Stat$401$–$Homework$7$Due$in$class$March$8$$We$will$be$enforcing$late$policy$–$papers$after$8$am$lose$20%$of$points$!1. Commercial!fertilizer!contains!nitrogen!that!helps!to!increase!corn!yield.!An!alternative!fertilizer!is!swine!manure,!which!also!contains!high!amounts!of!nitrogen.!Since!a!lot!of!corn!is!grown!each!year!in!Iowa!and!large!amounts!of!swine!manure!are!produced!each!year!in!Iowa,!it!makes!sense!to!examine!the!effectiveness!of!using!swine!manure!as!a!fertilizer!for!corn.!Researchers!conducted!an!experiment!to!evaluate!the!effectiveness!of!applying!swine!manure!to!soil!both!with!and!without!applying!commercial!fertilizer.!Three!levels!of!manure!were!considered!(none,!low,!high).!Two!levels!of!commercial!fertilizer!were!considered!(absent!and!present).!A!field!was!divided!into!30!plots!of!land.!!Five!plots!were!randomly!assigned!to!each!of!6!treatments!in!a!completely!randomized!design.!The!six!treatments!can!be!described!as!follows.!!Treatment$$Description$1!!no!manure!and!no!commercial!fertilizer!!2!!low!manure!and!no!commercial!fertilizer!!3!!high!manure!and!no!commercial!fertilizer!!4!!no!manure!with!commercial!fertilizer!(commercial!fertilizer!only)!5!!low!manure!with!commercial!fertilizer!6!!high!manure!with!commercial!fertilizer!!!Plots!were!treated!in!the!spring!and!yields!were!recorded!during!the!fall!harvest.!A!summary!of!the!grain!yield!data!(in!Mg/ha)!is!provided!below.!!Treatment$$1$2$3$4$5$6$Number!of!plots!!5!5!5!5!5!5!Sample!average!!8.3!11.3!14.0!11.0!13.9!14.1!Sample!variance!!2.6!3.3!3.1!3.3!2.9!2.8!!a) For!the!1Rway!ANOVA,!what!distribution!is!assumed!for!the!observations!Yij!in!group!i$?!Be!sure!to!include!appropriate!mathematical!notation.!!b) Calculate!an!estimate!for!!!!based!on!all!of!the!data.!c) Record!the!null!and!alternative!hypothesis!for!the!equality!of!means!test.!d) Provide!a!oneRway!ANOVA!table!for!this!data.!Include!SOURCE,!DF!(degrees!of!freedom),!SS!(sum!of!squares),!MS!(mean!square),!and!F!columns!in!your!table.!!e) Provide!degrees!of!freedom!and!a!pRvalue!for!the!FRstatistic!associated!with!your!ANOVA!table.!f) Based!on!the!pRvalue,!provide!a!scientific!conclusion!for!the!question!addressed!by!your!FRtest.!!!!2 2. Suppose!we!wanted!to!use!the!full!and!reduced!model!approach!to!testing!for!differences!among!treatment!means!in!Problem!1.!!You!will!want!to!refer!to!the!handout!called!“The!FRTest!as!a!Comparison!of!Full!and!Reduced!Models.”!a) What!specific!model!is!the!full!model!for!this!FRtest?!What!specific!model!is!the!reduced!model!for!this!FRtest?!b) For!the!full!model!and!the!reduced!model,!create!a!table!that!shows!the!population!means!for!each!group,!the!estimators!for!the!population!means!under!each!model!(the!estimator!is!the!symbol!for!the!estimated!mean!that!represents!what!you’ll!calculate!from!the!data),!and!the!values!for!the!mean!estimates.!!!!Full$Model$=$________________________________$model$from$(a)$Treatment!!!1!2!3!4!5!6!Population!mean!(greek!symbol)!!!!!!!Estimator!for!the!population!mean!(symbol!for!summary!statistic)!!!!!!!Estimate!(number)!!!!!!!!!!Reduced$Model$=$______________________________$model$from$(a)$Treatment!!!1!2!3!4!5!6!Population!mean!!!!!!!!!Estimator!for!the!population!mean!(symbol!for!summary!statistic)!!!!!!!Estimate!(number)!!!!!!!!!c) What!is!the!restriction!used!to!generate!the!special!case!of!the!reduced!model!from!the!more!general!full!model?!d) What!is!the!estimated!mean!for!group!3!under!the!full!model?!What!is!the!estimated!mean!for!group!3!under!the!reduced!model?!!Be!sure!to!use!appropriate!notation!in!expressing!your!answer!(i.e.,!insert!the!appropriate!value!for!i!in!your!notation).!!3 e) If!observation!2!in!group!3!was!equal!to!8!protein!units!(i.e.,!Y32!=!8),!what!is!the!value!of!the!residual!under!the!full!model?!!What!is!the!value!of!the!residual!under!the!reduced!model?!!Be!sure!to!use!appropriate!notation!in!expressing!your!answer!(i.e.,!insert!the!appropriate!value!for!i!in!your!notation).!f) Using!your!ANOVA!table!in!Problem!1,!fill!in!the!values!in!the!various!parts!of!the!FRstatistic!formula!in!the!handout,!and!calculate!the!FRstatistic!to!verify!it!is!the!same!value!as!the!one!given!in!the!ANOVA!table.!!!3. Consider!the!fatty!acid!data!from!problem!18!on!p.!142.!!We!will!assume!that!this!study!is!based!on!a!representative!sample,!that!all!observations!are!independent!of!each!other,!and!that!treatments!are!randomized!to!rats.!!!a) Create!a!sideRbyRside!stemRandRleaf!plot!by!hand.!!Comment!on!what!you!can!tell!from!this!plot.!b) Use!SAS!and!the!starter!program!posted!on!the!web,!calculate!the!1Rway!ANOVA!for!this!experiment.!!Record!the!ANOVA!table!in!your!homework.!c) Conduct!a!test!for!equality!of!means.!!Be!sure!to!include!the!hypotheses,!test!statistic!calculations,!degrees!of!freedom,!and!pRvalue.!d) Provide!a!conclusion!for!the!scientist!based!on!the!test!in!(c).!e) Using!SAS,!create!a!set!of!diagnostic!plots!that!include!a!plot!of!the!residuals!vs.!the!predicated!values,!a!normal!plot!for!the!residuals,!and!a!histogram!of!the!residuals.!!Include!these!plots!in!your!homework.!f)
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