STAT$401$–$Homework$1$–$Solutions$$!1. !!Diet!study!(a) Observational!study.!!Note!that!the!diet!type!(group!variable)!was!not!assigned!to!people.!(b) Because!diet!type!was!not!randomly!assigned!to!participants,!we!can’t!infer!cause!and!effect,!only!association.!!Other!reasons!cancer!rates!might!be!lower!in!the!organic!diet!is!that!they!may!care!more!about!health!and!take!care!of!themselves!(work!out,!visit!doctors,!etc)!more!than!those!who!eat!a!traditional!diet.!!That!is,!lifestyle!might!be!related!to!the!choice!of!diet!type!and!cancer!rate!(i.e.,!lifestyle!might!be!a!confounding!variable).!!!!2. !!House!sparrows!study!(units!in!1/1000!in.)!(a) 5Hnumber!summary!(and!calculating!the!fences!for!part!c)!!!Died$Survived$n"24!35!Minimum!659!!687!Lower!quartile!(Q1)!!716.5!!728!(n"–!1)/4!not!a!whole!number,!so!don’t!include!median!in!quartile!determination!Median!(M)!733.5!n"is!even,!so!take!average!of!2!middle!numbers!736!Upper!quartile!(Q3)!743.5!752!Maximum!765!780!!!!IQR!=!Q3!–!Q1!27!24!Lower!fence!!(Q1!–!1.5!IQR)!716.5!–!40.5!=!676!728!–!36!=!692!Upper!fence!(Q3!+!1.5!IQR)!743.5!+!40.5!=!784!752!+!36!=!788!!(b) Stem!and!leaf!plot.!!This!is!the!book’s!ordering,!with!the!smallest!number!on!top.!!It’s!ok!if!you!ordered!the!stem!so!that!small!values!are!at!the!bottom,!which!is!what!SAS!does.!!Died$Survived$65!9!65!!66!!66!!67!!67!!68!9!68!7!69!!69!!70!239!70!39!71!3!71!5!72!00669!72!13368889!73!167889!73!0033569!74!345!74!111139!75!224!75!12256!76!5!76!679!77!!77!0!78!!78!0!!(c) Box!plot!from!SAS!(fences!denoted!with!a!bold!red!dash$---)! --- | --- 780 + | | | | | | | | 760 + | | | | | | | +-----+ | +-----+ | | 740 + | | | + | | *-----* *-----* | | + | +-----+ | | | | 720 + | | | | +-----+ | | | | | | | 700 + | | | --- | | | 0 680 + | --- | | 660 + 0 ------------+-----------+----------- Died Survived (d) SAS!program! TITLE 'HW 01 Problem 2: Bumpus sparrows' ; FILENAME sparrows url "http://www.public.iastate.edu/~nusser/Stat401/sparrows.txt" ; * sparrows.txt columns length = humerus length in 1/1000 inches outcome = Died or Survived ; DATA a ; INFILE sparrows ; INPUT length outcome $ ; RUN; PROC SORT DATA=a ; BY outcome ; PROC UNIVARIATE DATA=a PLOTS ; BY outcome ; VAR length ; TITLE2 'Summary statistics and plots for checking parts a-c' ; RUN ; !(e) The!humerus!length!distribution!of!the!sparrows!who!survived!is!shifted!to!the!right!relative!to!the!distribution!for!sparrows!who!died.!!The!spread!of!each!distribution!is!fairly!similar,!and!both!distributions!appear!to!have!one!outlier.!!! !3. Chicago!Cubs!baseball!team!salaries!(a) Sample!median:!!M!=!(225!+!230)!/!!2!=!227.5!or!!$227,500!Sample!average:!!!!=!956.5625!or!!$956,562.50!(b) The!owner!would!be!using!the!sample!average!!!=$956,562.50,!which!makes!sense!in!that!the!owner!also!would!like!to!argue!against!paying!higher!salaries!(i.e.,!looks!good!for!the!owner!to!claim!.a!pitcher!earns!close!to!a!million!dollars!on!average").!!On!the!other!hand,!a!typical!pitcher!does!not!really!earn!close!to!a!million!dollars!since!the!median!is!227.5!(in!thousands!of!dollars);!that!means!50%!of!the!pitchers!earn!less!than!$227,500.!!Note!the!sample!average!is!much!higher!than!the!sample!median!because!the!distribution!of!salaries!(data)!is!skewed!right.!However,!overall,!the!owner!is!paying!out!in!salaries!close!to!a!million!dollars!per!pitcher.!!(c) Min!=!100,!!Q1!=!(114!+!165)!/!2!=!139:5,!!M!=!227:5;Q3!=!(1900!+!2100)!/!2!=!2000,!!max!=!3300!(d) Both!!!and!M!would!increase!by!200!units.!!Since!every!value!is!increased!by!200,!the!center!or!typical!value!in!the!data!increases!by!200.!!Consider!two!samples!for!illustration:!!Sample!A! ! ! ! 1!!!2!!!3!!Sample!B!!(Sample!A!+!200)! ! 201!!!202!!!203!The!sample!average!is!for!sample!A!is!!!!=!2.!!The!sample!average!is!for!B!is!!!!=!202!=!200!+!!!.!!!(e) The!spread!among!in!the!individual!observations!in!the!sample,!or!sample!standard!deviation!s,!would!NOT!change!by!adding!200!units!to!each!observation.!!Using!our!illustration!in!part!d,!the!SPREADS!among!observations!in!each!sample!are!the!same!(the!standard!deviation"s!=!1!for!both!data!sets).!!4. Age!distributions!(a)
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