STAT$401$–$Homework$2$–$Solutions$$!1. Effect!of!nematodes!on!protein!concentrations!in!soybeans!a. Because!the!investigators!have!control!of!group!assignments,!this!is!an!experiment.!!!b. Because!they!use!randomization!in!assigning!plants!to!the!treatments,!it!is!a!randomized!experiment.!!The!plants!were!not!randomly!selected!from!the!population!of!all!soybean!plants,!there!is!no!random!sampling.!!Thus,!the!study!belongs!in!the!lower!left!corner,!where!treatments!are!randomly!assigned,!but!the!experimental!units!are!not!randomly!selected!from!a!population.!!![Side!note:!!This!is!the!most!common!experimental!setBup.!!That!is,!you!work!with!the!experimental!units!that!are!available!(e.g.,!plants,!fields,!animals)!for!study,!but!it’s!not!a!random!sample!from!all!possible!experimental!units.!!Hopefully,!the!sample!is!representative!of!the!population!(e.g.,!used!standard!soybean!plants).!!The!experimental!setBup!with!randomly!selected!experimental!units,!to!which!treatments!are!randomly!assigned,!is!rare!(this!is!in!the!upper!left!corner!of!Display!1.5).]!c. Because!this!is!a!randomization,!causation!may!be!inferred.!!The!conclusion!may!state!whether!or!not!the!treatment!affected!the!outcome!variable.!d. The!plants!that!get!the!gelBlike!substance!with!no!nematodes!are!the!control!group,!and!the!gelBlike!substance!is!a!placebo.!!With!a!placebo,!plants!in!both!groups!are!treated!exactly!the!same!(i.e.,!both!are!treated!with!a!gelBlike!substance)!except!that!the!treatment!group!gets!nematodes!in!the!gel.!Differences!between!the!treatment!and!control!group!means!can!be!attributed!to!the!treatment!because!the!gel!application!is!not!a!confounding!factor.!e. Notation!for!the!treatment!means!(ok!if!you!labeled!the!1!and!2!differently):!μ1!=!mean!protein!concentration!for!plants!not!exposed!to!nematodes.!!!μ2!=!mean!protein!concentration!for!plants!exposed!to!nematodes!!Since!investigators!are!interested!in!whether!the!concentration!increases!with!exposure!to!nematodes,!the!hypotheses!are:!H0!:!!μ2!–!μ1!=!0!!Ha!:!!μ2!–!μ1!>!0!If!you!chose!a!twoBsided!alternative,!the!hypotheses!are:!H0!:!!μ2!–!μ1!=!0!Ha!:!!μ2!–!μ1!≠!0!!!f. The!test!statistic!for!the!randomization!test!is!!!!–!!!.!!For!the!notation!above,!!!!=!5!from!the!exposed!group!and!!!=!9!from!the!unexposed!group,!so!the!value!of!the!test!statistic!!!!–!!!!=!9!–!5!=!4.!g. All!possible!randomizations,!or!all!unique!ways!to!divide!4!values!into!2!groups!are:!!!!Units!for!infected!group!Units!for!uninfected!group!1,!2!3,!4!1,!3!2,!4!1,!4!2,!3!2,!3!1,!4!2,!4!1,!3!3,!4!1,!2!2 !h. The!difference!between!the!averages!for!each!partition!are:!!Data!for!infected!plants!(group!2)!Data!for!uninfected!plants!(group!1)!Difference!in!averages!12,!6!7,!3!9!–!5!=!! !4!12,!7!6,!3!9.5!–!4.5!=!! !5!12,!3!7,!6!7.5!–!6.5!=!! !1!7,!6!12,!3!6.5 –!7.5!=!! B1!6,!3!12,!7!4.5!–!9.5!=!! B5!7,!3!12,!6!5!–!9!=!! B4!!i. For!the!alternative!hypothesis!that!nematodes!lead!to!increased!protein!production!(oneBsided!test),!we!are!looking!for!the!number!of!samples!that!yield!differences!in!the!average!that!are!4!or!greater.!!Two!randomizations!(rows!1!and!2!in!the!table!above,!which!have!mean!differences!of!≥!4)!out!of!the!possible!6!randomizations!have!an!average!difference!that!is!at!least!as!great!as!what!we!observed!in!our!experimental!randomization.!!So!the!pBvalue!=!2/6!=!1/3.!!(If!you!used!a!2Bsided!alternative,!the!pBvalue!is!2/3.)!j. There!is!no!evidence!that!a!nematode!infection!increases!the!mean!protein!concentration!in!roots!relative!to!roots!that!have!not!been!infected!with!nematodes.!!2. Study!of!quiz!scores!in!relation!to!English!as!a!first!or!second!language!a. This!is!an!observational!study!because!group!membership!is!not!under!the!control!of!the!investigator.!!!b. This!stduy!belongs!in!the!lower!right!box!of!Display!1.5.!!A!random!sample!of!students!was!not!selected,!and!groups!are!not!under!the!control!of!the!investigator.!!!c. Results!cannot!be!generalized!beyond!the!group!that!was!studied.!!Also,!causation!can’t!be!inferred.!d. The!test!you!would!choose!is!a!permutation!test.!!A!hypothetical!framework!for!the!null!hypothesis!would!be!that!student!test!scores!are!randomly!distributed!across!students!of!both!types,!and!the!test!score!would!not!be!associated!with!whether!or!not!a!student!is!a!native!English!speaker.!First,!we!need!to!define!notation!for!the!group!means:!μ1!=!mean!quiz!score!for!native!English!speakers!!!μ2!=!mean!quiz!score!for!nonBnative!English!speakers!!Investigators!do!not!appear!to!be!interested!in!a!specific!direction.!!Thus,!we’ll!work!with!the!following!hypotheses:!H0!:!!μ2!–!μ1!=!0!Ha!:!!μ2!–!μ1!≠!0!!!! Next,!we!calculate!the!value!of!the!test!statistic,!!!!–!!!!=!8!–!4!=!4.!We!then!enumerate!all!possible!partitions!of!the!4!students!into!two!groups,!and!calculate!the!test!statistic!for!each!permutation.!!3 Data!for!nonBnative!!English!speakers!(group!2)!Data!for!native!English!speakers!(group!1)!Difference!in!averages!9,!7!5,!3!8!–!4!=!! !4!9,!5!7,!3!7!–!5!=!! !2!9,!3!7,!5!6!–!6!=!! !0!7,!5!9,!3!6!–!6!=!! !0!7,!3!9,!5!5!–!7!=!! B2!5,!3!9,!7!4!–!8!=!! B4!Finally,!we!tabulate!the!proportion!of!possible!samples!that!have!at!least!as!extreme!a!value!as!the!observed!value!of!the!test!statistic!(=!4).!!Because!we!have!a!2Bsided!alternative,!we!consider!the!absolute!value!of!the!difference!in!the!averages,!and!there!are!2!outcomes!that!are!as!extreme!as!the!observed!value!of!4,!out!of!the!total!of!6!possible!sample!partitions.!!Thus!our!pBvalue!is!2/6,!or!0.33.!!Under!a!permutation!test,!there!is!evidence!that!quiz!scores!appear!to!be!randomly!assigned!to!students.!e. There!is!no!evidence!that!being!a!native!or!nonBspeaker!is!associated!with!quiz!scores!for!this!set!of!students.!!(Note!that!because!this!is!not!a!randomized!experiment,!you!may!not!infer!causation,!and!because!this!not!a!random!sample!from!a!population,!you!may!not!infer!beyond!the!studied!students.)!!3. Writing!null!and!alternative!!hypotheses!!!!(a)!µ!=!mean!area!(in!sq!ft)!of!apartments!in!new!development!!H0:µ=1250!sq.!ft.!!(assume!claim!is!true)!!1250: <µaH!sq.!ft.!!(want!to!show!!!is!smaller!as!advertised)!!!!(b)!µ!=!mean!number!of!miles!per!gallon!(mpg)!for!Larry’s!car!on!new!motor!oil!!32:0=µHmpg!!(new!mean!is!same!as!old!one!under!old!oil)!!32: