statistics1历年考试真题分类汇编.docx
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statistics1历年考试真题分类汇编
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]
ThemeanandstandarddeviationforteamBareandrespectively.
(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
≤x<
17
85
≤x<
24
240
≤x<
19
190
≤x<
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
Themeantimewascalculatedtobeminutes.
(i)Formanequationinvolvingfandhenceshowthatthetotalnumberofchildreninthesurveywas26.[4]
(ii)Findthestandarddeviationofthesetimes.[2]
11Thefollowingback-to-backstem-and-leafdiagramshowsthecholesterolcountforagroupof45peoplewhoexercisedailyandforanothergroupof63whodonotexercise.Thefiguresinbracketsshowthenumberofpeoplecorrespondingtoeachsetofleaves.(05s)
(i)Giveoneusefulfeatureofastem-and-leafdiagram.[1]
(ii)Findthemedianandthequartilesofthecholesterolcountforthegroupwhodonotexercise.[3]
Youaregiventhatthelowerquartile,medianandupperquartileofthecholesterolcountforthegroupwhoexerciseare,andrespectively.
(iii)Onasinglediagramongraphpaper,drawtwobox-and-whiskerplotstoillustratethedata.[4]
12Agroupof10marriedcouplesand3singlemenfoundthatthemeanage
ofthe10womenwasyearsandthestandarddeviationofthewomen’sageswasyears.Forthe13men,themeanage
wasyearsandthestandarddeviationwasyears.(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(t35)=15Σ(t35)2=
Calculatethemeanandstandarddeviationofthesetimestakentodothecrossword.[4](07s)
20Thelengthsoftimeinminutestoswimacertaindistancebythemembersofaclassoftwelve9-year-oldsandbythemembersofaclassofeight16-year-oldsareshownbelow.(07s)
9-year-olds:
16-year-olds:
(i)Drawaback-to-backstem-and-leafdiagramtorepresenttheinformationabove.[4]
(ii)Anewpupiljoinedthe16-year-oldclassandswamthedistance.Themeantimefortheclassofninepupilswasnowminutes.Findthenew
pupil’stimetoswimthedistance.[3]
21Thestem-and-leafdiagrambelowrepresentsdatacollectedforthenumberofhitsonaninternetsiteoneachdayinMarch2007.Thereisonemissingvalue,denotedbyx.(08s)
(i)Findthemedianandlowerquartileforthenumberofhitseachday.[2]
(ii)Theinterquart