Stat$401$–$Homework$1$Due$in$class,$Friday,$January$25$!1. Suppose!researchers!randomly!select!a!large!sample!of!adults!in!the!United!States.!They!give!the!sampled!people!a!survey!to!determine!their!eating!habits!over!the!past!20!years.!They!classify!the!people!into!two!groups.!One!group!contains!those!people!who!have!been!eating!a!diet!consisting!mainly!of!organically!grown!fruits!and!vegetables.!The!other!group!consists!of!people!who!have!been!eating!a!more!traditional!diet.!The!researchers!find!that!the!rate!of!cancer!is!significantly!lower!among!people!in!the!first!group.!The!researchers!conclude!that!a!diet!consisting!of!organically!grown!fruits!and!vegetables!reduces!the!risk!of!cancer.!!(a) Is!this!an!experiment!or!an!observational!study?!(b) Comment!on!the!validity!of!the!researchers’!conclusion.!!Can!you!think!of!any!other!reasons!why!the!first!group!might!have!a!significantly!lower!rate!of!cancer?!!2. As!evidence!in!support!of!natural!selection!in!1898,!Hermon!Bumpus!presented!measurements!on!house!sparrows!brought!to!the!Anatomical!Laboratory!of!Brown!University!after!an!uncommonly!severe!winter!storm.!Bumpus!measured!the!length!of!each!bird’s!humerus,!which!is!the!wing!bone!closest!to!the!body.!!The!data!below!are!24!humerus!lengths!from!a!sample!of!adult!male!sparrows!who!died!in!the!storm,!and!35!humerus!lengths!from!a!sample!of!adult!male!sparrows!who!survived.!(a) Compute!the!five!number!summaries!for!each!group!by!hand.!!!(Do!not!use!SAS!for!this!part!of!the!problem.)!(b) Construct!a!stem!and!leaf!plot!by!hand!that!allows!you!to!visually!compare!the!distribution!of!humerus!lengths!for!the!survivors!and!nonRsurvivors.!!(Do!not!use!SAS!for!this!part!of!the!problem.)!(c) Create!a!sideRbyRside!box!plot!by!hand.!!In!addition!to!the!usual!items,!indicate!the!upper!and!lower!fence!on!your!plot!(see!SAS!Boxplot!Schematic!from!class!notes).!(d) After!making!hand!calculations,!use!SAS!to!check!your!answers!for!(a)!–!(c).!!The!dataset!sparrows.txt!will!be!posted!on!the!web.!!Attach!your!program!and!its!results!to!your!homework.!!!(e) Briefly!summarize!what!you’ve!learned!about!the!data!distributions!from!these!summary!statistics.!!Died!!!Survived!659!744!!687!739!689!745!!703!741!702!752!!709!741!703!752!!715!741!709!754!!721!741!713!765!!723!743!720!!!723!749!720!!!726!751!726!!!728!752!726!!!728!752!729!!!728!755!731!Units!are!1/1000!inch!(e.g.,!659!is!0.659!inches)!729!756!736!730!766!737!730!767!738!733!769!738!!!733!770!739!!!735!780!743!!!736!!3. Salaries!of!professional!athletes!receive!a!good!deal!of!attention!in!the!press.!The!1990!salaries!of!the!nonRpitchers!on!the!Chicago!Cubs!baseball!team!are!listed!below,!in!units!of!thousands!of!dollars!(e.g.,!the!value!of!100!units!below!means!100,000!dollars).!!100!!100!!111!!114!!165!!210!!225!!225!!230!!575!!1200!!1900!!2100!!2100!!2650!!3300!!(a) Calculate!the!median!and!the!sample!average!for!these!data.!(b) Suppose!the!owner!of!the!team!says!that!a!typical!player!earns!close!to!a!million!dollars!(1000!units!in!thousands!of!dollars).!Which!statistical!measure!for!the!center!of!the!distribution!is!the!owner!using?!Would!it!be!accurate!to!say!that!a!typical!player!earns!close!to!a!million!dollars?!Explain.!(c) Calculate!the!other!4!numbers!of!the!fiveRnumber!summary!for!these!data.!(d) If!each!player!received!an!extra!200,000!dollars!(or!200!units!in!thousands!of!dollars),!how!would!the!sample!average!and!median!change?!!!You!don’t!have!to!change!the!data!and!recalculate!the!sample!average!and!median.!Instead,!think!about!the!new!salary!for!each!player!and!how!the!central!tendency!is!affected!by!adding!a!constant.!(e) If!each!player!received!an!extra!200,000!dollars!(or!200!units!in!thousands!of!dollars),!how!would!the!standard!deviation!change?!!!NOTE:!!You!don’t!have!to!change!the!data!and!calculate!the!standard!deviation.!Instead,!think!about!what!it!measures!(i.e.,!spread)!and!how!that!is!affected!by!adding!a!constant.!!4. The!average!age!of!5!persons!in!a!room!is!30!years.!A!36RyearRold!person!walks!into!the!room.!!(a) What!is!now!the!average!age!of!the!persons!in!the!room?!!!(Think!about!how!this!observation!would!be!included!based!on!the!formula!for!a!sample!average!–!requires!good!working!knowledge!of!HW0.)!(b) Suppose!the!median!age!is!30!years!and!a!36RyearRold!person!enters!the!room.!Can!you!find!the!new!median!age!from!this!information?!!Why!or!why!not.!!5. Problems!on!summary!statistics:!(a) Provide!a!set!of!numbers!for!which!exactly!3!of!the!5!numbers!in!the!fiveRnumber!summary!are!zero.!!Is!there!more!than!one!answer!to!this!question?!(b) Provide!a!set!of!5!numbers!that!has!an!average!of!7!and!a!standard!deviation!of!0.!!Is!there!more!than!one!answer!to!this!question?!(c)!!For!which!data!set!is!th e!standard! deviatio n!l arger?!!Use!your!understanding!of!the!formul
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