΣ

findthestandarddeviationforthewholegroup.[5]

13Astudyoftheagesofcardriversinacertaincountryproducedtheresultsshowninthetable.（05w）

Percentageofdriversineachagegroup

Young

Middle-aged

Elderly

Males

40

35

25

Females

20

70

10

Illustratetheseresultsdiagrammatically.

14Thesalaries,inthousandsofdollars,of11people,chosenatrandominacertainoffice,werefoundtobe:

40,42,45,41,352,40,50,48,51,49,47.

Chooseandcalculateanappropriatemeasureofcentraltendency（mean,modeormedian）tosummarisethesesalaries.Explainbrieflywhytheothermeasuresarenotsuitable.[3]（06s）

15Eachfatherinarandomsampleoffatherswasaskedhowoldhewaswhenhisfirstchildwasborn.Thefollowinghistogramrepresentstheinformation.

（06s）

（i）Whatisthemodalagegroup?

[1]

（ii）Howmanyfatherswerebetween25and30yearsoldwhentheirfirstchildwasborn?

[2]

（iii）Howmanyfatherswereinthesample?

[2]

（iv）Findtheprobabilitythatafather,chosenatrandomfromthegroup,wasbetween25and30yearsoldwhenhisfirstchildwasborn,giventhathewasolderthan25years.[2]

1632teamsenterforaknockoutcompetition,inwhicheachmatchresultsinoneteamwinningandtheotherteamlosing.Aftereachmatchthewinningteamgoesontothenextround,andthelosingteamtakesnofurtherpartinthecompetition.Thus16teamsplayinthesecondround,8teamsplayinthethirdround,andsoon,until2teamsplayinthefinalround.（06s）

（i）Howmanyteamsplayinonly1match?

[1]

（ii）Howmanyteamsplayinexactly2matches?

[1]

（iii）Drawupafrequencytableforthenumbersofmatcheswhichtheteamsplay.[3]

（iv）Calculatethemeanandvarianceofthenumbersofmatcheswhichtheteamsplay.[4]

17Theweightsof30childreninaclass,tothenearestkilogram,wereasfollows.

50456153554752494651

60525447575942465153

56485051445249585545

Constructagroupedfrequencytableforthesedatasuchthattherearefiveequalclassintervalswiththefirstclasshavingalowerboundaryof41.5kgandthefifthclasshavinganupperboundaryof61.5kg.[4]（06w）

18Inasurvey,peoplewereaskedhowlongtheytooktotraveltoandfromwork,onaverage.Themediantimewas3hours36minutes,theupperquartilewas4hours42minutesandtheinterquartilerangewas3hours48minutes.Thelongesttimetakenwas5hours12minutesandtheshortesttimewas30minutes.（i）Findthelowerquartile.[2]（06w）

（ii）Representtheinformationbyabox-and-whiskerplot,usingascaleof2cmtorepresent60minutes.[4]

19Thelengthoftime,tminutes,takentodothecrosswordinacertainnewspaperwasobservedon12occasions.Theresultsaresummarisedbelow.

Σ（t−35）=−15Σ（t−35）2=82.23

Calculatethemeanandstandarddeviationofthesetimestakentodothecrossword.[4]（07s）

20Thelengthsoftimeinminutestoswimacertaindistancebythemembersofaclassoftwelve9-year-oldsandbythemembersofaclassofeight16-year-oldsareshownbelow.（07s）

9-year-olds:

13.016.116.014.415.915.114.213.716.716.415.013.2

16-year-olds:

14.813.011.411.716.513.712.812.9

（i）Drawaback-to-backstem-and-leafdiagramtorepresenttheinformationabove.[4]

（ii）Anewpupiljoinedthe16-year-oldclassandswamthedistance.Themeantimefortheclassofninepupilswasnow13.6minutes.Findthenew

pupil’stimetoswimthedistance.[3]

21Thestem-and-leafdiagrambelowrepresentsdatacollectedforthenum