STAT$401$–$Homework$5$–$Solutions$1. Study(of(metabolic(expenditure(in(relation(to(trauma((as(defined(by(having(multiple(fractures(when(admitted(to(the(hospital)(((a) Cause(and(effect(may(not(be(inferred(because(this(is(not(a(randomized(experiment.((In(other(words,(the(investigator(did(not(have(control(over(the(assignment(of(individual(patients(to(groups((trauma(or(nontrauma).(((b) Estimates(may(not(be(generalized(to(the(rest(of(the(hospital(patients(because(patients(were(recruited(as(available(from(the(hospital,(not(from(a(random(sample(of(all(patients.(((c) μ1((=(Mean(metabolic(expenditure((in(kcal/kg/day)(for(nontrauma(patients(μ2(=(Mean(metabolic(expenditure((in(kcal/kg/day)(for(trauma(patients(H0(:((μ2((–(μ1((=(0((vs.(Ha(:((μ2((–(μ1(≠(0(t(=((20.963(–(28.771)(/((4.7528(*!!+!!()=(S7.808(/(2.4598(=(S3.174((with(13(df((and(0.005(<(p(<(0.01(There(is(convincing(evidence(of(a(difference(in(the(metabolic(rate(of(trauma(patients(relative(to(the(metabolic(rate(for(nontrauma(patients.((For(patients(in(this(study,(the(metabolic(rate(for(trauma(patients(is(estimated(to(be(7.8(kcal/kg/day(higher(than(for(nontrauma(patients,(with(a(standard(error(of(2.5(kcal/kg/day.(SAS(program(to(check(answers:(title 'Ch 3 Prob 18 - Metabolic expenditure (kcal/kg/day) in relation to trauma' ; data a ; input metabexpend group $ ; datalines ; 20.1 nontrauma 22.9 nontrauma 18.8 nontrauma 20.9 nontrauma 20.9 nontrauma 22.7 nontrauma 21.4 nontrauma 20.0 nontrauma 38.5 trauma 25.8 trauma 22.0 trauma 23.0 trauma 37.6 trauma 30.0 trauma 24.5 trauma ; proc print data=a ; title2 'raw data' ; run ; proc sort data=a ; by group ; proc univariate data=a plots ; by group ; var metabexpend ; title2 'exploratory analyses' ; run ; proc ttest data=a ; class group ; var metabexpend ; title2 '2-sample t-test for difference between means' ; run ; proc sort data=a ; by metabexpend ; proc print data=a ; title2 'ordered data for ranking' ; run ; (d) It(is(clear(that(the(sample(standard(deviations(for(the(two(groups,(s1(=(1.3794(and(s2(=(6.8354,(are(quite(different.((It(is(more(difficult(to(judge(normality(because(sample(sizes(are(so(small.((However,(the(trauma(distribution(appears(to(have(some(large(values(that(suggest(it(may(be(skewed.((The(small(sample(sizes,(heterogeneous(standard(deviations(and(possible(nonSnormality(raise(questions(about(whether(the(tStest(will(be(robust.(((e) The(difference(between(sample(averages(is(not(resistant(because(it(is(a(function(of(statistics(that(are(not(resistant((sample(averages).(((2 ((f) The(ranked(data(are:((((((((((((((( ( ( Rank metabexpend group (((((((((((((((((((((((((((((((((((((((((((((( 1 18.8 nontrauma 2 20.0 nontrauma 3 20.1 nontrauma 4.5 20.9 nontrauma 4.5 20.9 nontrauma 6 21.4 nontrauma 7 22.0 trauma 8 22.7 nontrauma 9 22.9 nontrauma 10 23.0 trauma 11 24.5 trauma 12 25.8 trauma 13 30.0 trauma 14 37.6 trauma 15 38.5 trauma Average(of(ranks(for(group(1((nontrauma):(((1+2+3+4.5+4.5+6+8+9)(/(8(=(39(/(8(=(4.875(Average(of(ranks(for(group(2((trauma):(((7+10+11+12+13+14+15)(/(7(=(82(/(7(=(11.714((Difference(in(average(ranks(for(group(2(–(group(1(=(11.714(–(4.875(=(6.839((This(difference(would(be(resistant(because(deleting(large(values(would(have(little(effect(on(the(average(rank(for(a(group,(and(thus(would(not(impact(the(difference(between(average(ranks.(2.( (a)(( Thi s(is(an (ob servational(s tudy.(( Inferences(are(limited(to(statements(abo ut(the(areas((A(and(B)( studi ed.((No(cause(and(effect(may(be(inferred(about(differences(in(the(two(coastal(areas.((( (b)( To(estimate(ZAZBµµ−,(the(difference(in(population(means(in(the(logSscale,(we(use(the(difference(in(the(sample(averages(in(the(logSscale,(87.088.175.2 =−=−ABZZ(log(crabs.((( (c)(387.287.0==−eeABZZcrabs((( (d)((ABZZe−estimates(the(ratio(of(the(population(median(crab(count(for(coastal(area(B(to(the(population(median(crab(count(for(coastal(area(A,(or(!"!#$%&'"(!!"#$%&!!"#$!!"#$%!!"#!!"#$%#&!!"#!!!!"!#$%&'"(!!"#$%&!!"#$!!"#$%!!"#!!"#$%#&!!"#!!!=!!!!(.((( (e)(1511.122020)2.1()120()1.1()120(22=−+−+−=ps(with(df*=*n1*+*n2*.*2(=(38.(((( ( SE(ABZZ −)(=(⎟⎟⎠⎞⎜⎜⎝⎛+=+2012011511.11121nnsp=(0.36401(( ( Use()975.0(30t(=(2.042(since(df(=(38(is(not(in(Table(A.2.((( ()()975.0(38 ABABZZSEtZZ −±−=(0.87(±(2.042(0.3641)(=(0.87(±(0.7433(or((0.1267,(1.6133)(( ( The(95%(confidence(interval(for(ZAZBµµ−(is((0.13,(1.61)(log(crabs.((( (f)( Lower(limit(for(CI(=(e0.1267(=(1.1351((( ( Upper(limit(for(CI(=(e1.6133(=(5.0193((( ( 95%(CI(of((!"!#$%&'"(!!"#$%&!!"#$!!"#$%!!"#!!"#$%#&!!"#!!!!"!#$%&'"(!!"#$%&!!"#$!!"#$%!!"#!!"#$%#&!!"#!!!=!!!!((is((1.14,(5.02)(((3 ( (g)( There(is(at(least(modest(evidence(of(a(difference(between(the(median(crab(count(for(coastal(area(A(and(the(median(crab(count(for(coastal(area(B.((The(population(median(number(of(crabs(from(coastal(area(B(is(estimated(to(be(2.39(times(larger(than(the(population(median(number(of(crabs(from(coastal(area(A,(with(a(95%(confidence(interval(of((1.14,(5.02).(((3.( Estimating(CA(population(under(different(priming(treatments(–(note(that(this(is(an(example(of(a(study(that(does(not(provide(sufficient(information(to(fully(understand(the(((( (a)( The(study(is(a(randomized(experiment.((Cause(and(effect(can(be(inferred,(but(the(study(may(only(be(generalized(to(the(group(of(students(who(participated(in(the(study.((( (b)(