NAME DATE SECTION

Day1Two-VariableStatistics

Part1:Two-wayTables

UtensilsandPaperPreferences

Severalstudentsaresurveyedaboutwhethertheypreferwritingwithapenorapencilandtheyarealsoaskedwhethertheypreferlinedpaperorunlinedpaper.Someoftheresultsare:

• Thesurveyincluded100differentstudents.• 40studentssaidtheypreferusingpenmorethanpencil.• 45studentssaidtheypreferusingunlinedpapermorethanlinedpaper.• 10studentssaidtheypreferlinedpaperandpen.• 45studentssaidtheypreferpencilandlinedpaper.

Foreachpart,explainorshowyourreasoninginthespacebelow.

1. Howmanystudentspreferusingpencilmorethanpen?

2. Howmanystudentspreferusingpenandunlinedpaper?3. Howmanystudentspreferusingpencilandunlinedpaper?

FruitFlyMutations

Ascientististryingtodeterminetheroleofspecificgenesbylookingattraitsoffruitflies.Theoffspringoftwofruitfliesareexaminedtodeterminethecoloroftheireyesandwhethertheyhavecurledwingsorstandardwings.Eightyoffspringarerandomlyselected,andtheresultsarerecordedinthetwo-waytable.

curledwings standardwingsredeyes 17 45whiteeyes 5 13

1. Describewhatthe17inthetablerepresents.2. Howmanyselectedflyoffspringhadwhiteeyes?Explainorshowyourreasoning.3. Howmanyselectedflyoffspringhadstandardwings?Explainorshowyourreasoning.

Part2:RelativeFrequencyTables

NoticeandWonder:TeacherDegrees

Severaladultsinaschoolbuildingwereaskedabouttheirhighestdegreecompletedandwhethertheywereateacher.

Whatdoyounotice?Whatdoyouwonder?

teacher notateacherassociatedegree 4% 16%bachelor’sdegree 52% 64%master’sdegreeor

higher 44% 20%

ActualValues:

1. Isthepercentageofteacherswithbachelor’sdegreeshigherorlowerthanthatofnon-teacher’swithbachelordegrees?

Whydoyouthinkthisis?

2. Whatisthecategoricalvariableforthecategoriesthatlabeltherows?

3. Whatisthecategoricalvariablefor“teacher”and“notateacher”?

teacher notateacher

associatedegree

bachelor’sdegree

master’sdegreeorhigher

CityCat,CountryCat

200peoplewereaskediftheypreferdogsorcats,andwhethertheyliveinaruralorurbansetting.

Theactualvaluescollectedfromthesurveyareinthefirsttable.

urban rural totalcat 54 42 96dog 80 24 104total 134 66 200

Thenexttableshowswhatpercentageofthe200totalpeopleincludedarerepresentedbyeachcombinationofcategories.Thesegmentedbargraphrepresentsthesameinformationgraphically.

urban ruralcat 27% 21%dog 40% 12%

Thenexttableshowsthepercentageofeachcolumnthathadacertainpetpreferenceinacolumnrelativefrequencytable.Thesegmentedbargraphrepresentsthesameinformationgraphically.

urban ruralcat 40% 64%dog 60% 36%

Thelasttableshowsthepercentageofeachrowthatliveinacertainareainarowrelativefrequencytable.Thesegmentedbargraphrepresentsthesameinformationgraphically.

urban ruralcat 56% 44%dog 77% 23%

1. Foreachrelativefrequencytable,selectapercentageandexplainhownumbersfromtheoriginaltablewereusedtogetthepercentage.Writeyouranswersnexttoeachfrequencytable.

2. Whatpercentageofthosesurveyedliveinanurbanareaandpreferdogs?______________

3. Amongthepeoplesurveyedwhopreferdogs,whatpercentageofthemliveinanurbansetting?_________________

4. Whatpercentageofpeoplesurveyedwholiveinanurbansettingpreferdogs?_________5. Howmanyofthepeoplerespondedthattheypreferdogsandliveinanurban

setting?_____________6. Amongthepeoplesurveyed,aretheremorepeoplewhopreferdogsorcats?___________7. Yourpetfoodcompanyhasaccesstoabillboardinaruralsetting.Wouldyou

recommendadvertisingdogfoodorcatfoodonthisbillboard?8. Whichtabledidyouusetomakethisdecision?Explainyourreasoning.

AnalyzingaStudyWithTwoTreatmentsInanexperimenttotesttheeffectivenessofvitaminConthelengthofcolds,twogroupsofpeoplewithcoldsaregivenapilltotakeonceaday.Thepillforoneofthegroupscontains1,000mgofvitaminC,whiletheothergrouptakesaplacebopill.Theresearchersrecordtheresultsinatable.

groupA groupBcoldlastslessthanaweek 16 27coldlastsaweekormore 17 53

1. First,theresearcherswanttoknowwhatpercentage(tothenearestwholepercent)ofpeopleareineachcombinationofcategories.FourteenpercentofalltheparticipantshadacoldthatlastedlessthanaweekandwereingroupA.WhatpercentageofalltheparticipantshadacoldthatlastedlessthanaweekandwereingroupB?Completetherestoftherelativefrequencytablewiththecorrespondingpercentages.

2. Next,theresearchersnoticethat,amongparticipantswhohadcoldsthatlastedlessthanaweek,37%wereingroupA.Amongparticipantswhohadcoldsthatlastedaweekormore,whatpercentagewereingroupB?Completethetablewiththecorrespondingpercentages.

3. Finally,theresearchersnoticethat,amongtheparticipantsingroupA,48%hadcoldsthatlastedlessthanoneweek.AmongtheparticipantsingroupB,howmanyhadcoldsthatlastedaweekormore?Completethetablewiththecorrespondingpercentages.

4. Tounderstandtheresults,theresearcherswanttoknow:Amongpeoplewhosecoldslastedlessthanaweek,whatpercentageareingroupB?Explainyourreasoning.

5. IftheresearchersbelievethatvitaminChasasmalleffectonthelengthofacold,whichgroupmostlikelygotthepillscontainingvitaminC?Explainyourreasoning.

groupA groupBcoldlastslessthanaweek 14%( !"

!"!!"!!"!!"≈ 0.14)

coldlastsaweekormore

groupA groupBcoldlastslessthanaweek 37%( !"

!"!!"≈ 0.37)

coldlastsaweekormore

groupA groupBcoldlastslessthanaweek 48%( !"

!"!!"≈ 0.48)

coldlastsaweekormore

Day1Summary

Part1

Instatistics,avariableisacharacteristicthatcantakeondifferentvalues.Acategoricalvariableisavariablethattakesonvalueswhichcanbedividedintogroupsorcategories.Datafromtwocategoricalvariablesaboutasinglesubjectcanbeorganizedusingatwo-waytable.

Forexample,thistwo-waytableshowstheresultsfrom170responsestoasurveyaskingpeopletheiragegroupandwhethertheyhaveacellphoneornot.

hasacellphone doesnothaveacellphone10–12yearsold 25 3513–15yearsold 38 1216–18yearsold 52 8

The38inthetablemeansthat38ofthe170peoplesurveyedareinboththe13–15yearsoldagecategoryandhaveacellphone.Thetwo-waytablealsoshowsthat55ofthepeoplesurveyeddonothavecellphones,since35+ 12+ 8 = 55.

Thecategoriesforasinglevariableshouldnotoverlap(apersoncannotbe10–12yearsoldand13–15yearsoldatthesametime);eachindividualisincludedinonlyoneofthecellsinthetableratherthaninmultipleplaces.

Part2

Convertingtwo-waytablestorelativefrequencytablescanhelprevealpatternsinpairedcategoricalvariables.Relativefrequencytablesarecreatedbydividingthevalueineachcellinatwo-waytablebythetotalnumberofresponsesintheentiretable,orthetotalresponsesinaroworacolumn.Dependingonwhatpatternsareimportant,differenttypesofrelativefrequencytablesareused.Toexaminehowindividualcombinationsofthecategoricalvariablesrelatetothewholegroup,divideeachvalueinatwo-waytablebythetotalnumberofresponsesintheentiretabletofindtherelativefrequency.

Forexample,thistwo-waytabledisplaystheconditionofacertaintextbookanditspricefor120ofthebooksatacollegebookstore.

$10orless morethan10𝑏𝑢𝑡𝑙𝑒𝑠𝑠𝑡ℎ𝑎𝑛30 $30ormorenew 3 9 27used 33 36 12

Atwo-wayrelativefrequencytableiscreatedbydividingeachnumberinthetwo-waytableby120,becausethereare120values(3+ 9+ 28+ 32+ 35+ 13)inthisdataset.

Theresultingtwo-wayrelativefrequencytablecanberepresentedusingfractionsordecimals.

$10orless morethan10𝑏𝑢𝑡𝑙𝑒𝑠𝑠𝑡ℎ𝑎𝑛30 $30ormorenew 0.025 0.075 0.225used 0.275 0.300 0.100

Thistwo-wayrelativefrequencytableallowsyoutoseewhatproportionofthetotalisrepresentedbyeachnumberinthetwo-waytable.Thenumber33intheoriginaltwo-waytablerepresentsthenumberofusedbooksthatalsosellfor$10orless,whichis27.5%ofallthebooksinthedataset.Usingthistwo-wayrelativefrequencytable,youcanseethatthereareveryfew(2.5%)newbooksthatarealsoinexpensiveandthat10%ofthebooksinthebookstorearebothexpensiveandinusedcondition.

Inothersituations,itmakessensetoexamineroworcolumnproportionsinarelativefrequencytable.Forexample,toconverttheoriginaltwo-waytabletoacolumnrelativefrequencytableusingcolumnproportions,divideeachvaluebythesumofthecolumn.

$10orless morethan10𝑏𝑢𝑡𝑙𝑒𝑠𝑠𝑡ℎ𝑎𝑛30 $30ormorenew 0.08 0.2 0.692used 0.917 0.8 0.308

Thisshowsthatabout91.7%( !!!!!!

≈ 0.917)ofthebooksthataresoldfor$10orlessareinusedcondition.Noticethateachcolumnofthiscolumnrelativefrequencytablerevealstheproportionsofthebooksineachpricecategorythatareineachconditionandtherelativefrequenciesineachcolumnsumto1.Inparticular,thisshowsthatmostoftheinexpensiveandmoderatelypricedbooksareused,andmostoftheexpensivebooksarenew.