501
Major
Men-ExpectedMajor
Men-ActualMajor
Arts&Humanities
11.0%
600
BiologicalSciences
6.7%
330
Business
22.7%
1130
Education
5.8%
305
Engineering
15.6%
800
PhysicalSciences
3.6%
175
Professional
9.3%
460
SocialSciences
7.6%
370
Technical
1.8%
90
Other
8.2%
400
Undecided
6.6%
340
Table11.23
Exercise11.13.9
(Solutiononp.519.)
ArecentdebateaboutwhereintheUnitedStatesskiersbelievetheskiingisbestpromptedthe
followingsurvey.Testtoseeifthebestskiareaisindependentoftheleveloftheskier.
U.S.SkiArea
Beginner
Intermediate
Advanced
Tahoe
20
30
40
Utah
10
30
60
Colorado
10
40
50
Table11.24
Exercise11.13.10
Carmanufacturersareinterestedinwhetherthereisarelationshipbetweenthesizeofcaran
individualdrivesandthenumberofpeopleinthedriver’sfamily(thatis,whethercarsizeand
familysizeareindependent). Totestthis,supposethat800carownerswererandomlysurveyed
withthefollowingresults.Conductatestforindependence.
FamilySize
Sub&Compact
Mid-size
Full-size
Van&Truck
1
20
35
40
35
2
20
50
70
80
3-4
20
50
100
90
5+
20
30
70
70
Table11.25
Exercise11.13.11
(Solutiononp.519.)
Collegestudentsmaybeinterestedinwhetherornot theirmajorshaveanyeffectonstarting
salariesafter graduation. . Suppose e that 300recentgraduateswere surveyedastotheirmajors
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CHAPTER11. THECHI-SQUAREDISTRIBUTION
incollegeandtheir startingsalariesaftergraduation. . Beloware e thedata. . Conduct t atest for
independence.
Major
<$50,000
$50,000-$68,999
$69,000+
English
5
20
5
Engineering
10
30
60
Nursing
10
15
15
Business
10
20
30
Psychology
20
30
20
Table11.26
Exercise11.13.12
Sometravelagentsclaimthathoneymoonhotspotsvaryaccordingtoageofthebrideandgroom.
Supposethat280EastCoastrecentbrideswereinterviewedastowheretheyspenttheirhoney-
moons.Theinformationisgivenbelow.Conductatestforindependence.
Location
20-29
30-39
40-49
50andover
NiagaraFalls
15
25
25
20
Poconos
15
25
25
10
Europe
10
25
15
5
VirginIslands
20
25
15
5
Table11.27
Exercise11.13.13
(Solutiononp.519.)
Amanagerofasportsclub keepsinformationconcerningthe mainsport inwhichmembers
participateandtheirages. Totestwhetherthereisarelationshipbetweentheageofamember
andhisorherchoiceofsport,643membersofthesportsclubarerandomlyselected. Conducta
testforindependence.
Sport
18-25
26-30
31-40
41andover
racquetball
42
58
30
46
tennis
58
76
38
65
swimming
72
60
65
33
Table11.28
Exercise11.13.14
AmajorfoodmanufacturerisconcernedthatthesalesforitsskinnyFrenchfrieshavebeende-
creasing.Asapartofafeasibilitystudy,thecompanyconductsresearchintothetypesoffriessold
acrossthecountrytodetermineifthetypeoffriessoldisindependentoftheareaofthecountry.
Theresultsofthestudyarebelow.Conductatestforindependence.
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503
TypeofFries
Northeast
South
Central
West
skinnyfries
70
50
20
25
curlyfries
100
60
15
30
steakfries
20
40
10
10
Table11.29
Exercise11.13.15
(Solutiononp.519.)
AccordingtoDanLenard,anindependentinsuranceagentintheBuffalo,N.Y.area,thefollowing
isabreakdownoftheamountoflifeinsurancepurchasedbymalesinthefollowingagegroups.
Heisinterestedinwhethertheageofthemaleandtheamountoflifeinsurancepurchasedare
independentevents.Conductatestforindependence.
AgeofMales
None
<$200,000
$200,000-$400,000
$401,001-$1,000,000
$1,000,000+
20-29
40
15
40
0
5
30-39
35
5
20
20
10
40-49
20
0
30
0
30
50+
40
30
15
15
10
Table11.30
Exercise11.13.16
Supposethat600thirty–year–oldsweresurveyedtodeterminewhetherornotthereisarelation-
shipbetweenthelevelofeducationanindividualhasandsalary.Conductatestforindependence.
AnnualSalary
Notahighschool
graduate
Highschoolgrad-
uate
Collegegraduate
Mastersordoctor-
ate
<$30,000
15
25
10
5
$30,000-$40,000
20
40
70
30
$40,000-$50,000
10
20
40
55
$50,000-$60,000
5
10
20
60
$60,000+
0
5
10
150
Table11.31
Exercise11.13.17
(Solutiononp.519.)
APsychologistisinterestedintestingwhetherthereisadifferenceinthedistributionofpersonal-
itytypesforbusinessmajorsandsocialsciencemajors.Theresultsofthestudyareshownbelow.
ConductaTestofHomogeneity.Testata5%levelofsignificance.
Open
Conscientious
Extrovert
Agreeable Neurotic
Business
41
52
46
61 58
SocialScience
72
75
63
80 65
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504
CHAPTER11. THECHI-SQUAREDISTRIBUTION
Table11.32
Exercise11.13.18
(Solutiononp.519.)
Domenandwomenselectdifferentbreakfasts?Thebreakfastorderedbyrandomlyselectedmen
andwomenatapopularbreakfastplaceisshownbelow.Conductatestofhomogeneity.Testata
5%levelofsignificance
FrenchToast
Pancakes
Waffles
Omelettes
Men
47
35
28
53
Women
65
59
55
60
Table11.33
Exercise11.13.19
(Solutiononp.520.)
Isthereadifferencebetweenthedistributionofcommunitycollegestatisticsstudentsandthe
distributionofuniversitystatisticsstudentsinwhattechnologytheyuseontheirhomework? Of
therandomlyselected communitycollegestudents43usedacomputer, 102used acalculator
withbuiltinstatisticsfunctions,and65usedatablefromthetextbook.Oftherandomlyselected
universitystudents28usedacomputer,33usedacalculatorwithbuiltinstatisticsfunctions,and
40usedatablefromthetextbook. Conductanappropriatehypothesistestusinga0.05levelof
significance.
Exercise11.13.20
(Solutiononp.520.)
AfishermanisinterestedinwhetherthedistributionoffishcaughtinGreenValleyLakeisthe
sameasthedistributionoffishcaughtinEchoLake. Ofthe191randomlyselectedfishcaughtin
GreenValleyLake,105wererainbowtrout,27wereothertrout,35werebass,and24werecatfish.
Ofthe293randomlyselectedfishcaughtinEchoLake, 115wererainbowtrout,58were other
trout,67werebass,and53werecatfish.Performthehypothesistestata5%levelofsignificance.
Exercise11.13.21
(Solutiononp.520.)
Aplantmanagerisconcernedherequipmentmayneedrecalibrating. Itseemsthattheactual
weightofthe15oz. cerealboxesitfillshasbeenfluctuating. . Thestandarddeviationshouldbe
atmost
1
2
oz. Inordertodetermineifthemachineneedstoberecalibrated,84randomlyselected
boxesofcerealfromthenextday’sproductionwereweighed. Thestandarddeviationofthe84
boxeswas0.54.Doesthemachineneedtoberecalibrated?
Exercise11.13.22
Consumersmaybeinterestedinwhetherthecostofaparticularcalculatorvariesfromstoreto
store. Basedonsurveying43stores,whichyieldedasamplemeanof$84andasamplestandard
deviationof$12,testtheclaimthatthestandarddeviationisgreaterthan$15.
Exercise11.13.23
(Solutiononp.520.)
Isabella,anaccomplishedBaytoBreakersrunner,claimsthatthestandarddeviationforhertime
torunthe7½mileraceisatmost3minutes. Totestherclaim,Rupinderlooksup5ofherrace
times.Theyare55minutes,61minutes,58minutes,63minutes,and57minutes.
Exercise11.13.24
Airlinecompaniesareinterestedintheconsistencyofthenumberofbabiesoneachflight,sothat
theyhaveadequatesafetyequipment. Theyarealsointerestedinthevariationofthenumberof
babies.Supposethatanairlineexecutivebelievestheaveragenumberofbabiesonflightsis6with
avarianceof9atmost. Theairlineconductsasurvey. Theresultsofthe18flightssurveyedgive
asampleaverageof6.4withasamplestandarddeviationof3.9.Conductahypothesistestofthe
airlineexecutive’sbelief.
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505
Exercise11.13.25
(Solutiononp.520.)
ThenumberofbirthsperwomaninChinais1.6downfrom5.91in1966(SourceWorldBank,
6/5/12). Thisfertilityratehasbeenattributedtothelawpassedin1979restrictingbirthstoone
perwoman. Supposethatagroupofstudentsstudiedwhetherornotthestandarddeviationof
birthsperwomanwasgreaterthan0.75.Theyasked50womenacrossChinathenumberofbirths
theyhad. Belowaretheresults. Doesthestudents’surveyindicatethatthestandarddeviationis
greaterthan0.75?
#ofbirths
Frequency
0
5
1
30
2
10
3
5
Table11.34
Exercise11.13.26
Accordingtoanavidaquariest, theaveragenumberoffishina20–gallontankis10, witha
standarddeviationof2.Hisfriend,alsoanaquariest,doesnotbelievethatthestandarddeviation
is2.Shecountsthenumberoffishin15other20–gallontanks.Basedontheresultsthatfollow,do
youthinkthatthestandarddeviationisdifferentfrom2?Data:11;10;9;10;10;11;11;10;12;9;7;
9;11;10;11
Exercise11.13.27
(Solutiononp.520.)
Themanagerof"Frenchies"isconcernedthatpatronsare notconsistentlyreceivingthesame
amountofFrenchfrieswitheachorder. Thechefclaimsthatthestandard d deviationfora10–
ounceorderoffriesisatmost1.5oz.,butthemanagerthinksthatitmaybehigher.Herandomly
weighs49ordersoffries,whichyieldsameanof11oz.andastandarddeviationof2oz.
11.13.2Trythesetrue/falsequestions.
Exercise11.13.28
(Solutiononp.521.)
Asthedegreesoffreedomincrease,thegraphofthechi-squaredistributionlooksmoreandmore
symmetrical.
Exercise11.13.29
(Solutiononp.521.)
Thestandarddeviationofthechi-squaredistributionistwicethemean.
Exercise11.13.30
(Solutiononp.521.)
Themeanandthemedianofthechi-squaredistributionarethesameifdf=24.
Exercise11.13.31
(Solutiononp.521.)
InaGoodness-of-Fittest,theexpectedvaluesarethevalueswewouldexpectifthenullhypoth-
esisweretrue.
Exercise11.13.32
(Solutiononp.521.)
Ingeneral, iftheobservedvaluesandexpectedvaluesofaGoodness-of-Fittestarenotclose
together,thentheteststatisticcangetverylargeandonagraphwillbewayoutintherighttail.
Exercise11.13.33
(Solutiononp.521.)
ThedegreesoffreedomforaTestforIndependenceareequaltothesamplesizeminus1.
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506
CHAPTER11. THECHI-SQUAREDISTRIBUTION
Exercise11.13.34
(Solutiononp.521.)
UseaGoodness-of-Fittesttodetermineifhighschoolprincipalsbelievethatstudentsareabsent
equallyduringtheweekornot.
Exercise11.13.35
(Solutiononp.521.)
TheTestforIndependenceusestablesofobservedandexpecteddatavalues.
Exercise11.13.36
(Solutiononp.521.)
Thetesttousewhendeterminingifthecollegeoruniversityastudentchoosestoattendisrelated
tohis/hersocioeconomicstatusisaTestforIndependence.
Exercise11.13.37
(Solutiononp.521.)
Thetesttousetodetermineifasix-sideddieisfairisaGoodness-of-Fittest.
Exercise11.13.38
(Solutiononp.521.)
InaTestofIndependence,theexpectednumberisequaltotherowtotalmultipliedbythecolumn
totaldividedbythetotalsurveyed.
Exercise11.13.39
(Solutiononp.521.)
InaGoodness-ofFittest,ifthep-valueis0.0113,ingeneral,donotrejectthenullhypothesis.
Exercise11.13.40
(Solutiononp.521.)
ForaChi-Squaredistributionwithdegreesoffreedomof17,theprobabilitythatavalueisgreater
than20is0.7258.
Exercise11.13.41
(Solutiononp.521.)
Ifdf=2,thechi-squaredistributionhasashapethatremindsusoftheexponential.
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507
11.14Review
14
Thenexttwoquestionsrefertothefollowingrealstudy:
ArecentsurveyofU.S.teenagepregnancywasansweredby720girls,age12-19.6%ofthegirlssurveyed
saidtheyhavebeenpregnant.(ParadeMagazine)WeareinterestedinthetrueproportionofU.S.girls,age
12-19,whohavebeenpregnant.
Exercise11.14.1
(Solutiononp.521.)
Findthe95%confidenceintervalforthetrueproportionofU.S.girls,age12-19,whohavebeen
pregnant.
Exercise11.14.2
(Solutiononp.521.)
The report alsostatedthatthe resultsofthesurveyareaccurateto within3.7%at the95%
confidencelevel.Supposethatanewstudyistobedone.Itisdesiredtobeaccuratetowithin2%
ofthe95%confidencelevel.Whatistheminimumnumberthatshouldbesurveyed?
Exercise11.14.3
Given:XExp
1
3
.Sketchthegraphthatdepicts:P(x>1).
Thenextfourquestionsrefertothefollowinginformation:
Supposethatthetimethatownerskeeptheircars(purchasednew)isnormallydistributedwithamean
of7yearsandastandarddeviationof2years. Weareinterestedinhowlonganindividualkeepshiscar
(purchasednew).Ourpopulationispeoplewhobuytheircarsnew.
Exercise11.14.4
(Solutiononp.521.)
60%ofindividualskeeptheircarsatmosthowmanyyears?
Exercise11.14.5
(Solutiononp.521.)
Supposethatwerandomlysurveyoneperson.Findtheprobabilitythatpersonkeepshis/hercar
lessthan2.5years.
Exercise11.14.6
(Solutiononp.521.)
Ifwearetopickindividuals10atatime,findthedistributionforthemeancarlengthownership.
Exercise11.14.7
(Solutiononp.521.)
Ifwearetopick10individuals,findtheprobabilitythatthesumoftheirownershiptimeismore
than55years.
Exercise11.14.8
(Solutiononp.521.)
Forwhichdistributionisthemediannotequaltothemean?
A. Uniform
B. Exponential
C. Normal
D. Student-t
Exercise11.14.9
(Solutiononp.521.)
Comparethestandardnormaldistributiontothestudent-tdistribution,centeredat0. Explain
whichofthefollowingaretrueandwhicharefalse.
a. Asthenumbersurveyedincreases,theareatotheleftof-1forthestudent-tdistributionap-
proachestheareaforthestandardnormaldistribution.
b. Asthedegreesoffreedomdecrease,thegraphofthestudent-tdistributionlooksmorelikethe
graphofthestandardnormaldistribution.
c. Ifthenumbersurveyedis15,thenormaldistributionshouldneverbeused.
14
Thiscontentisavailableonlineat<http://cnx.org/content/m17057/1.11/>.
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508
CHAPTER11. THECHI-SQUAREDISTRIBUTION
Thenextfivequestionsrefertothefollowinginformation:
We are interestedinthe checkingaccount balanceofatwenty-year-oldcollegestudent. . We e randomly
survey16twenty-year-old college students. . We e obtainasamplemeanof$640and asample standard
deviationof$150.LetX=checkingaccountbalanceofanindividualtwentyyearoldcollegestudent.
Exercise11.14.10
ExplainwhywecannotdeterminethedistributionofX.
Exercise11.14.11
(Solutiononp.522.)
Ifyouweretocreateaconfidenceintervalorperformahypothesistestforthepopulationmean
checkingaccountbalanceof20-yearoldcollegestudents,whatdistributionwouldyouuse?
Exercise11.14.12
(Solutiononp.522.)
Findthe95%confidenceintervalforthetruemeancheckingaccountbalanceofatwenty-year-old
collegestudent.
Exercise11.14.13
(Solutiononp.522.)
Whattypeofdataisthebalanceofthecheckingaccountconsideredtobe?
Exercise11.14.14
(Solutiononp.522.)
Whattypeofdataisthenumberof20yearoldsconsideredtobe?
Exercise11.14.15
(Solutiononp.522.)
Onaverage,abusyemergencyroomgetsapatientwithashotgunwoundaboutonceperweek.
Weareinterestedinthenumberofpatientswithashotgunwoundtheemergencyroomgetsper
28days.
a. DefinetherandomvariableX.
b. StatethedistributionforX.
c. Findtheprobabilitythattheemergencyroomgetsnopatientswithshotgunwoundsinthenext
28days.
Thenexttwoquestionsrefertothefollowinginformation:
Theprobabilitythatacertainslotmachinewillpaybackmoneywhenaquarterisinsertedis0.30.Assume
thateachplayoftheslotmachineisindependentfromeachother.Apersonputsin15quartersfor15plays.
Exercise11.14.16
(Solutiononp.522.)
Istheexpectednumberofplaysoftheslotmachinethatwillpaybackmoneygreaterthan,less
thanorthesameasthemedian?Explainyouranswer.
Exercise11.14.17
(Solutiononp.522.)
Isitlikelythatexactly8ofthe15playswouldpaybackmoney?Justifyyouranswernumerically.
Exercise11.14.18
(Solutiononp.522.)
Agameisplayedwiththefollowingrules:
 itcosts$10toenter
 afaircoinistossed4times
 ifyoudonotget4headsor4tails,youloseyour$10
 ifyouget4headsor4tails,yougetbackyour$10,plus$30more
Overthelongrunofplayingthisgame,whatareyourexpectedearnings?
Exercise11.14.19
(Solutiononp.522.)
 ThemeangradeonamathexaminRachel’sclasswas74,withastandarddeviationof5.
Rachelearnedan80.
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509
 ThemeangradeonamathexaminBecca’sclasswas47, withastandarddeviationof2.
Beccaearneda51.
 ThemeangradeonamathexaminMatt’sclasswas70,withastandarddeviationof8.Matt
earnedan83.
Findwhosescorewasthebest,comparedtohisorherownclass.Justifyyouranswernumerically.
Thenexttwoquestionsrefertothefollowinginformation:
Arandomsampleof70compulsivegamblerswereaskedthenumberofdaystheygotocasinosperweek.
Theresultsaregiveninthefollowinggraph:
Figure11.3
Exercise11.14.20
(Solutiononp.522.)
Findthenumberofresponsesthatwere“5".
Exercise11.14.21
(Solutiononp.522.)
Findthemean,standarddeviation,themedian,thefirstquartile,thethirdquartileandtheIQR.
Exercise11.14.22
(Solutiononp.522.)
BaseduponresearchatDeAnzaCollege,itisbelievedthatabout19%ofthestudentpopulation
speaksalanguageotherthanEnglishathome.
Supposethatastudywasdonethisyeartoseeifthatpercenthasdecreased.Ninety-eightstudents
wererandomlysurveyedwiththefollowingresults. Fourteensaidthattheyspeakalanguage
otherthanEnglishathome.
a. Stateanappropriatenullhypothesis.
b. Stateanappropriatealternatehypothesis.
c. DefinetheRandomVariable,P’.
d. Calculatetheteststatistic.
e. Calculatethep-value.
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510
CHAPTER11. THECHI-SQUAREDISTRIBUTION
f. Atthe5%levelofdecision,whatisyourdecisionaboutthenullhypothesis?
g. WhatistheTypeIerror?
h. WhatistheTypeIIerror?
Exercise11.14.23
(Solutiononp.522.)
Assumethatyouareanemergencyparamediccalledintorescuevictimsofanaccident. You
needtohelpapatientwhoisbleedingprofusely. Thepatientisalsoconsideredtobeahighrisk
forcontractingAIDS.AssumethatthenullhypothesisisthatthepatientdoesnothavetheHIV
virus.WhatisaTypeIerror?
Exercise11.14.24
(Solutiononp.522.)
It isoftensaidthat Californiansaremore casualthantherest ofAmericans. . Suppose e that a
surveywasdonetoseeiftheproportionofCalifornianprofessionalsthatwearjeanstoworkis
greaterthantheproportionofnon-Californianprofessionals.Fiftyofeachwassurveyedwiththe
followingresults.15Californianswearjeanstoworkand6non-Californianswearjeanstowork.
 C=Californianprofessional
 NC=non-Californianprofessional
a. Stateappropriatenullandalternatehypotheses.
b. DefinetheRandomVariable.
c. Calculatetheteststatisticandp-value.
d. Atthe5%significancelevel,whatisyourdecision?
e. WhatistheTypeIerror?
f. WhatistheTypeIIerror?
Thenexttwoquestionsrefertothefollowinginformation:
AgroupofStatisticsstudentshavedevelopedatechniquethattheyfeelwilllowertheiranxietylevelon
statisticsexams. Theymeasuredtheiranxietylevelatthestartofthequarterandagainattheendofthe
quarter. Recordedisthepaireddatainthatorder: : (1000,900);(1200,1050);(600,700);(1300,1100);(1000,
900);(900,900).
Exercise11.14.25
(Solutiononp.522.)
Thisisatestof(pickthebestanswer):
A. largesamples,independentmeans
B. smallsamples,independentmeans
C. dependentmeans
Exercise11.14.26
(Solutiononp.522.)
Statethedistributiontouseforthetest.
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