1"Homework(11(Solutions("1.""Manatee"part"2""(a) The"predicted"number"of"manatee"deaths"in"a"future"year"with"800,000"boat"registrations"(X"="800)"is""Pred(Y|X=800)"="µˆ(Y|X=800)"="D"41.43044"+"0.12486"(800)"="58.4589"≈"58.5"manatees"killed"per"year.""(b) The"95%"prediction"interval"is"(46.8,"70.1)"manatees"killed"in"a"future"year,"calculated"using"SE["Pred(Y|X=800)"]"="! 1 + !!!+(!!!)!(! !!)!!!"="4.2764 1 +!!"+(!""!!"#.!)!(!"!!)!".!"#!!"="5.3476"≈"5.3"HalfDwidth"="t(nD2),(1Dα/2)"SE[Pred(Y|X=800)]"="2.179"(5.3476)"="11.6523"Predication"interval"(PI)"is"(46.8066,"70.1112)"manatees"killed"per"year,"or"in"rounded"numbers,""(46.8,"70.1)"manatees"killed"in"a"future"year""" "2." pH"in"steer"muscle"in"relation"to"log"hours"since"slaughter""(a) "The"plot"of"muscle"pH"(Y)"vs."log"hours"since"slaughter"(X)"shows"that"the"variation"among"sample"averages"for"pH"(iY)"for"a"specific"log"time"is"larger"than"the"residual"variation"among"responses"for"a"specific"time"point.""There"is"also"a"clear"negative"trend"in"pH"as"time"since"slaughter"increases"that"is"pretty"linear"for"lower"log"times.""The"asymptote"(leveling"off)"for"the"response"at"higher"log"times"suggests"that"there"may"also"be"a"lack"of"fit"in"the"linear"model.""(b) 1Dway"ANOVA"with"linear"and"lack"of"fit"(LOF)"test"is"uses"the"1Dway"ANOVA,"with"only"the"SSREG"term"coming"from"the"SLR"ANOVA."""""""""""(c) Ho:""µ(Y"|"log"1)"="µ(Y"|"log"2)"="µ(Y"|"log"4)"="µ(Y"|"log"6)"="µ(Y"|"log"8)"="µ(Y"|"log"24)""Ha:"at"least"one"mean"is"different"from"the"others""The"1Dway"ANOVA"overall"FDtest"can"be"used"evaluate"these"hypotheses.""The"FDstatistic"has"df1"="5"and"df2"="6"is"equal"to"79.59,"with"pDvalue"<"0.0001.""Thus"we"have"convincing"evidence"that"at"least"one"mean"is"different"the"others"(i.e.,"we"choose"the"separate"means"model"over"the"common"mean"model).""(d) Ho:""µ(Y"|"log"X)"="β0""""Ha:""µ(Y"|"log"X)"="β0"+"β1log(X)""for"X"="1,"2,"4,"6,"8,"24"hours"and"some"β1"not"equal"to"0""Using"the"1Dway"ANOVA"framework"in"(b),"use"the"FDtest"on"the"line"that"says"“Linear”."""The"FDstatistic"has"df1"="1"and"df2"="6"is"equal"to"357.59,"with"pDvalue"<"0.0001.""We"have"convincing"evidence"of"a"SOURCE"DF"SS"MS"F"PDValue"Between"5"3.91637"0.783275"79.59"<"0.0001"""""Linear""1"3.51940"3.51940"357.59"<"0.0001"""Lack"of"Fit"4"0.39697"0.09924"10.08"<"0.01"Within"6"0.05905"0.009842"""Total"11"3.97542""""2"linear"relationship"between"the"mean"pH"of"steer"muscle"and"logDhours"since"slaughter"(i.e.,"we"choose"the"linear"regression"model"over"the"common"mean"model).""""(e) Ho:""µ(Y"|"log"X)"="β0"+"β1log(X)""for"X"="1,"2,"4,"6,"8,"24"hours"and"some"β1"not"equal"to"0"Ha:""the"relationship"between"log"time"and"the"mean"pH"at"a"time"is"not"described"by"a"linear"model"(i.e.,"something"other"than"a"linear"relationship"is"needed)""Using"the"1Dway"ANOVA"framework"in"(b),"use"the"FDtest"on"the"line"that"says"“Lack"of"Fit”."""The"FDstatistic"has"df1"="4"and"df2"="6"is"equal"to"10.08,"with"pDvalue"<"0.01.""""Alternatively,"you"can"use"the"full"and"reduced"model"framework."Reduced"model:""mean"pH"a"linear"function"of"log"time"Full"model:""separate"means"for"each"log"time"From"1Dway"ANOVA:""RSS(full)"="SSW"="0.0590"with"df(full)"="dfW"="n"–"I"="6"From"SLR"ANOVA:""RSS(reduced)"="SSE"from"SLR"ANOVE"="0.45602"with"df(red)"="dfE"from"SLR"="n"–"2"="10"The"FDtest"for"lack"of"fit"is""F"="[SSEREG! SSW ] / [(n ! 2) ! (n ! I )]SSW / (n ! I )=[0.45602 ! 0.05905] / (10 ! 6)0.0590 / 6=0.39697 / 40.00984167=.09924250.00984167"="10.08,""with"df1"="dfSSE(REG)"D"dfW"="4"and"df2"="n"–"I"="6"and"a"pDvalue"between"0.001"and"0.01"(using"FDtable)""There"is"strong"evidence"of"lack"of"fit"in"the"linear"model"relating"the"mean"pH"of"steer"muscle"to"logDhours"since"slaughter.""The"linear"model"is"not"sufficient"to"fully"describe"the"relationship"between"pH"and"logDhours"(i.e.,"we"choose"the"separate"means"model"over"the"linear"regression"model).""""Note:""looking"at"the"scatterplot"of"Y"log"X"for"the"12"pairs"of"(Y,"log"X),"this"finding"makes"sense.""There"is"a"curved,"not"fully"linear,"relationship"due"to"the"pH"values"when"log"X"="3.17805"(or"X"="24"hours).""The"pH"levels"at"X"="24"hours"don’t"seem"to"follow"the"linear"relationship"apparent"among"the"values"of"pH"(or"the"YDvalues)"for"the"smaller"times"on"the"log"(hours).""BOTTOMLINE:""Of"the"three"models,"estimating"the"mean"pH"of"the"steer"muscle"for"each"time"point"separately"is"the"best"approach"to"describing"the"relationship"between"time"since"slaughter"and"mean."There"is"a"difference"among"means,"as"indicated"by"the"overall"1Dway"ANOVA"FDtest.""The"differences"among"means"are"not"adequately"described"by"a"model"that"states"that"the"mean"pH"is"a"linear"function"of"logDhours,"as"indicated"by"the"lackDofDfit"test.""Thus,"the"separate"means"model"for"estimating"the"mean"pH"after"a"period"of"time"is"the"best"model.""""""""(f)"
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