1""Stat$401$–$Homework$12$Due"at"8"am"in"class"on"Friday"April"26"1. Animal"ecologists"wish"to"track"the"health"of"bear"populations"in"the"wild."By"obtaining"physical"measurements"of"bears,"researchers"can"gauge"the"effects"of"changes"taking"place"in"and"around"bear"habitat"(e.g.,"nearby"housing"and"business"development,"recreational"activities,"introduction"of"nonCnative"plant"and"animal"species,"severe"storms,"harsh"winters,"etc.)."Researchers"have"studied"bears"by"anesthetizing"them"in"order"to"obtain"vital"measurements,"such"as"age,"gender,"length,"and"width."A"bear’s"weight"is"another"important"variable"that"is"quite"difficult"to"obtain"in"the"wild"because"most"bears"are"heavy"and"difficult"to"lift."The"scientific"problem"is"to"develop"a"method"for"predicting"the"weight"of"a"bear,"given"other"more"easily"obtained"measurements."A"good"method"might"alleviate"the"need"to"weigh"bears"in"the"wild"and"greatly"simplify"the"data"collection"process."In"one"detailed"study,"researchers"were"able"to"obtain"weight"measurements"along"with"several"other"variables"for"each"of"54"bears."The"data"are"in"a"starter"program"on"the"web."The"data"set"contains"54"rows"and"9"columns."There"is"one"row"for"each"bear"that"was"anesthetized"and"measured"carefully"using"a"tape"measure"and"scale."There"is"one"column"for"each"variable"in"the"data"set."The"variables"(from"left"to"right)"are"age" " age"in"months"length" " length"of"body"in"inches"sex" " 1=male"2=female"weight"" weight"in"pounds"chest" " chest"circumference"in"inches"headlth" length"of"head"in"inches"headwth" width"of"head"in"inches"month"" month"of"capture"(1=January,"2=February,"etc.)"neck" " neck"circumference"in"inches"(a) Answer"the"following:"i. Which"response"that"best"describes"the"design"used"in"this"study."• Randomized"experiment"with"convenience"sample"• Randomized"experiment"with"random"sample"• Observational"study"with"convenience"sample"• Observational"study"with"random"sample"ii. Can"the"researcher"infer"causation"from"this"study?"""""iii. To"what"group"can"the"results"be"generalized?"""(b) Record"the"mean"model"for"these"data"for"response"variable"weight"and"explanatory"variables"length,"chest"and"neck."(c) Express"the"4"assumptions"for"the"fitting"the"multiple"linear"regression"model"in"(b).""Try"using"the"shorthand"description"of"the"distribution"of"weight"given"length,"chest"and"neck"(i.e."use"statistical"notation).""2""(d) Using"weight"as"the"response"variable"and"chest,"length,"and"neck"as"explanatory"variables,"estimate"the"model"parameters"(intercept,"partial"regression"coefficients)"and"the"standard"deviation"of"Y"given"the"explanatory"variables)."Record"the"estimated"regression"model."(e) Record"the"estimated"standard"deviation"of"bear"weights"for"any"given"values"of"the"explanatory"variables."""(f) Conduct"a"test"of"the"null"hypothesis"that"the"partial"regression"coefficients"for"chest,"length,"and"neck"are"all"zero"against"the"alternative"that"at"least"one"partial"regression"coefficient"is"not"zero."State"the"null"and"alternative"hypotheses,"the"test"statistic,"the"degrees"of"freedom,"pCvalue"and"conclusion."(g) Provide"an"interpretation"of"the"partial"regression"coefficient"associated"with"the"variable"chest.""(See"initial"handout"on"MLR.)"(h) Compute"a"95%"confidence"interval"for"the"partial"regression"coefficient"associated"with"the"variable"length."Is"the"partial"regression"coefficient"associated"with"the"variable"length"significantly"different"from"zero?"Explain"how"your"confidence"interval"can"be"used"to"answer"this"question."""(i) What"is"the"proportion"(or"percentage)"of"variation"in"bear"weights"explained"by"the"multiple"regression"of"weight"on"chest,"length,"and"neck?"(j) Provide"an"estimate"of"the"mean"weight"of"a"captured"bear"that"is"60"inches"long,"with"chest"circumference"of"35"inches,"and"neck"circumference"24"inches."(k) Provide"a"95%"confidence"interval"for"the"mean"estimated"in"(j)."(l) Researchers"are"interested"in"predicting"the"weight"of"an"individual"bear"based"on"body"measurements."Provide"a"predicted"weight"of"a"captured"bear"that"is"60"inches"long,"with"chest"circumference"of"35"inches,"and"neck"circumference"24"inches."(m) Provide"a"95%"prediction"interval"for"the"weight"of"the"bear"with"the"measurements"described"in"part"(l)."(n) Examine"the"diagnostic"plots"associated"with"this"regression."Comment"on"whether"the"plots"suggest"that"the"normality,"linearity"and/or"homogenous"variance"assumptions"of"multiple"linear"regression"are"violated.""[On"your"own:""You"might"consider"what"your"next"step"would"be.""Try"log"transforming"weight,"chest"and"neck.""ReCrun"the"model"with"the"logCtransformed"variables"and"check"the"diagnostics.]"2. In"this"problem,"we"use"data"to"consider"the"association"between"yield"and"amount"of"fertilizer"for"three"varieties"of"corn"(A,"B,"C)."These"data"come"from"the"same"type"of"corn"yield"and"fertilizer"experiment"we"discussed"in"class."SAS"code"and"output"for"a"regression"analysis"that"involves"indicator"variables"for"variety"and"interactions"between"variety"indicators"and"fertilizer"amount"are"on"the"last"pages"of"this"homework.""You"will"want"to"be"familiar"with"the"material"on"Indicators"and"Interactions"for"parts"of"this"homework."(a) Examine"the"code"on"PAGE"1"of"the"SAS"pages"(p."4"of"HW)."Record"the"population"regression"model"for"the"mean"of"Y."Be"sure"to"define"notation"for"the"response"variable"Y"and"each"explanatory"variable,"X1,"X2,"X3,"X4,"and"X5."(b) Test"the"null"hypothesis"that"says"that"the"mean"μ(Y"|"X1,"X2,"X3,"X4,"X5)"does"not"depend"on"X1,"X2,"X3,"X4,"and"X5."Be"sure"to"state"the"hypotheses,"test"statistics,"pCvalue"and"conclusion."(c) Using"the"output,"record"the"least"squares"regression"that"estimates"the"population"regression"model"in"(a)."3""(d) Use"the"population"mean"of"Y"(yield)"given"X1,"X2,"X3,"X4,"and"X5"to"write"a"separate"model"for"each"variety"(A,"B"and"C)."Note"that"there"will"be"three"population"regression"models,"one"for"each"variety"of"corn."(e) Suppose"we"wish"to"test"the"null"hypothesis"that"says"that"slopes"of"the"regression"lines"relating"yield"to"amount"of"fertilizer"for"each"variety"are"identical."You"could"think"of"performing"this"hypothesis"test"as"a"comparison"between"full"and"reduced"models."Make"a"table"like"the"one"below,"and"fill"in"the"cells"in"the"table"as"follows:"•
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