Day 1 Two-Variable Statistics - SECTION 1A PRE-ALGEBRA · NAME DATE SECTION Day 1 Two-Variable...
Transcript of Day 1 Two-Variable Statistics - SECTION 1A PRE-ALGEBRA · NAME DATE SECTION Day 1 Two-Variable...
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.