1" Stat$401$–$Homework$2$Due$in$class,$Friday,$February$1$"1. Researchers"are"interested"in"whether"the"concentration"of"a"particular"soybean"protein"produced"in"the"root"tissue"of"soybean"plants"increases"in"response"to"the"presence"of"nematodes"(nematodes"are"small"worms"that"can"infect"plant"roots).""The"researchers"have"4"soybean"plants"available"for"their"study.""The"researchers"randomly"selected"2"plants"from"the"4"plants."These"two"plants"are"infected"with"soybean"cyst"nematodes"by"spreading"a"gel?like"substance"containing"the"nematodes"over"their"roots."The"roots"of"the"other"2"plants"are"treated"with"a"gel?like"substance"that"contained"no"nematodes.""""""(a) Is"this"an"observational"study"or"an"experiment?"Explain.""(b) Examine"Display"1.5"in"your"text."Into"which"of"the"4"boxes"might"this"investigation"be"classified?""Briefly"explain"why.""(c) Based"on"your"answer"in"(b),"what"kind"of"inferences"that"would"be"valid"for"this"design.""(d) Explain"the"purpose"of"treating"the"2"plants"with"the"gel?like"substance"that"contained"no"nematodes.""(e) The"researchers"plan"to"conduct"a"randomization"test"using"the"results"of"the"experiment.""Record"a"null"and"an"alternative"hypothesis"for"comparing"treatment"means"and"explain"why"this"set"of"hypotheses"is"of"interest.""Be"sure"to"define"your"notation.""Data"on"protein"concentrations"from"this"investigation"are"provided"in"the"first"table"below."Units"on"the"protein"concentrations"are"not"provided,"but"the"larger"numbers"are"associated"with"higher"protein"concentrations."The"table"below"gives"the"data"from"this"experiment,"along"with"the"sample"average"of"the"infected"and"uninfected"plants.""Treatment$(i)$Plant$id$(j)$Observed$$protein$concentration$(Yij)$Sample$average$$protein$concentration$Infected"plants"1"12"9""2"6""Uninfected"plants"3"7"5""4"3"""(f) Define"the"test"statistic"for"a"randomization"test"and"verify"that"its"value"is"4"units"of"protein.""""(g) Conduct"a"randomization"test"for"your"hypotheses.""Because"the"sample"size"is"so"small,"we"can"list"all"the"possible"ways"that"the"4"plant"protein"values"could"be"divided"into"infected"and"uninfected"groups,"with"two"plants"per"group.""There"are"6"possible"randomizations"(in"listing"all"randomizations,"note"that"having"1"and"2"in"the"same"group"is"the"same"as"having"2"and"1"in"the"same"group)."Make"a"list"of"all"possible"randomizations"for"this"experimental"design.""(h) For"each"of"the"samples"in"(g),"calculate"the"sample"average"for"the"infected"and"uninfected"groups,"and"determine"the"difference"in"the"two"sample"averages.""""2" (i) Determine"the"p?value"for"your"test.""Show"how"you"calculated"the"p?value.""(j) Write"a"conclusion"for"the"study"based"on"your"randomization"test"assuming"your"audience"is"a"scientist"(i.e.,"answer"the"scientific"question"as"if"you"were"talking"to"a"scientist,"not"a"statistician).""Be"sure"to"cite"the"strength"of"evidence.""2. An"elementary"school"teacher"wants"to"examine"the"effect"of"language"skills"on"student"academic"performance.""There"are"4"students"in"the"study.""Two"of"these"students"are"recent"immigrants"(non?native"English"speakers)"and"2"are"native"English"speakers.""The"quiz"scores"for"the"4"students"are"below.""""Group$(i)$Quiz$scores$(Yij)$Non?native"English"speakers"9"""""7"Native"English"speakers"5"""""3""(a) Is"this"an"observational"study"or"an"experiment?"Explain.""(b) Examine"Display"1.5"in"your"text."Into"which"of"the"4"boxes"might"this"investigation"be"classified?""Briefly"explain"why.""(c) Based"on"your"answer"in"(b),"what"kind"of"inferences"that"would"be"valid"for"this"design.""(d) Conduct"a"permutation"test"to"compare"the"mean"quiz"scores"of"the"non?native"English"speaking"students"with"the"native"English"speaking"students.""Show"all"of"your"work.""Be"sure"to"define"your"notation,"hypotheses,"a"test"statistic"and"its"value,"and"your"work"in"calculating"a"p?value.""(e) Write"a"conclusion"for"the"study"based"on"your"permutation"test"assuming"your"audience"is"an"educator.""3. When"we"identify"hypotheses"for"μ2"–"μ1","it"is"like"testing"a"hypothesis"for"a"single"parameter"δ"="μ2"–"μ1".""This"question"asks"you"to"consider"construct"hypotheses"for"studies"with"only"one"group.""The"group"has"a"single"population"mean"μ","and"we"need"to"test"whether"the"mean"is"has"a"specific"value"c"(e.g.,"μ"="1)"or"has"some"alternative"value"(e.g.,"μ"≠"1,"or"μ">"1,"or"μ"<"1).""For"each"part,"define"μ,"the"specific"value"c,"and"the"null"and"alternative"hypotheses"for"the"scientific"question.""(Use"ideas"we"discussed"in"the"2nd"week"of"class.)""(a) The"mean"area"of"the"several"thousand"apartments"in"a"new"development"is"advertised"to"be"1250"square"feet.""A"tenant"group"thinks"that"the"apartments"are"smaller"than"advertised.""They"hire"an"engineer"to"measure"a"sample"of"apartments"to"test"their"suspicion."""(b) Larry’s"car"averages"32"miles"per"gallon"on"the"highway.""He"now"switches"to"a"new"motor"oil"that"is"advertised"as"increasing"gas"mileage.""After"driving"3000"highway"miles"with"the"new"oil,"he"wants"to"determine"if"his"gas"mileage"actually"has"increased.""(c)
View Full Document