111
$4,000withastandarddeviationof$2,500. Themeancostforaguitaris$500withastandard
deviationof$200.Themeancostfordrumsis$700withastandarddeviationof$100.Whichcost
isthelowest,whencomparedtootherinstrumentsofthesametype? Whichcostisthehighest
whencomparedtootherinstrumentsofthesametype.Justifyyouranswernumerically.
Exercise2.13.33
(Solutiononp.120.)
Supposethatapublisherconductedasurveyaskingadultconsumersthenumberoffictionpa-
perbackbookstheyhadpurchasedinthepreviousmonth.Theresultsaresummarizedinthetable
below.(NotethatthisisthedatapresentedforpublisherBinhomeworkexercise13).
PublisherB
#ofbooks
Freq.
Rel.Freq.
0
18
1
24
2
24
3
22
4
15
5
10
7
5
9
1
Table2.27
a. Arethereanyoutliersinthedata? ? UseanappropriatenumericaltestinvolvingtheIQRto
identifyoutliers,ifany,andclearlystateyourconclusion.
b. Ifadatavalueisidentifiedasanoutlier,whatshouldbedoneaboutit?
c. Areanydatavaluesfurtherthan2standarddeviationsawayfromthemean?Insomesitu-
ations,statisticiansmayusethiscriteriatoidentifydatavaluesthatareunusual,compared
totheotherdatavalues. (Notethatthiscriteriaismostappropriatetousefordatathatis
mound-shapedandsymmetric,ratherthanforskeweddata.)
d. Doparts(a)and(c)ofthisproblemgivethesameanswer?
e. Examinetheshapeofthedata.Whichpart,(a)or(c),ofthisquestiongivesamoreappropri-
ateresultforthisdata?
f. Basedontheshapeofthedatawhichisthemostappropriatemeasureofcenterforthisdata:
mean,medianormode?
**Exercises32and33contributedbyRobertaBloom
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112
CHAPTER2. DESCRIPTIVESTATISTICS
2.14Lab:DescriptiveStatistics
17
ClassTime:
Names:
2.14.1StudentLearningOutcomes
 Thestudentwillconstructahistogramandaboxplot.
 Thestudentwillcalculateunivariatestatistics.
 Thestudentwillexaminethegraphstointerpretwhatthedataimplies.
2.14.2CollecttheData
Recordthenumberofpairsofshoesyouown:
1. Randomlysurvey30classmates.Recordtheirvalues.
SurveyResults
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
Table2.28
2. Constructahistogram.Make5-6intervals.Sketchthegraphusingarulerandpencil.Scaletheaxes.
17
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113
Figure2.4
3. Calculatethefollowing:
x=
 s=
4. Arethedatadiscreteorcontinuous?Howdoyouknow?
5. Describetheshapeofthehistogram.Usecompletesentences.
6. Arethereanypotentialoutliers? ? Whichvalue(s)is(are)it(they)? ? Useaformulatochecktheend
valuestodetermineiftheyarepotentialoutliers.
2.14.3AnalyzetheData
1. Determinethefollowing:
 Minimumvalue=
 Median=
 Maximumvalue=
 Firstquartile=
 Thirdquartile=
 IQR=
2. Constructaboxplotofdata
3. Whatdoestheshapeoftheboxplotimplyabouttheconcentrationofdata?Usecompletesentences.
4. Usingtheboxplot,howcanyoudetermineiftherearepotentialoutliers?
5. Howdoesthestandarddeviationhelpyoutodetermineconcentrationofthedataandwhetherornot
therearepotentialoutliers?
6. WhatdoestheIQRrepresentinthisproblem?
7. Showyourworktofindthevaluethatis1.5standarddeviations:
a. Abovethemean:
b. Belowthemean:
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114
CHAPTER2. DESCRIPTIVESTATISTICS
SolutionstoExercisesinChapter2
SolutiontoExample2.2,Problem(p.61)
Thevalue12.3maybeanoutlier.Valuesappeartoconcentrateat3and4kilometers.
Stem
Leaf
1
15
2
357
3
23358
4
025578
5
56
6
57
7
8
9
10
11
12
3
Table2.29
SolutiontoExample2.7,Problem(p.66)
 3.5to4.5
 4.5to5.5
 6
 5.5to6.5
SolutiontoExample2.9,Problem(p.70)
FirstDataSet
 Xmin 32
 Q56
 74.5
 Q82.5
 Xmax 99
SecondDataSet
 Xmin 25.5
 Q78
 81
 Q89
 Xmax 98
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115
SolutiontoExample2.11,Problem(p.72)
FortheIQRs,seetheanswertothetestscoresexample(SolutiontoExample2.9:p.114).Thefirstdataset
hasthelargerIQR,sothescoresbetweenQ3andQ1(middle50%)forthefirstdatasetaremorespreadout
andnotclusteredaboutthemedian.
FirstDataSet
3
2
 (IQR) =
3
2
 (26.5) = 39.75
 Xmax  Q99  82.5 16.5
 Q Xmin 56  32 24
3
2
 (IQR) = 39.75islargerthan16.5andlargerthan24,sothefirstsethasnooutliers.
SecondDataSet
3
2
 (IQR) =
3
2
 (11) = 16.5
 Xmax  Q98  89 9
 Q Xmin 78  25.5 52.5
3
2
 (IQR) = 16.5islargerthan9butsmallerthan52.5,soforthesecondset45and25.5areoutliers.
Tofindthepercentiles,createafrequency,relativefrequency,andcumulativerelativefrequencychart(see
"Frequency"fromtheSamplingandDataChapter(Section1.9)).Getthepercentilesfromthatchart.
FirstDataSet
 30th%ile(betweenthe6thand7thvalues) =
(56+59)
2
57.5
 80th%ile(betweenthe16thand17thvalues) =
(84+84.5)
2
84.25
SecondDataSet
 30th%ile(7thvalue) 78
 80th%ile(18thvalue) 90
30%ofthedatafallsbelowthe30th%ile,and20%fallsabovethe80th%ile.
SolutiontoExample2.13,Problem(p.73)
1.
(8+9)
2
8.5
Lookwherecum.rel.freq.=0.80.80%ofthedatais8orless.80th%ileisbetweenthelast8andfirst
9.
2. 9
3. 6
4. FirstQuartile=25th%ile
SolutiontoExercise2.6.1(p.75)
a. Forrunnersinaraceitismoredesirabletohavealowpercentileforfinishtime.Alowpercentilemeans
ashorttime,whichisfaster.
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116
CHAPTER2. DESCRIPTIVESTATISTICS
b. INTERPRETATION:20%ofrunnersfinishedtheracein5.2minutesorless.80%ofrunnersfinishedthe
racein5.2minutesorlonger.
c. Heisamongthe slowest cyclists(90%ofcyclistswere fasterthanhim.) ) INTERPRETATION:90%of
cyclistshadafinishtimeof1hour,12minutesorless.Only10%ofcyclistshadafinishtimeof1hour,
12minutesorlonger
SolutiontoExercise2.6.2(p.75)
a. Forrunnersinaraceitismoredesirabletohaveahighpercentileforspeed.Ahighpercentilemeansa
higherspeed,whichisfaster.
b. INTERPRETATION:40%ofrunnersranatspeedsof7.5milesperhourorless(slower).60%ofrunners
ranatspeedsof7.5milesperhourormore(faster).
SolutiontoExercise2.6.3(p.75)
Onanexamyouwouldpreferahighpercentile;higherpercentilescorrespondtohighergradesonthe
exam.
SolutiontoExercise2.6.4(p.75)
WhenwaitinginlineattheDMV,the85thpercentilewouldbealongwaittimecomparedtotheother
peoplewaiting. 85%ofpeoplehadshorterwaittimesthanyoudid. Inthiscontext,youwouldprefera
waittimecorrespondingtoalowerpercentile. INTERPRETATION:85%ofpeopleattheDMVwaited32
minutesorless.15%ofpeopleattheDMVwaited32minutesorlonger.
SolutiontoExercise2.6.5(p.75)
Lishouldbepleased. Hersalaryisrelativelyhighcomparedtootherrecentcollegegrads. 78%ofrecent
collegegraduatesearnlessthanLidoes.22%ofrecentcollegegraduatesearnmorethanLidoes.
SolutiontoExercise2.6.6(p.75)
Themanufacturerandtheconsumerwouldbeupset.Thisisalargerepaircostforthedamages,compared
totheothercarsinthesample. INTERPRETATION:90%ofthecrashtestedcarshaddamagerepaircosts
of$1700orless;only10%haddamagerepaircostsof$1700ormore.
SolutiontoExercise2.6.7(p.75)
a. Thetop12%ofstudentsarethosewhoareatorabovethe 88thpercentileofadmissionsindexscores.
b. The top4%ofstudents’GPAsareatorabovethe96thpercentile,makingthetop4%ofstudents"eligible
inthelocalcontext".
SolutiontoExercise2.6.8(p.75)
Youcanafford34%ofhouses.66%ofthehousesaretooexpensiveforyourbudget.INTERPRETATION:
34%ofhousescost$240,000orless.66%ofhousescost$240,000ormore.
SolutionstoPractice1:CenteroftheData
SolutiontoExercise2.11.1(p.90)
65
SolutiontoExercise2.11.2(p.90)
1
SolutiontoExercise2.11.5(p.91)
4.75
SolutiontoExercise2.11.6(p.91)
1.39
SolutiontoExercise2.11.7(p.91)
65
SolutiontoExercise2.11.8(p.91)
4
SolutiontoExercise2.11.9(p.91)
4
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117
SolutiontoExercise2.11.10(p.91)
4
SolutiontoExercise2.11.11(p.91)
4
SolutiontoExercise2.11.12(p.91)
6
SolutiontoExercise2.11.13(p.91)
6 4 2
SolutiontoExercise2.11.14(p.91)
3
SolutiontoExercise2.11.15(p.91)
6
SolutiontoExercise2.11.16(p.92)
a. 8.93
b. 0.58
SolutionstoPractice2:SpreadoftheData
SolutiontoExercise2.12.1(p.93)
6
SolutiontoExercise2.12.2(p.93)
a. 1447.5
b. 528.5
SolutiontoExercise2.12.3(p.93)
474FTES
SolutiontoExercise2.12.4(p.93)
50%
SolutiontoExercise2.12.5(p.93)
919
SolutiontoExercise2.12.6(p.93)
0.03
SolutiontoExercise2.12.7(p.94)
mean=1809.3
median=1812.5
standarddeviation=151.2
Firstquartile=1690
Thirdquartile=1935
IQR=245
SolutiontoExercise2.12.9(p.94)
Hint:Thinkaboutthenumberofyearscoveredbyeachtimeperiodandwhathappenedtohighereduca-
tionduringthoseperiods.
SolutionstoHomework
SolutiontoExercise2.13.1(p.95)
a. 1.48
b. 1.12
e. 1
f. 1
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118
CHAPTER2. DESCRIPTIVESTATISTICS
g. 2
h.
i. 80%
j. 1
k. 3
SolutiontoExercise2.13.3(p.95)
a. 3.78
b. 1.29
e. 3
f. 4
g. 5
h.
i. 32.5%
j. 4
k. 5
SolutiontoExercise2.13.5(p.97)
b. 241
c. 205.5
d. 272.5
e.
f. 205.5,272.5
g. sample
h. population
i. i. 236.34
ii. 37.50
iii. 161.34
iv. 0.84std.dev.belowthemean
j. Young
SolutiontoExercise2.13.9(p.98)
Kamala
SolutiontoExercise2.13.15(p.103)
a. True
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119
b. True
c. True
d. False
SolutiontoExercise2.13.17(p.104)
b. 4,3,5
c. 4
d. 3
e.
f. 3,5
g. 3.94
h. 1.28
i. 3
j. mode
SolutiontoExercise2.13.19(p.105)
c. Maybe
SolutiontoExercise2.13.21(p.106)
a. morechildren
b. 62.4%
SolutiontoExercise2.13.23(p.107)
b. 51,99
SolutiontoExercise2.13.24(p.107)
A
SolutiontoExercise2.13.25(p.108)
A
SolutiontoExercise2.13.26(p.108)
B
SolutiontoExercise2.13.27(p.108)
D
SolutiontoExercise2.13.28(p.108)
A
SolutiontoExercise2.13.29(p.109)
C
SolutiontoExercise2.13.30(p.109)
D
SolutiontoExercise2.13.31(p.109)
ExamplesolutionforbusingtherandomnumbergeneratorfortheTi-84Plustogenerateasimplerandom
sampleof8states.Instructionsarebelow.
Numbertheentriesinthetable1-51(IncludesWashington,DC;Numberedvertically)
PressMATH
ArrowovertoPRB
Press5:randInt(
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120
CHAPTER2. DESCRIPTIVESTATISTICS
Enter51,1,8)
Eight numbers are generated (use the right arrow w key y to scroll l through the e numbers).
The
numbers correspond d to o the e numbered d states s (for r this example: : {47 7 21 1 9 9 23 3 51 13 3 25 5 4}.
If
any numbers are repeated, , generate e a a different t number by using 5:randInt(51,1)).
Here, the
states (and Washington DC) ) are e {Arkansas, , Washington n DC, Idaho, , Maryland, Michigan, , Missis-
sippi, Virginia, , Wyoming}.
Corresponding percents s are {28.7 21.8 24.5 26 6 28.9 9 32.8 25 24.6}.
SolutiontoExercise2.13.32(p.110)
Forpianos,thecostofthepianois0.4standarddeviationsBELOWthemean. Forguitars,thecostofthe
guitaris0.25standarddeviationsABOVEthemean. Fordrums,thecostofthedrumsetis1.0standard
deviationsBELOWthemean. Ofthethree,thedrumscostthelowestincomparisontothecostofother
instrumentsofthesametype.Theguitarcostthemostincomparisontothecostofotherinstrumentsofthe
sametype.
SolutiontoExercise2.13.33(p.111)
 IQR=4–1=3;Q1–1.5*IQR=1–1.5(3)=-3.5;Q3+1.5*IQR=4+1.5(3)=8.5;Thedatavalueof9is
largerthan8.5.Thepurchaseof9booksinonemonthisanoutlier.
 Theoutliershouldbeinvestigatedtoseeifthereisanerrororsomeotherprobleminthedata;thena
decisionwhethertoincludeorexcludeitshouldbemadebasedontheparticularsituation.Ifitwas
acorrectvaluethenthedatavalueshouldremaininthedataset.Ifthereisaproblemwiththisdata
value,thenitshouldbecorrectedorremovedfromthedata. Forexample: Ifthedatawasrecorded
incorrectly(perhapsa9wasmiscodedandthecorrectvaluewas6)thenthedatashouldbecorrected.
Ifitwasanerrorbutthecorrectvalueisnotknownitshouldberemovedfromthedataset.
 xbar–2s=2.45–2*1.88=-1.31;xbar+2s=2.45+2*1.88=6.21;Usingthismethod,thefivedatavalues
of7bookspurchasedandtheonedatavalueof9bookspurchasedwouldbeconsideredunusual.
 No:part(a)identifiesonlythevalueof9tobeanoutlierbutpart(c)identifiesboth7and9.
 Thedataisskewed(totheright).ItwouldbemoreappropriatetousethemethodinvolvingtheIQR
inpart(a),identifyingonlytheonevalueof9bookspurchasedasanoutlier.Notethatpart(c)remarks
thatidentifyingunusualdatavaluesbyusingthecriteriaofbeingfurtherthan2standarddeviations
awayfromthemeanismostappropriatewhenthedataaremound-shapedandsymmetric.
 Thedataareskewedtotheright. Forskeweddataitismoreappropriatetouse e themedianasa
measureofcenter.
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