274 Chapter 13. Chapter 13


 Charles Hunter
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1 74 hpter 3 hpter 3 3. () ounts will be obtined from the smples so th problem bout compring proportions. (b) h n observtionl study compring rndom smples selected from two independent popultions. 3. () cores will be obtined from the smples so th problem bout compring mens (verge scores). (b) h n experiment becuse the reserchers n imposing tretment nd mesuring response vrible. ince these re volunteers we will not be ble to generlize the results to ll gmers. 3.3 () wo smples. he two segments re used by two independent groups of children. (b) Pired dt. he two segments re both used by ech child. 3.4 () ingle smple. he smple men will be compred with the known concentrtion. (b) wo smples. he men concentrtion in bekers with the new method will be compred to the men concentrtion in different bekers with the old method 3.5 () H : = : >, where nd re the men improvement of reding bility of the tretment nd control group respectively. (b) he tretment group slightly leftskewed with greter men nd smller stndrd devition ( x =5.48, s=.) thn the control group ( x =4.5, s= 7.5). he htogrms below show no serious deprtures from ormlity for the tretment group (on the left) nd one unusully lrge score for the control group (on the right) ount 3 ount DRP score (tretment group) DRP score (control group) 8 he boxplot (on the left below) lso shows tht the medin DRP score higher for the tretment group nd the IQR higher for the control group. otice tht the unusully high score not identified s n outlier by initb. he combined orml probbility plot (on the right below) shows n overll liner trend for both sets of scores, so the orml condition stfied for both groups.
2 ompring wo Popultion Prmeters retment Percent ontrol DRP score DRP score 6 8 (c) Rndomiztion ws not possible, becuse exting clsses were used. he resercher could not rndomly ssign the students to the two groups without drupting clsses. 3.6 () he two popultions re brestfeeding women nd other women. We wnt to test H : = : <, where nd re the men percent chnge in minerl content of the spines over three months for brestfeeding nd other mothers, respectively. (b) Dotplots (on the left) nd boxplots (on the right) re shown below. oth dtributions pper to be resonbly orml. rest rest Other inerl content (% chnge).6 3. Other inerl content (% chnge) restfeeding mothers hve lower men minerl content ( x = 3.587, s=.56) with more vribility thn other mothers ( x =.34, s=.97). (c) h n observtionl study so we cnnot mke cuse nd effect conclusion, but th effect certinly worth investigting becuse there ppers to be difference in the two groups of mothers for some reson () he hypotheses should involve nd (popultion mens) rther thn x nd x (smple mens). (b) he smples re not independent. We would need to compre the scores of the boys to the scores for the girls. (c) We need the Pvlue to be smll (for exmple, less thn.5) to reject. A lrge Pvlue like th gives no reson to doubt H. H 3.8 () Answers will vry. Exmine rndom digits, if the digit even then use Design A, otherwe use Design. Once you use design 3 dys, stop nd use the other design for the remining dys in the study. he first three digits re even, so the first three dys for using Design A would be dys,, nd 3. (ote, if Design A used when the digit odd, then the first three dys for using Design A re dy 5, dy 6, nd dy 8.) (b) Use twosided lterntive ( H : A = : A ), becuse we (presumbly) hve no prior suspicion tht one
3 76 hpter 3 design will be better thn the other. (c) oth smple sizes re the sme ( n = n = 3 ), so the pproprite degrees of freedom would be df = 3 = 9. (d) ecuse.45 < t <.5, nd the lterntive twosided, ble tells us tht.4 < Pvlue <.5. (oftwre gives P =.485.) We would reject H nd conclude tht there difference in the men dily sles for the two designs. 3.9 () We wnt to test H : = : >. he test stttic t =.3,. < Pvlue <. with df = (I clcultor gives P vlue =.3 with df = nd initb gives Pvlue =.3 with df=37). At the 5% significnce level, it does not mtter which method you use to obtin the Pvlue. he Pvlue (rounded to.3) less thn.5, so the dt give good evidence tht the new ctivities improve the men DRP score. (b) A 95% confidence intervl for ± = (.97, 8.94) with df = ; (.33, 8.68) on I clcultor with df = 37.86; nd (.637, ) using initb with df = 37. We estimte the men improvement in reding bility using the new reding ctivities compred to not using them over n 8week period to be between.3 nd 8.68 points. 3. We wnt to test H : = : <. he test stttic t = 8.5, Pvlue <.5 with df = (the I clcultor nd initb give Pvlues very close to ). he smll Pvlue less thn ny resonble significnce level, sy %, so the dt give very strong evidence tht nursing mothers on verge lose more bone minerl thn other mothers. (b) A 95% confidence intervl for O ± = ( 4.86,.95) with df = ; ( 4.86,.986) on I clcultor with df =66. (see the screen shots below); nd ( 4.863,.98633) using initb with df = 66. We estimte the difference in the men chnge in bone minerl for brestfeeding mothers when compred to other mothers to be between bout 3% nd 5%, with brestfeeding mothers losing more bone density. 3. () ecuse the smple sizes re so lrge, the t procedures re robust ginst non ormlity in the popultions. (b) A 9% confidence intervl for F ± = ($4.68, ) using df = ; ($43.54, $634.7) using df = 6; ($43.6, ) using df = 49.. We re 9% confident
4 ompring wo Popultion Prmeters 77 tht the difference in men summer ernings between $43.6 nd $ higher for men. (c) he smple not relly rndom, but there no reson to expect tht the method used should introduce ny bis. h known s systemtic smpling. (d) tudents without employment were excluded, so the survey results cn only (possibly) extend to employed undergrdutes. Knowing the number of unreturned questionnires would lso be useful. hese students re from one college, so it would be very helpful to know if th student body representtive of some lrger group of students. It very unlikely tht you will be ble to generlize these results to ll undergrdutes. 3. Answers will vry. 3.3 () We wnt to test H : R = W : R > W, where R nd W re the men percent chnge in polyphenols for men who drink red nd white wine respectively. he test stttic t = 3.8 with df = 8 nd.5 < Pvlue <.5. (b) he vlue of the test stttic the sme, but df = 4.97 nd the Pvlue.85 (initb gives. with df = 4). he more complicted degrees of freedom give smller nd less conservtive Pvlue. (c) h study ppers to hve been welldesigned experiment, so it does provide evidence of custion. 3.4 () A 95% confidence intervl for R W ± = * (.8%, 8.45%). (b) With df = 4.97, t =.3 nd the confidence intervl.3% to 8.%. (initb gives.34% to 8.9% with df = 4.) here very little difference in the resulting confidence intervls. 3.5 () We wnt to test H : = : >, where nd re the men knee velocities for skilled nd novice femle competitive rowers, respectively. he test stttic t = nd the Pvlue =.5. ote tht the twosided Pvlue provided on the A output, so to get the pproprite Pvlue for the onesided test use.4/ =.5. ince.5 <., we reject H t the % level nd conclude tht the men knee velocity higher for skilled rowers. (b) Using df = 9., the criticl vlue t* =.86 nd the resulting confidence intervl for (.498,.8475). With 9% confidence, we estimte tht skilled femle rowers hve men ngulr knee velocity of between.498 nd.847 units higher thn tht of novice femle rowers. (c) king the conservtive pproch with ble, df = 7 nd the criticl vlue t* =.895. ince.895 >.86, the mrgin of error would be lrger, so the confidence intervl would be slightly wider. 3.6 () he msing t stttic t = (b) We wnt to test H : = :, where nd re the men weights of skilled nd novice femle competitive rowers, respectively. he test stttic t =.543 nd the Pvlue =.665. ince.665 >.5, we cnnot reject H t the 5% level. here no significnt
5 78 hpter 3 difference in the men weights for skilled nd novice rowers. (c) he more conservtive pproch would use df = 7. he t dtribution with df = 7 hs slightly hevier tils thn the t dtribution with df =., so the conservtive Pvlue would be lrger. 3.7 () wosmple t test. (b) Pired t test. (c) Pired t test. (d) wosmple t test. (e) Pired t test. 3.8 () he summry tble shown below. he only vlues not given directly re the stndrd devitions, which re found by computing s= E. (b) Use df = 9. Group retment n x s IDX Untreted (c) h completely rndomized design with one control group nd one tretment group. he esiest wy to crry out the rndomiztion might be to number the hmsters (or their individul cges) from to. Use the R pplet nd put blls in the popultion hopper. elect blls from the hopper. he hmsters with these numbers will be injected with IDX. he other hmsters will serve s the control group. 3.9 () Yes, the test stttic for testing H: = : > t = With either df = 9 or df =.5, we hve significnt result (Pvlue <. or Pvlue <.5, respectively), so there strong evidence tht IDX prolongs life. (b) If using df = 9, the 95% confidence intervl for ± = (4., 4.88). With 95% confidence we estimte tht IDX hmsters live, on verge, between 4. nd 4.88 dys longer thn untreted hmsters. If using df =.5, the intervl (4.49, 4.5). 3. () h twosmple t stttic, compring two independent groups (supplemented nd control). (b) Using the conservtive df = 5, t =.5 would hve Pvlue between.3 nd.4, which (s the report sid) not significnt. 3. We wnt to test H : = :. he test stttic 4..3 t = 3.74 nd the Pvlue between. nd. (df = 5) or (df =.95), greeing with the stted conclusion ( significnt difference). 3. () hese re pired t stttics: For ech bird, the number of dys behind the cterpillr pek ws observed, nd the t vlues were computed bsed on the pirwe differences between the first nd second yers. (b) For the control group, df = 5, nd for the supplemented group, df = 6. (c) he control t not significnt (so the birds in tht group did not dvnce their lying dte in the second yer ), while the supplemented group t significnt with onesided Pvlue =.95 (so those birds did chnge their lying dte).
6 ompring wo Popultion Prmeters Answers will vry, but here n exmple. he difference between verge femle (55.5) nd mle (57.9) selfconcept scores ws so smll tht it cn be ttributed to chnce vrition in the smples (t =.83, df = 6.8, Pvlue =.4). In other words, bsed on th smple, we hve no evidence tht men selfconcept scores differ by gender. 3.4 () If the loggers hd known tht study would be done, they might hve (consciously or subconsciously) cut down fewer trees, in order to reduce the impct of logging. (b) Rndom ssignment llows us to mke cuse nd effect conclusion. (c) We wnt to test H : U = L : U > L, where U nd L re the men number of species in unlogged nd logged plots respectively. he test stttic t =. with df = 8 nd.5 < P vlue <.5. Logging does significntly reduce the men number of species in plot fter 8 yers t the 5% level, but not t the % level. (d) A 9% confidence intervl for U L ± = (.46, 7.). (initb gives n intervl from to 7.73.) We re 9% confident tht the difference in the mens for unlogged nd logged plots between.46 nd 7. species. 3.5 Let p denote the proportion of mice redy to breed in good corn yers nd p denote the proportion of mice redy to breed in bd corn yers. he smple proportions re p ˆ = 54 7 =.75 nd p ˆ = 7 =.588, nd the stndrd error E = A 9% confidence intervl for p p ± = (.58,.3753). With 9% confidence, we estimte tht the percent of mice redy to breed in the good corn yers between 5.% lower nd 37.5% higher thn in the bd yers. hese methods cn be used becuse the popultions of mice re certinly more thn times s lrge s the smples, nd the counts of successes nd filures re t lest 5 in both smples. We must view the trpped mice s n R of ll mice in the two res () he smple proportion of women who felt vulnerble p ˆW =.48, nd the corresponding smple proportion for men p ˆ =.73. (b) A 95% confidence intervl for the difference p pw (.73.48) ±.96 + = (.773,.487). With 95% confidence, we estimte the percent of men who feel vulnerble in th re to be bout.8 to.4 bove the proportion of women who feel vulnerble. otice tht not included in our confidence intervl, so there significnt difference between these proportions t the 5% level.
7 8 hpter () A 95% confidence intervl for p ±.96 = (.435,.4486) With 95% confidence, we estimte the percent of crs tht go fster thn 65 mph when no rdr present between 43.5% nd 44.86%. (b) A 95% confidence intervl for p pr (.44.3) ±.96 + = (.,.38). With 95% confidence, we estimte the percent of crs going over 65 mph between.% nd 3.8% higher when no rdr present compred to when rdr present. (c) In cluster of crs, where one driver s behvior might ffect the others, we do not hve independence; one of the importnt properties of rndom smple A 95% confidence intervl for p ±.96 = (.693,.657). We re % confident tht between 6% nd 65% of ll dults use the internet. (b) A 95% confidence intervl for pu p (.79.38) ±.96 + = (.3693,.456). We re % confident tht the difference in the proportion of internet users nd nonusers who expect businesses to hve Web sites between.37 nd Let p = the proportion of students who use illegl drugs in schools with drug testing progrm nd p = the proportion of students who use illegl drugs in schools without drug testing progrm. We wnt to test H: p = p : p < p. he combined smple 7+ 7 proportion pˆ c =.3 nd the test stttic z = 3.53, with Pvlue =.. ince. <.,.3(.3)( ) we reject H. here extremely strong evidence tht drug use mong thletes lower in schools tht test for drugs. here should be some concern expressed bout the condition of two independent simple rndom smples, becuse these two smples my not be representtive of similr schools. 3.3 () he ptients were rndomly ssigned to two groups. he first group of 649 ptients received only spirin nd the second group of 65 ptients received spirin nd dipyridmole. (b) We wnt to test H: p = pversus H : p p. he combined smple proportion pˆ c =.nd the test stttic z =.73, with (.)( ) Pvlue =.64. ince.64 <., there very strong evidence tht there significnt difference in the proportion of strokes between spirin only nd spirin plus dipyridmole. (c) A 95% confidence intervl for p p
8 ompring wo Popultion Prmeters (.4.) ±.96 + = (.3,.97). We re 95% confident tht the difference in the proportion of deths for the two tretment groups between. nd.. otice tht in the confidence intervl, so we do not hve evidence of significnt difference in the proportion of deths for these two tretments t the 5% level. (d) A ype I error committed if the reserchers conclude tht there significnt difference in the proportions of strokes with these two tretments, when in fct there no difference. A ype II error committed if the reserchers conclude tht there no difference in the proportions of strokes with these two tretments, when in fct there difference. A ype II error more serious becuse no ptients would be hrmed with ype I error, but ptients suffer unnecessrily from strokes if the best tretment not recommended. 3.3 For computer ccess t home, we wnt to test H : p = pw : p pw. he combined smple proportion pˆ c =.65 nd the test stttic z =., with Pvlue =.34. he sme hypotheses re.65(.65)( 3+ 96) used for the proportions with computer ccess t work. he combined smple proportion pˆ c =.6 nd the test stttic z = 3.9, (.6)( 3+ 96) with Pvlue <.4. ince the Pvlue below ny resonble significnce level, sy %, we hve very strong evidence of difference in the proportion of blcks nd whites who hve computer ccess t work. 3.3 () Let p = the proportion of women got pregnnt fter in vitro fertiliztion nd intercessory pryer nd p = the proportion of women in the control group who got pregnnt fter in vitro fertiliztion. We wnt to test H: p = p : p p. he combined smple 44 + proportion pˆ c =.3846 nd the test stttic z = 3., with Pvlue =.4. ince.4 <., we.3846(.3846)( ) reject H. h very strong evidence tht the observed difference in the proportions of women who got pregnnt not due to chnce. (b) h study shows tht intercessory pryer my cuse n increse in pregnncy. However, it uncler if the women knew tht they were in tretment group. If they found out tht other people were prying for them, then their behviors my hve chnged nd there could be mny other fctors to explin the difference in the two proportions. (c) A ype I error would be committed if reserchers concluded tht the proportions of pregnncies re different, when in fct they re the sme. h my led mny couples to seek intercessory pryer. A ype II error would be committed if reserchers concluded tht the proportions re not different, when in fct they re different. ouples would fil to tke dvntge of helpful technique to improve their chnces of hving bby. For couples who re interested in hving bby, ype II error clerly more serious.
9 8 hpter () H should refer to popultion proportions p nd p, not smple proportions. (b) onfidence intervls ccount only for smpling error () Let p = the proportion of households where no messge ws left nd contct ws eventully mde nd p = the proportion of household where messge ws left nd contct ws eventully mde. We wnt to test H: p = p : p < p. he combined smple 58 + proportion pˆ c =.66 nd the test stttic z =.95, with Pvlue =.56. Yes, t the 5% level, there.66(.66)( + 9) good evidence tht leving messge increses the proportion of households tht re eventully contcted. (b) Let p = the proportion of households where no messge ws left but the survey ws completed nd p = the proportion of household where messge ws left nd the survey ws completed. We wnt to test H: p = p : p < p. he combined smple proportion pˆ c =.47 nd the test stttic z =.8, with Pvlue =.3. Yes, t the 5% level,.47(.47)( + 9) there good evidence tht leving messge increses the proportion of households who complete the survey. (c) A 95% confidence intervl for the difference p pwhen deling with eventul contct (.8,.3). A 95% confidence intervl for the difference p pwhen deling with completed surveys (.39,.). Although these effects do not pper to be lrge, when you re deling with hundreds (or thousnds) of surveys nything you cn do to improve nonresponse in the rndom smple useful () H: p = p : p > p where p the proportion of ll HIV ptients tking plcebo tht develop AID nd p the proportion of ll HIV ptients tking AZ tht develop AID. he popultions re much lrger thn the smples, nd npˆ, n( pˆ ), npˆ, n ( p) c c c 38 7 re ll t lest 5. (b) he smple proportions re p ˆ = =.874, p ˆ = =.39, nd p ˆc =.63. he test stttic z =.93, with Pvlue.63(.63)( ) of.7. here very strong evidence tht significntly smller proportion of ptients tking AZ develop AID thn if they took control. (c) either the subjects nor the reserchers who hd contct with them knew which subjects were getting which drug A ype I error would be committed if reserchers concluded tht the tretment more effective thn plcebo, when in fct it not. A consequence tht ptients would be tking AZ nd perhps suffering from side effects from the mediction tht not helpful. A ype II error would be committed if reserchers conclude tht there no difference in the success of ˆc
10 ompring wo Popultion Prmeters 83 AZ nd plcebo, when in fct there difference. he consequence tht ptients would not get the best possible tretment. A ype II error more serious in th sitution becuse we wnt ptients to get the best possible tretment () he number of orders completed in 5 dys or less before the chnges ws X =.6 = 3. With p ˆ =.6 nd Ep ˆ.59, the 95% confidence intervl for p (.9,.8). (b) After the chnges, X =.9 = 8. With p ˆ =.9 nd Ep ˆ., the 95% confidence intervl for p (.8584,.946). (c) he stndrd error of the difference in the proportions Ep ˆ nd the 95% confidence intervl for pˆ.335 p p (.6743,.857) or bout 67.4% to 8.6%. o, the confidence intervls re not directly relted. Ech intervl bsed on different smpling dtribution. Properties of the smpling dtribution of the difference cn be obtined from properties of the individul smpling dtributions in prts () nd (b), but the upper nd lower limits of the intervls re not directly relted () We must hve two simple rndom smples of highschool students from Illino; one for freshmn nd one for seniors. (b) he smple proportion of freshmn who hve used 34 nbolic steroids p ˆ F =.3. ince the number of successes (34) nd the number of 679 filures (645) re both t lest, the z confidence intervl cn be used. A 95% confidence intervl for p F.3±.96 = (.35,.7). We re 95% confident 679 tht between.35% nd.7% of highschool freshmn in Illinios hve used nbolic steroids. 4 (c) he smple proportion of seniors who hve used nbolic steroids p ˆ = otice tht.76 flls in the 95% confidence intervl for plusible vlues of pf from prt (b), so there no evidence of significnt difference in the two proportions. he test stttic for forml hypothes test z =.54 with Pvlue = We wnt to test H: p = p : p p. From the output, z = 3.45 with P vlue =.6, showing significnt difference in the proportion of children in the two ge groups who sorted the products correctly. A 95% confidence intervl for p p (.5579, ). With 95% confidence we estimte tht between 5.4% nd 5.3% more 6 to 7 yerolds cn sort new products into the correct ctegory thn 4 to 5yerolds () he two smple proportions re p ˆW =.3 nd p ˆ =.467. (b) We 53 8 wnt to test H : pw = p : pw p. he combined smple proportion pˆ c =.368 nd the test stttic z = 3.89, (.368)( ) with Pvlue <.. ince the Pvlue less thn ny resonble significnce level, sy
11 84 hpter 3 %, we reject H. We hve very strong evidence tht there significnt difference between the proportions of injured inline skters who sustin wrt injuries with nd without wrt gurds. (hese re Rs of ll people injured while inline skting with nd without wrt gurds, so we cn only mke our inference to these popultions.) (c) he proportion of nonresponse 45/6 =.84 or bout.84%. (d) Yes. uppose tht ll 45 people who were not interviewed were injured while wering wrt gurds. (h unlikely, but we re looking t the extreme cse to see if our nswer could chnge.) he proportion of injuries with wrt gurds now pˆ W =.54. he test stttic would become z =.49 with P vlue of.36, which not significnt. AE LOED! () ) We wnt to test H: = : >, where the men drivethru service time for cdonld s (urger King) in nd the men percent drivethru service time for cdonld s (urger King) in yer fter the incentive/rewrds progrms were implemented. () Using initb with df = 93, the 95% confidence intervl for ± = (6.574,.36). With 95% confidence we estimte tht the verge drivethru service time decresed between 6.5 nd. seconds fter the incentive/rewrds progrm ws implemented t cdonld s. (3) We wnt to test H : = : >, where the men drivethru service time for cdonld s in 4 nd the men percent drivethru service time for co ell in he test stttic t = 4.6 with df = (the lrgest vlue below in ble ) or 6 (with initb) nd Pvlue <.5. Yes, these dt provide extremely strong evidence tht drivethru service times t co ell were significntly fster thn those t cdonld s. (4) We wnt to test H: p = p : p < p where p the proportion of ll orders in tht were filled ccurtely nd correct chnge ws given nd p the proportion of orders in tht were filled ccurtely nd correct chnge ws given. he popultions re npˆ, n pˆ, npˆ, n p re ll t lest 5. (b) he much lrger thn the smples, nd c c c smple proportions re p ˆ =.8, p ˆ =.884, nd pˆ c = =.848. he test stttic z = 3.43, with Pvlue of.3. Yes, there.848(.848)( ) ws significnt improvement in ccurcy between nd. In short, the difference observed from these two independent smples (or something more extreme) would only occur bout 3 times in, trils. We hve very convincing evidence tht the observed difference not due to chnce, but to some other fctor, perhps better trining by the mngers! (5) Let p denote the proportion of inccurte for hickfila in nd p denote the proportion inccurte orders t cdonld s in. A 95% confidence intervl for ±.96 + = (.95,.57). We re 95% ˆc p p
12 ompring wo Popultion Prmeters 85 confident tht the difference in the proportion of inccurte orders in for the two fst food resturnts between.9 nd.. otice tht not in the confidence intervl, so there significnt difference in the proportion of inccurte orders t the two resturnts. 3.4 () h twosmple t test. he two groups of women re (presumbly) independent. (b) df = 45 = 44. (c) he smple sizes re lrge enough, n = n = 45, tht the verges will be pproximtely orml, so the fct tht the individul responses do not follow orml dtribution hs little effect on the relibility of the t procedure. 3.4 () h n observtionl study becuse the reserchers simply observed the rndom smples of women; they did not impose ny tretments. (b) We wnt to test H : p = p versus H : p > p. he combined smple proportion pˆ c =.7448 nd the test stttic z = 5., with Pvlue <.. We hve very strong.7448(.7448)( + 7) evidence tht smller proportion of femle Hpnic drivers wer set belts in oston thn in ew York We wnt to test H : ph = pw : ph pw. he combined smple proportion pˆ c =.545 nd the test stttic z =.86, (.545)( ) with Pvlue = ince.3898 >.5, there not significnt difference between Hpnic nd white drivers. For the size of the difference, construct 95% (or other level) confidence intervl. A 95% confidence intervl for ph pw ( ) ±.96 + = (.8,.398). With 95% confidence we estimte the difference in the proportions for Hpnic nd white drivers who were set belts to be between. nd.4. otice tht in the 95% confidence intervl, so we would conclude tht there no difference t the 5% significnce level We wnt to test H : = : >, where the men difference (post pre) for the tretment group nd the men difference (post pre) for the control group. he boxplots (on the left below) show tht the dtributions re roughly symmetric with no outlier, nd the orml probbility plots (on the right below) show liner trends which indicte tht the orml dtribution resonble for these dt.
13 86 hpter diff Percent diff Differences (Post  Pre) Differences (Post  Pre) 5 he test stttic t = , with.5 < Pvlue <.5 nd df = 7 (initb gives Pvlue of.39 with df=3). he Pvlue less thn.5, so the dt give good evidence tht the positive subliminl messge brought bout greter improvement in mth scores thn the control. (b) A 9% confidence intervl for ± = (.3, 6.7) with df = 7; (.35, 6.65) using initb with df = 3. With 9% confidence, we estimte the men difference in gins to be.35 to 6.65 points better for the tretment group. (c) h ctully repeted mesures design, where two mesurements (repeted mesures) re tken on the sme individuls. ny students will probbly describe th design s completely rndomized design for two groups, with twt insted of mesuring one response vrible on ech individul, two mesurements re mde nd we compre the differences (improvements) () A 99% confidence intervl for p pw ( ) ± = (.465,.3359). Yes, becuse the 99% confidence intervl does not contin. (b) We wnt to test H : = W versus H : W. he test stttic t =.87, with Pvlue close to (initb reports Pvlue of.387 with df = 777.) ince.4 >., the difference between the men scores of men nd women not significnt t the % level () tched pirs t. (b) wosmple t. (c) wosmple t. (d) tched pirs t. (e) tched pirs t () A 99% confidence intervl for OP WI ± = (6.55, 69.45). (b) he fct tht the smple sizes re both so lrge (36 nd 395).
14 ompring wo Popultion Prmeters () We wnt to test H : P = : P >. he test stttic t =.7, with Pvlue close to.5. (initb reports Pvlue of with df = 44.) ince.5 >.5, we do not hve strong evidence tht pets hve higher men cholesterol thn clinic dogs. (b) A 95% confidence intervl for P ± = ( 4.579, 5.579). initb gives ( , ). With 95% confidence, we estimte the difference in the men cholesterol levels between pets nd clinic doges to be between 4 nd 53 mg/dl. (c) A 95% confidence intervl 68 for p 93±.6 = (65.58,.479). initb gives (65.534,.466). With 6 95% confidence, we estimte the men cholesterol level in pets to be between 65.5 nd.5 mg/dl. (d) We must hve two independent rndom smples to mke the inferences in prts () nd (b) nd rndom smple of pets for prt (c). It unlikely tht we hve rndom smples from either popultion () he two smple proportions re p ˆ = for residents of congested streets nd p ˆ = for residents of bypss streets. he difference pˆ ˆ p =.5 with stndrd error of E = (b) he hypotheses re H : p = p : p < p. he lterntive reflects the resonble expecttion tht reducing pollution might decrese wheezing. (c) he combined smple proportion pˆ c =.6 nd the test stttic z = 4.85, (.6)( ) with Pvlue <.. A sketch of the dtribution of the test stttic, ssuming H true, shown below orml density curve est stttic (z).5 5. otice tht reference line provided t 4.85 to illustrte how fr down in the lower til of the dtribution tht th vlue of the test stttic locted. he Pvlue tells us the chnce of observing test stttic of 4.85 or something smller if H true. As you cn see there lmost no chnce of th hppening, so we hve very convincing evidence tht the percent of residents reporting improvement from wheezing higher for residents of bypss streets. (d) he 95% confidence intervl, using the stndrd error from prt (b), hs mrgin of error =.68. hus, the 95% confidence intervl.5 ±.68 = (.,
15 88 hpter 3.838). he percentge reporting improvement ws between 8% nd % higher for bypss residents. (e) here my be geogrphic fctors (e.g., wether) or culturl fctors (e.g., diet) tht limit how much we cn generlize the conclusions. 3.5 () A 99% confidence intervl for ph p (.7.4) ± = (.99,.49). With 99% confidence, the percentge of blcks between 4.9% nd 9.9% higher for nonhousehold providers. Yes, the difference significnt t the % level becuse the 99% confidence intervl does not contin. (b) A 99% confidence intervl for H.6. ± = (.7944,.456), using df =. (initb gives (.7948,.4588) with df =456.) With 99% confidence, the men number of yers of school for nonhousehold workers between.4 nd.79 yers higher thn household providers. Yes, the difference significnt t the % level, becuse not included in the 99% confidence intervl.
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