8: Large-Sample Estimation

Transcription

1 8: Lrge-Smple Estimtio 8. The mrgi of error i estimtio provides prcticl upper boud to the differece betwee prticulr estimte d the prmeter which it estimtes. I this chpter, the mrgi of error is.96 (stdrd error of the estimtor). σ 8.3 For the estimte of μ give s x, the mrgi of error is.96 SE = = b = c = 8.4 Refer to Exercise 8.3. As the popultio vriceσ icreses, the mrgi of error lso icreses. σ 8.5 The mrgi of error is.96 SE =.96, where σ c be estimted by the smple stdrd devitio s for lrge vlues of = b = c = 8.6 Refer to Exercise 8.5. As the smple size icreses, the mrgi of error decreses. o 8.3 The poit estimte of μ is x = 39.8 d the mrgi of error with s = 7. d = 5 is σ s SE = =.96 = x 8.7 The poit estimte for p is give s pˆ = =.5 d the mrgi of error is pproximtely pq ˆˆ.5(.49) = 9 = σ σ 8.4 x ± z.5 = x ± ±.58 = 34 ±.45 or 3.55 < μ < σ σ 5 x ± z = x ± ±.645 = 49 ±.457 or < μ < b.5 σ σ.48 c x ± z.5 = x ± ±.96 = 66.3 ±.37 or < μ < σ 8.7 The width of 95% cofidece itervl for μ is give s.96. ece, Whe b Whe c Whe Refer to Exercise =.96 = 3.9. =, the width is.96 =.386 =.77. =, the width is.96 =.98 = =, the width is

2 Whe the smple size is doubled, the width is decresed by. b Whe the smple size is qudrupled, the width is decresed by 4 =. σ 8.9 A 9% cofidece itervl for μ is x ±.645. ece, its width is σ.645 =.645 = (.645) = 3.9 σ b A 99% cofidece itervl for μ is x ±.58. ece, its width is σ.58 =.58 = (.58) = 5.6 c Notice tht s the cofidece coefficiet icreses, so does the width of the cofidece itervl. If we wish to be more cofidet of eclosig the ukow prmeter, we must mke the itervl wider. 8.3 A pproximte 95% cofidece itervl for p is pq ˆˆ.54(.46) pˆ ±.96 =.54 ±.96 =.54 ±.49 4 or.49 < p <.589. b A pproximte 95% cofidece itervl for p is pq ˆˆ.3(.7) pˆ ±.96 =.3 ±.96 =.3 ± or.5 < p < The 9% cofidece itervl for p is pq ˆˆ.39(.6) pˆ ±.645 =.39 ±.645 =.39 ±.5 or.365 < p <.45. b The 9% cofidece itervl for p is pq ˆˆ.53(.47) pˆ ±.645 =.53 ±.645 =.53 ±.6 or.54 < p < The 99% cofidece itervl for μ is s.73 x ±.58 = 98.5 ±.58 = 98.5 ±.65 or < μ < b Sice the possible vlues for μ give i the cofidece itervl does ot iclude the vlue μ = 98.6, it is ot likely tht the true verge body temperture for helthy hums is 98.6, the usul verge temperture cited by physicis d others. 8.4 Similr to previous exercises. The 9% cofidece itervl for μ μis pproximtely s s ( x x) ± ± ±.33 or.3 < <.368 ( μ μ ) Itervls costructed i this mer will eclose μ μ 9% of the time. ece, we re firly μ μ. certi tht this prticulr itervl ecloses

3 8.44 Similr to previous exercises. The 95% cofidece itervl for μ μis pproximtely ( x x ) s s ±.96 (.6) (.9) ± ±.5 or < < 9.5 ( μ μ) Itervls costructed i this mer will eclose ( μ μ) ece, we re firly certi tht this prticulr itervl ecloses ( μ μ ) 95% of the time i repeted smplig The poit estimte of the differece μ μis x x = 53, 659 5, 4 = 67 d the mrgi of error is σ σ = b Sice the mrgi of error does ot llow the estimte of the differece μ μto be egtive the lower limit is = 74.9 it is likely tht the me for chemicl egieerig mjors is lrger th the me for computer sciece mjors The 95% cofidece itervl for μ μis pproximtely s s ( x x) ± ± ±.48 or.58 < <.3 ( μ μ ) b Sice the cofidece itervl i prt hs two egtive edpoits, it does ot coti the vlue μ μ =. ece, it is ot likely tht the mes re equl. It ppers tht there is rel differece i the me tempertures for mles d femles. x x 8.54 Clculte p ˆ = =.4d p ˆ = =.44. The pproximte 95% cofidece itervl is pq ˆˆ pq ˆˆ ( pˆ pˆ) ±.96.4(.59).44(.56).4.44 ± ±.4 or.7 < <. ( p p ) Sice the vlue p p = is i the cofidece itervl, it is possible tht p = p. You should ot coclude tht there is differece i the proportio of Republics d Democrts who fvor metioed the ecoomy s importt issue i the electios.

4 8.56 Clculte p 4 ˆ = = d p ˆ = 55 =.98. The pproximte 95% cofidece itervl is 55 pq ˆˆ pq ˆˆ ( pˆ pˆ) ±.96.99(.9).98(.8) ± ±.35 or.44 < <.6 ( p p ) Sice the vlue p p = is i the cofidece itervl, it is possible tht p = p. You should ot coclude tht there is differece i the proportio of fs versus o-fs who fvor mdtory drug testig. x The poit estimte for p is give s pˆ = = =.56 d the mrgi of error is pproximtely 4 pq ˆˆ = 4 = 3 b Clculte p ˆ = =.35 d p ˆ = =.56. The pproximte 95% cofidece itervl is 3 4 pq ˆˆ pq ˆˆ ( pˆ pˆ) ±.96.35(.6875).56(.439) (.35.56) ± ±. or.4696 < ( p p) < It is ecessry to fid the smple size required to estimte certi prmeter to withi give boud with cofidece ( α ). Recll from Sectio 8.5 tht we my estimte prmeter with ( α ) cofidece withi the itervl (estimtor) ± z α (std error of estimtor). Thus, z α (std error of estimtor) provides the mrgi of error with ( α ) cofidece. The experimeter will specify give boud B. If we let z α (std error of estimtor) B, we will be ( α ) tht the estimtor will lie withi B uits of the prmeter of iterest. cofidet For this exercise, the prmeter of iterest is μ, B =.6 d α =.95. ece, we must hve σ (.7) = or 43 pq 8.69 For this exercise, B =.4for the biomil estimtor ˆp, where SE ( pˆ ) =. Assumig mximum vritio, which occurs if p =.3 (sice we suspect tht. < p <.3 ) d z.5 =.96, we hve pq.96σ pˆ B.96 B.96.3(.7) or 55.4

5 8.7 I this exercise, the prmeter of iterest is μ μ, = =, d σ σ 7.8. The we must hve z (std error of x x ) B α σ σ or = = Similr to Exercise 8.7. (.5)(.5) (.5)(.5) pq pq z or = = The prmeter of iterest is μ μ, the differece i grde-poit verges for the two popultios of studets. Assume tht = =, d σ σ.6 =.36 d tht the desired boud is.. σ σ The or = = 7 studets should be icluded i ech group.

6 9: Lrge-Smple Tests of ypotheses 9.3 The criticl vlue tht seprtes the rejectio d orejectio regios for right-tiled test bsed o z-sttistic will be vlue of z (clled z α ) such tht P( z > z α ) = α =.. Tht is, z. =.33 (see the figure below). The ull hypothesis will be rejected if z >.33. b For two-tiled test with α =.5, the criticl vlue for the rejectio regio cuts off α =.5 i the two tils of the z distributio i Figure 9., so tht z.5 =.96. The ull hypothesis will be rejected if z >.96 or z <.96 (which you c lso write s z >.96 ). c Similr to prt, with the rejectio regio i the lower til of the z distributio. The ull hypothesis will be rejected if z <.33. d Similr to prt b, with α =.5. The ull hypothesis will be rejected if z >.58 or z <.58 (which you c lso write s z >.58 ). 9.4 The p-vlue for right-tiled test is the re to the right of the observed test sttistic z =.5 or p-vlue = P( z >.5) =.8749 =.5 This is the shded re i the figure below.

7 b For two-tiled test, the p-vlue is the probbility of beig s lrge or lrger th the observed test sttistic i either til of the smplig distributio. As show i the figure below, the p-vlue for z =.78 is p-vlue = P z >.78 =.7 =.54 c The p-vlue for left-tiled test is the re to the left of the observed test sttistic z =.8or p-vlue = P z <.8 = Use the guidelies for sttisticl sigificce i Sectio 9.3. The smller the p-vlue, the more evidece there is i fvor of rejectig. For prt, p -vlue =.5 is ot sttisticlly sigifict; is ot rejected. For prt b, p -vlue =.54 is less th. d the results re highly sigifict; should be rejected. For prt c, p -vlue =.35is betwee. d.5. The results re sigifict t the 5% level, but ot t the % level (P <.5). 9.6 I this exercise, the prmeter of iterest is μ, the popultio me. The objective of the experimet is to show tht the me exceeds.3. We wt to prove the ltertive hypothesis tht μ is, i fct, greter the.3. ece, the ltertive hypothesis is d the ull hypothesis is : μ >.3 : μ =.3. b The best estimtor for μ is the smple verge x, d the test sttistic is

8 x μ z = σ which represets the distce (mesured i uits of stdrd devitios) from x to the hypothesized me μ. ece, if this vlue is lrge i bsolute vlue, oe of two coclusios my be drw. Either very ulikely evet hs occurred, or the hypothesized me is icorrect. Refer to prt. If α =.5, the criticl vlue of z tht seprtes the rejectio d o-rejectio regios will be vlue (deoted by z ) such tht P( z > z ) = α =.5 Tht is, z =.645 (see below). ece, will be rejected if z >.645. c The stdrd error of the me is foud usig the smple stdrd devitio s to pproximte the popultio stdrd devitio σ : σ s.9 SE = = = d To coduct the test, clculte the vlue of the test sttistic usig the iformtio cotied i the smple. Note tht the vlue of the true stdrd devitio, σ, is pproximted usig the smple stdrd devitio s. x μ x μ.4.3 z = = =.4 σ s.49 The observed vlue of the test sttistic, z =.4, flls i the rejectio regio d the ull hypothesis is rejected. There is sufficiet evidece to idicte tht μ > Sice this is right-tiled test, the p-vlue is the re uder the stdrd orml distributio to the right of z =.4 : p-vlue = P z >.4 =.9793 =.7 b The p-vlue,.7, is less th α =.5, d the ull hypothesis is rejected t the 5% level of sigificce. There is sufficiet evidece to idicte tht μ >.3. c The coclusios reched usig the criticl vlue pproch d the p-vlue pproch re ideticl. 9.8 Refer to Exercise 9.6, i which the rejectio regio ws give s z >.645 where

9 x μ x.3 z = = s.9 35 Solvig for x we obti the criticl vlue of x ecessry for rejectio of. x.3.9 >.645 x > = b-c The probbility of Type II error is defied s β = P ( ccept whe is flse) Sice the cceptce regio is x.38 from prt, β c be rewritte s β = P x.38 whe is flse = P x.38 whe μ >.3 Severl ltertive vlues of μ re give i this exercise. For μ =.4,.38.4 β = P( x.38 whe μ =.4) = P z.9 35 = P z.4 =.349 For μ =.3, For μ =.5, For μ =.6,.38.3 β = P( x.38 whe μ =.3) = P z.9 35 = P z.63 = β = P( x.38 whe μ =.5) = P z.9 35 = P z.45 = β = P( x.38 whe μ =.6) = P z.9 35 = P z 4.49 d The power curve is grphed usig the vlues clculted bove d is show below Power μ.5.6

10 9. If the irlie is to determie whether or ot the flight is uprofitble, they re iterested i fidig out whether or ot μ < 6 (sice flight is profitble if μ is t lest 6). ece, the ltertive hypothesis is : μ < 6d the ull hypothesis is : μ = 6. b Sice oly smll vlues of x (d hece, egtive vlues of z) would ted to disprove i fvor of, this is oe-tiled test. c For this exercise, =, x = 58, d s =. ece, the test sttistic is x μ x μ 58 6 z = = =.99 σ s The rejectio regio with.5 P z < z =.5. This vlue is z =.645 d will be rejected if z <.645 (compre the right-tiled rejectio regio i Exercise 9.6). The observed vlue of z flls i the rejectio regio d is rejected. The flight is uprofitble. α = is determied by criticl vlue of z such tht 9.3 -b We wt to test the ull hypothesis tht μ is, i fct, 8% gist the ltertive tht it is ot: : μ = 8 versus : μ 8 Sice the exercise does ot specify μ < 8 or μ > 8, we re iterested i two directiol ltertive, μ 8. c The test sttistic is x μ x μ z = = = 3.75 σ s.8 The rejectio regio with α =.5 is determied by criticl vlue of z such tht α α P( z < z) P( z > z ) = =.5 This vlue is z =.96 (see the figure i Exercise 9.3b). ece, will be rejected if z >.96 or z <.96. The observed vlue, z = 3.75, flls i the rejectio regio d is rejected. There is sufficiet evidece to refute the mufcturer s clim. The probbility tht we hve mde icorrect decisio is α = The hypothesis to be tested is : μ = 7.4 versus : μ > 7.4 d the test sttistic is x μ x μ z = = =.63 σ s.9 with p P( z ) -vlue = >.63 =.9957 =.43. To drw coclusio from the p-vlue, use the guidelies for sttisticl sigificce i Sectio 9.3. Sice the p-vlue is less th., the test results re highly sigifict. We c reject t both the % d 5% levels of sigificce. b You could clim tht you work sigifictly fewer hours th those without college eductio. c If you were ot college grdute, you might just report tht you work verge of more th 7.4 hours per week The hypothesis to be tested is d the test sttistic is : μ = 98.6 versus : μ 98.6

11 with p P( z ) x μ x μ z = = = 5.47 σ s.73 3 p-vlue = P z < 5.47 P( z > 5.47) () =. Altertively, we could write -vlue = < 5.47 < (.) =.4 With α =.5, the p-vlue is less th α d is rejected. There is sufficiet evidece to idicte tht the verge body temperture for helthy hums is differet from b-c Usig the criticl vlue pproch, we set the ull d ltertive hypotheses d clculte the test sttistic s i prt. The rejectio regio with α =.5 is z >.96. The observed vlue, z = 5.47, does fll i the rejectio regio d is rejected. The coclusio is the sme is i prt. d ow did the doctor record millio tempertures i 868? The techology vilble t tht time mkes this difficult if ot impossible tsk. It my lso hve bee tht the istrumets used for this reserch were ot etirely ccurte b The hypothesis of iterest is oe-tiled: : μ μ = versus : μ μ > c The test sttistic, clculted uder the ssumptio tht μ μ =, is ( x x) ( μ μ) z = σ σ with σ d σ kow, or estimted by x x s d z = =.9 s s 8 8 s, respectively. For this exercise, vlue which lies slightly more th two stdrd devitios from the hypothesized differece of zero. This would be somewht ulikely observtio, if is true. d The p-vlue for this oe-tiled test is p-vlue = P z >.9 =.987 =.83 Sice the p-vlue is ot less th α =., the ull hypothesis cot be rejected t the % level. There is isufficiet evidece to coclude tht μ μ >. e Usig the criticl vlue pproch, the rejectio regio, with α =., is z >.33 (see Exercise 9.3). Sice the observed vlue of z does ot fll i the rejectio regio, is ot rejected. There is isufficiet evidece to idicte tht μ μ >, or μ > μ. 9. The probbility tht you re mkig icorrect decisio is iflueced by the fct tht if μ μ =, it is just s likely tht x x will be positive s tht it will be egtive. ece, two-tiled rejectio regio must be used. Choosig oe-tiled regio fter determiig the sig of x x simply tells us which of the two pieces of the rejectio regio is beig used. ece, α = P( reject whe true) = P( z >.645 or z <.645 whe true) = α α =.5.5 =. which is twice wht the experimeter thiks it is. ece, oe cot choose the rejectio regio fter the test is performed. 9. The hypothesis of iterest is oe-tiled:

12 : μ μ = versus : μ μ > The test sttistic, clculted uder the ssumptio tht μ μ =, is ( x x) z = = 5.33 s s ( 5) ( 8) 4 4 The rejectio regio with α =., is z >.33 d is rejected. There is evidece to idicte tht μ μ >, or μ > μ. The verge per-cpit beef cosumptio hs decresed i the lst te yers. (Altertively, the p-vlue for this test is the re to the right of z = 5.33 which is very close to zero d less th α =..) b For the differece μ μi the popultio mes this yer d te yers go, the 99% lower cofidece boud uses z. =.33 d is clculted s s s 5 8 ( x x).33 = ( 73 63) or μ μ > 5.63 Sice the differece i the mes is positive, you c gi coclude tht there hs bee decrese i verge per-cpit beef cosumptio over the lst te yers. I dditio, it is likely tht the verge cosumptio hs decresed by more th 5.63 pouds per yer. 9.4 The hypothesis of iterest is two-tiled: : μ μ = versus : μ μ d the test sttistic, clculted uder the ssumptio tht μ μ =, is ( x x) 53, 659 5, 4 z = = 5.69 s s The rejectio regio, with α =.5, is z >.96 d is rejected. There is evidece to idicte differece i the mes for the grdutes i chemicl egieerig d computer sciece. b The coclusios re the sme. 9.9 The hypothesis of iterest is two-tiled: : μ μ = versus : μ μ d the test sttistic is ( x x) z = =. s s with p P( z ) -vlue = >. =.9868 =.64. Sice the p-vlue is betwee. d.5, the ull hypothesis is rejected, d the results re sigifict. There is evidece to idicte differece i the me tempertures for me versus wome. b Sice the p-vlue =.64, we c reject t the 5% level (p-vlue <.5), but ot t the % level (p-vlue >.). Usig the guidelies for sigificce give i Sectio 9.3 of the text, we declre the results sttisticlly sigifict, but ot highly sigifict. 9.3 The hypothesis of iterest cocers the biomil prmeter p d is oe-tiled: : p =.3 versus : p <.3 b The rejectio regio is oe-tiled, with α =.5, or z <.645.

13 x 79 c It is give tht x = 79 d =, so tht pˆ = = =.79. The test sttistic is the pˆ p.79.3 z = = =.449 pq.3(.7) Sice the observed vlue does ot fll i the rejectio regio, is ot rejected. We cot coclude tht p < The hypothesis to be tested ivolves the biomil prmeter p: : p =.5 versus : p <.5 where p is the proportio of prets who describe their childre s overweight. For this test, x 68 x = 68 d = 75, so tht pˆ = = =.9, the test sttistic is 75 pˆ p.9.5 z = = = 4.53 pq.5(.85) 75 b The rejectio regio is oe-tiled, with z <.645 with α =.5. Sice the test sttistic flls i the rejectio regio, the ull hypothesis is rejected. There is sufficiet evidece to idicte tht the proportio of prets who describe their childre s overweight is less th the ctul proportio reported by the Americ Obesity Associtio. c The p-vlue is clculted s p-vlue = P( z < 4.53 ) <. or p-vlue. Sice the p-vlue is less th.5, the ull hypothesis is rejected s i prt b b Sice the survivl rte without screeig is p = 3, the survivl rte with effective progrm my be greter th /3. ece, the hypothesis to be tested is c With : p = 3 versus : p > 3 x 64 pˆ = = =.8, the test sttistic is pˆ p.8 3 z = = = 4.6 pq ( 3)( 3) The rejectio regio is oe-tiled, with α =.5 or z >.645 d is rejected. The screeig progrm seems to icrese the survivl rte. d For the oe-tiled test, p-vlue = P( z > 4.6) <.9998 =. Tht is, c be rejected for y vlue of α.. The results re highly sigifict. 9.4 The hypothesis of iterest is : p =.35 versus : p.35 x 3 with pˆ = = =.4, the test sttistic is 3 pˆ p.4.35 z = = =.7 pq.35(.65) 3

14 The rejectio regio with α =. is z >.58 d the ull hypothesis is ot rejected. (Altertively, we could clculte p-vlue = P( z <.7) = (.5) =.3. Sice this p-vlue is greter th., the ull hypothesis is ot rejected.) There is isufficiet evidece to idicte tht the percetge of dults who sy tht they lwys vote is differet from the percetge reported i Time. 9.4 Sice it is ecessry to detect either p > por p < p, two-tiled test is ecessry: : p p = versus : p p b The stdrd error of pˆ ˆ pis pq pq I order to evlute the stdrd error, estimtes for p d p must be obtied, usig the ssumptio tht p p =. Becuse we re ssumig tht p p =, the best estimte for this commo vlue will be x x 74 8 pˆ = = = d the estimted stdrd error is pq ˆˆ =.554(.446) = c Clculte p ˆ = =.59 d p ˆ = =.579. The test sttistic, bsed o the smple dt 4 4 will be pˆ pˆ ( p p) pˆ pˆ z = = =.84 pq.594 pq pq ˆˆ This is likely observtio if is true, sice it lies less th oe stdrd devitio below p p =. d Clculte the two tiled p P( z ) -vlue = >.84 =.5 =.4. Sice this p-vlue is greter th., is ot rejected. There is o evidece of differece i the two popultio proportios. e The rejectio regio with α =., or z >.58 d is ot rejected. There is o evidece of differece i the two popultio proportios The hypothesis of iterest is: : p p = versus : p p < pˆ pˆ 8 3 Clculte p ˆ =.36, p ˆ =.6 d pˆ = = =.48. The test sttistic is the 5 5 pˆ ˆ p.36.6 = = = z.4.48(.5)( 5 5) pq ˆˆ The rejectio regio, with α =.5, is z <.645 d is rejected. There is evidece of differece i the proportio of survivors for the two groups. b From Sectio 8.7, the pproximte 95% cofidece itervl is

15 pq ˆˆ pq ˆˆ ( pˆ pˆ) ± (.36.6) ± ±.9 or.43 < <.5 ( p p ) 9.48 The hypothesis of iterest is : p p = versus : p p > 4 x x 4 Clculte p ˆ = =.8, p ˆ = =.9, d pˆ = = =.3. The test sttistic is the pˆ pˆ.8.9 z = = =.67.3(.987)( 66 66) pq ˆˆ The rejectio regio, with α =., is z >.33 d is rejected. There is sufficiet evidece to idicte tht the risk of demeti is higher for ptiets usig Prempro. 9.5 Sice the two tretmets were rdomly ssiged, the rdomiztio procedure c be implemeted s ech ptiet becomes vilble for tretmet. Choose rdom umber betwee d 9 for ech ptiet. If the ptiet receives umber betwee d 4, the ssiged drug is spiri. If the ptiet receives umber betwee 5 d 9, the ssiged drug is clopidogrel. b Assume tht = 77 d = 778. It is give tht p ˆ =.54, p ˆ =.38, so tht pˆ pˆ 77 (.54) 778 (.38) pˆ = = =.46. 5, 5 The test sttistic is the pˆ pˆ z = = = (.954)( ) pq ˆˆ with p P( z ) -vlue = > 4.75 < (.) =.4. Sice the p-vlue is less th., the results re sttisticlly sigifict. There is sufficiet evidece to idicte differece i the proportios for the two tretmet groups. c Clopidogrel would be the preferred tretmet, s log s there re o dgerous side effects The hypothesis to be tested is d the test sttistic is : μ = 5 versus : μ > 5 x μ x μ 7. 5 z = = =.9 σ s The rejectio regio with α =.is z >.33. Sice the observed vlue, z =.9, does ot fll i the rejectio regio d is ot rejected. The dt do ot provide sufficiet evidece to idicte tht the me ppm of PCBs i the popultio of gme birds exceeds the FDA s recommeded limit of 5 ppm Refer to Exercise 9.75, i which the rejectio regio ws give s z >.33 where

16 x μ x.3 z = = s.9 35 Solvig for x we obti the criticl vlue of x ecessry for rejectio of. x 5 6. >.33 x >.33 5 = The probbility of Type II error is defied s β = P ccept whe is flse Sice the cceptce regio is x 7.34 from prt, β c be rewritte s β = P x 7.34 whe is flse = P x 7.34 whe μ > 5 Severl ltertive vlues of μ re give i this exercise. For μ = 6, β = P( x 7.34 whe μ = 6) = P z = P( z.33 ) =.98 d β =.98 =.98. b For μ = 7, β = P( x 7.34 whe μ = 7) = P z = P( z.34) =.633 d β =.633 = c For μ = 8, β = P( x 7.34 whe μ = 8) = P z = P( z.66) =.7454 For μ = 9, β = P( x 7.34 whe μ = 9) = P z = P z.65 =.955 For μ =, β = P x 7.34 whe μ = 7.34 = P z = P z.64 =.9959 For μ =,

17 ( μ ) β = P x 7.34 whe = 7.34 = P z = P z 4.63 d The power curve is show o the ext pge...8 Power Me You c see tht the power becomes greter th or equl to.9 for vlue of μ little smller th μ = 9. To fid the exct vlue, we eed to solve for μ i the equtio: 7.34 μ β = P( x 7.34) = P z = 7.34 μ or P z = From Tble 3, the vlue of z tht cuts off. i the lower til of the z-distributio is z =.8, so tht 7.34 μ =.8 6. / μ = =

18 : Iferece from Smll Smples. Refer to Tble 4, Appedix I, idexig df log the left or right mrgi d t α cross the top. t.5 =.5 with 5 df b t.5 =.36 with 8 df c t. =.33 with 8 df c t.5.96 with 3 df. The vlue P t > t = is the tbled etry for prticulr umber of degrees of freedom. For two-tiled test with α =., the criticl vlue for the rejectio regio cuts off α =.5 i the two tils of the t distributio show below, so tht t.5 = The ull hypothesis will be rejected if t > 3.55 or t < 3.55 (which you c lso write s t > 3.55 ). b For right-tiled test, the criticl vlue tht seprtes the rejectio d orejectio regios for right tiled test bsed o t-sttistic will be vlue of t (clled t α ) such tht P( t > t α ) = α =.5 d df = 6. Tht is, t.5 =.746. The ull hypothesis will be rejected if.746 t >. c For two-tiled test with α =.5 d df = 5, will be rejected if t >.6. d For left-tiled test with α =.d df = 7, will be rejected if t < The p-vlue for two-tiled test is defied s p-vlue = P( t >.43) = P( t >.43) so tht P( t >.43) = p-vlue Refer to Tble 4, Appedix I, with df =. The exct probbility, P( t >.43) is uvilble; however, it is evidet tht t =.43 flls betwee t.5 =.79 d t. =.68. Therefore, the re to the right of t =.43 must be betwee. d.5. Sice. < -vlue.5 p < the p-vlue c be pproximted s. < p-vlue <.5

19 b For right-tiled test, p-vlue P( t 3.) = > with df = 6. Sice the vlue t = 3. is lrger th t.5 =.9, the re to its right must be less th.5 d you c boud the p-vlue s p -vlue <.5 c For two-tiled test, p-vlue = P( t >.9) = P( t >.9), so tht P( t ) From Tble 4 with df = 5, t =.9 is smller th t. =.36 so tht -vlue. d -vlue. p > p > d For left-tiled test, p-vlue P( t 8.77) P( t 8.77) >.9 = p-vlue. = < = > with df = 7. Sice the vlue t = 8.77 is lrger th t.5 = 3.499, the re to its right must be less th.5 d you c boud the p-vlue s p -vlue <.5.9 Similr to previous exercises. The hypothesis to be tested is : μ = versus : μ < x Clculte x = i = = xi xi 65, s = = =.565 d s = The test sttistic is x μ t = = = 3.44 s The criticl vlue of t with α =.d = 9degrees of freedom is t. =.539 d the rejectio regio is t <.539. The ull hypothesis is rejected d we coclude tht μ is less th DL. b The 95% upper oe-sided cofidece boud, bsed o = 9degrees of freedom, is s x t μ < This cofirms the results of prt i which we cocluded tht the me is less th DL..6 Aswers will vry. A typicl histogrm geerted by Miitb shows tht the dt re pproximtely moud-shped.

20 6 4 Frequecy Serum-Cholesterol 35 x 348 b Clculte x = i = = xi 348 xi 3,56,896 s = = 5 = d s = Tble 4 does ot give vlue of t with re.5 to its right. If we re coservtive, d use the vlue of t with df = 9, the vlue of t will be t.5 =.45, d the pproximte 95% cofidece itervl is s x ± t ± ± 3.54 or 33.4 < μ < Refer to Exercise.6. If we use the lrge smple method of Chpter 8, the lrge smple cofidece itervl is s x ± z ± ±.98 or < μ < The itervls re firly similr, which is why we choose to pproximte x μ the smplig distributio of with z distributio whe > 3. s /.4 If the tiplque rise is effective, the plque buildup should be less for the group usig the tiplque rise. ece, the hypothesis to be tested is : μ μ = versus : μ μ > b The pooled estimtor of σ is clculted s s s s = = = d the test sttistic is ( x x).6.78 t = = =.86 s The rejectio regio is oe-tiled, bsed o = degrees of freedom. With α =.5, from Tble 4, the rejectio regio is t > t.5 =.78 d is rejected. There is evidece to idicte tht the rise is effective.

21 c The p-vlue is p-vlue = P( t >.86) From Tble 4 with df =, t =.86 is betwee two tbled etries t.5 = 3.55 d t. =.68, we c coclude tht.5 < p-vlue <..7 Check the rtio of the two vrices usig the rule of thumb give i this sectio: lrger s.7895 = = 6. smller s.743 which is greter th three. Therefore, it is ot resoble to ssume tht the two popultio vrices re equl. b You should use the upooled t test with Stterthwite s pproximtio to the degrees of freedom for testig : μ μ = versus : μ μ The test sttistic is ( x x) t = = =.4 s s 5 5 with s s s s ( ) df = = = With df 5, the p-vlue for this test is bouded betwee. d.5 so tht c be rejected t the 5% level of sigificce. There is evidece of differece i the me umber of ucotmited eggplts for the two disifectts..8 Use your scietific clcultor or the computig formuls to fid: x =.5 s =.78 s =.59 x =.38 s =.3733 s =.93 Sice the rtio of the vrices is less th 3, you c use the pooled t test, clcultig s s 9 (.78) 9 (.3733) s = = =.36 8 d the test sttistic is t ( x x ).5.38 = = = s s.68 For two-tiled test with df = 8, the p-vlue c be bouded usig Tble 4 so tht.5 < -vlue. or. -vlue. p < < p < Sice the p-vlue is greter th., : μ μ = is ot rejected. There is isufficiet evidece to idicte tht there is differece i the me titium cotets for the two methods. μ μ is give s b A 95% cofidece itervl for

22 ( x x ) ± t.5 s.5.38 ±. s.3 ±.6 or.9 < <.3 ( μ μ ) Sice μ μ = flls i the cofidece itervl, the coclusio of prt is cofirmed. This prticulr dt set is very susceptible to roudig error. You eed to crry s much ccurcy s possible to obti ccurte results.

23 .9 The Miitb stem d lef plots re show below. Notice the mouded shpes which justify the ssumptio of ormlity. Stem-d-Lef Disply: Geeric, Sumid Stem-d-lef of Geeric N = 4 Stem-d-lef of Sumid N = 4 Lef Uit =. Lef Uit = (5) b Use your scietific clcultor or the computig formuls to fid: x = 6.4 s = s =.5 x = 6.43 s = s =.43 Sice the rtio of the vrices is greter th 3, you must use the upooled t test with Stterthwite s pproximte df. s s df = 9 s s c For testig : μ μ = versus : μ μ, the test sttistic is ( x x) t = = =. s s 4 4 For two-tiled test with df = 9, the p-vlue c be bouded usig Tble 4 so tht -vlue. or -vlue. p > p > Sice the p-vlue is greter th., : μ μ = is ot rejected. There is isufficiet evidece to idicte tht there is differece i the me umber of risis per box..3 If swimmer is fster, his(her) verge time should be less th the verge time for swimmer. Therefore, the hypothesis of iterest is : μ μ = versus : μ μ > d the prelimiry clcultios re s follows: The Swimmer Swimmer x i = xi = x i = xi = = =

24 s ( x ) ( x ) x x = i i i i ( ) ( 596.7) = = Also, x = = d x = = The test sttistic is ( x x) t = = =.4 s.347 For oe-tiled test with df = = 8, the p-vlue c be bouded usig Tble 4 so tht p -vlue >., d is ot rejected. There is isufficiet evidece to idicte tht swimmer s verge time is still fster th the verge time for swimmer..38 A pired-differece test is used, sice the two smples re ot idepedet (for y give city, Allstte d st Cetury premiums will be relted). b The hypothesis of iterest is : μ μ = or : μd = : μ μ or : μd where μ is the verge for Allstte isurce d μ is the verge cost for st Cetury isurce. The tble of differeces, log with the clcultio of d d s d, is preseted below. City 3 4 Totls d d i i d 5,3 4,849 49,84 7,449 37,93 i 75 d = = = d ( di ) ( 75) di 37,93 s 4 d = = = = The test sttistic is d μd t = = = 6.34 s d 4 with = 3degrees of freedom. The rejectio regio with α =. is t > t.5 = 5.84, d is rejected. There is sufficiet evidece to idicte differece i the verge premiums for Allstte d st Cetury. c p-vlue = P( t > 6.34) = P( t > 6.34). Sice t = 6.34 is greter th t.5 = 5.84, p-vlue < (.5) p-vlue <.. d A 99% cofidece itervl for μ μ = μd is s d ± t d ± ± 7.556

25 or ( μ μ ) 3.94 < < e The four cities i the study were ot ecessrily rdom smple of cities from throughout the Uited Sttes. Therefore, you cot mke vlid comprisos betwee Allstte d st Cetury for the Uited Sttes i geerl..43 A pired-differece test is used, sice the two smples re ot rdom d idepedet (t y loctio, the groud d ir tempertures re relted). The hypothesis of iterest is : μ μ = : μ μ The tble of differeces, log with the clcultio of d d s d, is preseted below. Loctio Totl d i d i 7.9 d = = =.58 5 di 7.9 di 5.5 s 5 d = = =.667 d s d = d the test sttistic is d μd.58 t = = = 4.36 s.867 d 5 A rejectio regio with α =.5 d df = = 4is t > t.5 =.776, d is rejected t the 5% level of sigificce. We coclude tht the ir-bsed temperture redigs re bised. b The 95% cofidece itervl for μ μ = μd is s.867 d ± t d.5.58 ± ± < μ μ <.566. or c The iequlity to be solved is t α SE B We eed to estimte the differece i me tempertures betwee groud-bsed d ir-bsed sesors to withi. degrees cetigrde with 95% cofidece. Sice this is pired experimet, the iequlity becomes s t d.5. With s d =.867 d represets the umber of pirs of observtios, cosider the smple size obtied by replcig t.5 by z.5 = = 64.3 or = 65 Sice the vlue of is greter th 3, the use of z α for t α is justified..44 A pired-differece test is used, sice the two smples re ot rdom d idepedet (withi y smple, the dye d dye mesuremets re relted). The hypothesis of iterest is : μ μ = : μ μ The tble of differeces, log with the clcultio of d d s d, is preseted below.

26 d i Smple Totl d 3 i d = = =. 9 di di 8 s 9 d = = =. d s d = d the test sttistic is d μd. t = = =.9 s d 9 A rejectio regio with α =.5 d df = = 8is t > t.5 =.36, d is ot rejected t the 5% level of sigificce. We cot coclude tht there is differece i the me brightess scores.

27 4: Alysis of Ctegoricl Dt 4. See Sectio 4. of the text. 4. Idex Tble 5, Appedix I, with χ.5 = 7.8 b c χ.5 = 3.83 d χ α d the pproprite degrees of freedom. χ. =.9 χ. = For test of specified cell probbilities, the degrees of freedom re k. Use Tble 5, Appedix I: df = 6; χ =.59; reject if X >.59 b c d.5 df = 9; χ. =.666; reject if df = 3; χ.5 = 9.84; reject if df = ; χ = 5.99; reject if.5 X >.666 X > X > Three hudred resposes were ech clssified ito oe of five ctegories. The objective is to determie whether or ot oe ctegory is preferred over other. To see if the five ctegories re eqully likely to occur, the hypothesis of iterest is : p = p = p3 = p4 = p5 = 5 versus the ltertive tht t lest oe of the cell probbilities is differet from /5. b The umber of degrees of freedom is equl to the umber of cells, k, less oe degree of freedom for ech lierly idepedet restrictio plced o p, p,, pk. For this exercise, k = 5 d oe degree of freedom is lost becuse of the restrictio tht p i = ece, X hs k = 4degrees of freedom. c The rejectio regio for this test is locted i the upper til of the chi-squre distributio with df = 4. From Tble 5, the pproprite upper-tiled rejectio regio is X > χ.5 = d The test sttistic is ( O E ) i i X = Ei which, whe is lrge, possesses pproximte chi-squre distributio i repeted smplig. The vlues of Oi re the ctul couts observed i the experimet, d E = p = 3 5 = 6. i i A tble of observed d expected cell couts follows: Ctegory O i E i The X = = = 8. 6 e Sice the observed vlue of X does ot fll i the rejectio regio, we cot coclude tht there is differece i the preferece for the five ctegories.

28 4.9 If the frequecy of occurrece of hert ttck is the sme for ech dy of the week, the whe hert ttck occurs, the probbility tht it flls i oe cell (dy) is the sme s for y other cell (dy). ece, : p = p = = p7 = 7 vs. : t lest oe p i is differet from the others, or equivletly, : pi pj for some pir i j Sice E = p = 7 = d the test sttistic is =, i i ( ) ( ) X = = = The degrees of freedom for this test of specified cell probbilities is k = 7 = 6d the upper tiled rejectio regio is X > χ =.59.5 is ot rejected. There is isufficiet evidece to idicte differece i frequecy of occurrece from dy to dy. 4. It is ecessry to determie whether proportios t give hospitl differ from the popultio proportios. A tble of observed d expected cell couts follows: Disese A B C D Other Totls O i E i The ull hypothesis to be tested is : p =.5; p =.; p3 =.8; p4 =.4 gist the ltertive tht t lest oe of these probbilities is icorrect. The test sttistic is X = = The umber of degrees of freedom is k = 4d, sice the observed vlue of X = 3.77 is greter th χ.5, the p-vlue is less th.5 d the results re declred highly sigifict. We reject d coclude tht the proportios of people dyig of diseses A, B, C, d D t this hospitl differ from the proportios for the lrger popultio. 4. Similr to previous exercises. The hypothesis to be tested is : p = p = = p = versus : t lest oe p i is differet from the others with Ei = pi = 4( ) = The test sttistic is ( ) ( ) X = = The upper tiled rejectio regio is with α =.5 d k = df is X > χ.5 = The ull hypothesis is ot rejected d we cot coclude tht the proportio of cses vries from moth to moth.

29 4.6 The experimet is lyzed s 3 4cotigecy tble. ece, the expected cell couts must be obtied for ech of the cells. Sice vlues for the cell probbilities re ot specified by the ull hypothesis, they must be estimted, d the pproprite estimtor is rc ˆ i j Eij =, where r i is the totl for row i d c j is the totl for colum j (see Sectio 4.4). The cotigecy tble, icludig colum d row totls d the estimted expected cell couts (i pretheses) follows. Colum Row 3 4 Totl (67.68) (66.79) (67.97) (58.56) (84.7) (83.7) (84.64) (7.9) (78.5) (77.3) (78.39) (67.53) Totl The test sttistic is ( O ˆ ij Eij ) ( 67.68) ( ) X = = =.7 Eˆ ij usig the two-deciml ccurcy give bove. The degrees of freedom re df = ( r )( c ) = ( 3 )( 4 ) = 6 b Similr to prt. The estimted expected cell couts re clculted s ˆ rc i j Eij =, d re show i pretheses i the tble below. Colum Row 3 Totl (37.84) (6.37) (7.8) (7.6) (8.63) (9.) Totl The test sttistic is clculted (usig clcultor ccurcy rther th the two-deciml ccurcy give i prt () s ( ) ( 6 9.) X = = r c = 3 =. The degrees of freedom re 4.8 Sice r = d 3 r c = =. b The experimet is lyzed s 3cotigecy tble. The cotigecy tble, icludig colum d row totls d the estimted expected cell couts, follows. Colum Row 3 Totl (4.3) (37.3) (84.46) (6.77) (53.69) (.54) Totl c =, the totl degrees of freedom re

30 The estimted expected cell couts were clculted s: ˆ rc 64 ( 3) E = = = ˆ rc 64 ( 9) E = = = 37.3 d so o. 4 The X = = c With α =., oe-tiled rejectio regio is foud usig Tble 5 to be X > χ. = 9.. d-e The observed vlue of X = 3.59 does ot fll i the rejectio regio. ece, is ot rejected. There is o reso to expect depedece betwee rows d colums. I fct, X = 3.59 hs p -vlue >.. 4. The hypothesis to be tested is : opiio is idepedet of politicl ffilitio : opiio is depedet o politicl ffilitio d the Miitb pritout below shows the observed d estimted expected cell couts. Chi-Squre Test: Support, Oppose, Usure Expected couts re prited below observed couts Chi-Squre cotributios re prited below expected couts Support Oppose Usure Totl Totl Chi-Sq = 8.93, DF = 4, P-Vlue =.64 The test sttistic is the chi-squre sttistic give i the pritout s X = 8.93with p -vlue =.64. Sice the p-vlue is greter th.5, is ot rejected. There is isufficiet evidece to idicte tht there is differece i perso s opiio depedig o the politicl prty with which he is ffilited. b Eve though there re o sigifict differeces, we c still look t the coditiol distributios of opiios for the three groups, show i the tble below. Support Oppose Usure Democrts = =.5 44 = Idepedets = = = Republics = = = You c see tht Democrts d Idepedets hve lmost ideticl opiios o mdtory helthcre, while Republics re less likely to be supportive.

31 4. The hypothesis of idepedece betwee ttchmet ptter d child cre time is tested usig the chi-squre sttistic. The cotigecy tble, icludig colum d row totls d the estimted expected cell couts, follows.

32 The test sttistic is Child Cre Attchmet Low Moderte igh Totl Secure (4.9) (3.97) (8.95) Axious 8 9 (.9) (4.3) (4.5) Totl ( ) ( ) ( ) X = = d the rejectio regio is X > χ.5 = 5.99with df. is rejected. There is evidece of depedece betwee ttchmet ptter d child cre time. b The vlue X = 7.67 is betwee χ.5 d χ.5 so tht.5 < p-vlue <.5. The results re sigifict. 4.4 The hypothesis of idepedece betwee opiio d politicl ffilitio is tested usig the chi-squre sttistic. The cotigecy tble, icludig colum d row totls d the estimted expected cell couts, follows. Opiio Politicl Affilitio Kow ll fcts Cover up Not sure Totl Democrts (53.48) (84.38) (44.4) Republics (49.84) (65.) (4.4) Idepedets (.68) (.6) (8.7) Totl The test sttistic is ( ) ( ) ( 7 8.7) X = = The test sttistic is very lrge, compred to the lrgest vlue i Tble 5 ( χ.5 = 4.86), so tht p -vlue <.5 d is rejected. There is evidece of differece i the distributio of opiios depedig o politicl ffilitio. b You c see tht the percetge of Democrts who thik there ws cover-up is higher th the sme percetges for either Republics or Idepedets. 4.9 Becuse set umber of Americs i ech sub-popultio were ech fixed t, we hve cotigecy tble with fixed rows. The tble, with estimted expected cell couts pperig i pretheses, follows.

33 The test sttistic is Yes No Totl White-Americ 4 6 (6) (38) Afric-Americ (6) (38) ispic-americ 68 3 (6) (38) Asi-Americ 84 6 (6) (38) Totl ( ) ( ) ( ) X = = d the rejectio regio with 3 df is X > is rejected d we coclude tht the icidece of pretl support is depedet o the sub-popultio of Americs The umber of observtios per colum were selected prior to the experimet. The test procedure is ideticl to tht used for r ccotigecy tble. The cotigecy tble geerted by Miitb, icludig colum d row totls d the estimted expected cell couts, follows. Chi-Squre Test: s, 3s, 4s, 5s, 6 Expected couts re prited below observed couts Chi-Squre cotributios re prited below expected couts s 3s 4s 5s 6 Totl Totl 5 Chi-Sq =.34, DF = 4, P-Vlue =.3 The observed vlue of the test sttistic is X =.34 with p-vlue =.3 d the ull hypothesis is rejected t the 5% level of sigificce. There is sufficiet evidece to idicte tht the proportio of dults who tted church regulrly differs depedig o ge. b The percetge who tted church icreses with ge To test for homogeeity of the four biomil popultios, we use chi-squre sttistic d the 4cotigecy tble show below. Rhode Colordo Clifori Florid Totl Isld Prticipte (63.3) (78.63) (97.88) (97.88) Do ot prticipte (3.38) (6.37) (.) (.) Totl The test sttistic is X = = With df = 3, the p-vlue is less th.5 d is rejected. There is differece i the proportios for the four sttes. The differece c be see by cosiderig the proportio of people prticiptig i ech of the four sttes: Rhode Isld Colordo Clifori Florid Prticipte = = =.4 3 =

34 4.45 Similr to previous exercises. The cotigecy tble, icludig colum d row totls d the estimted expected cell couts, follows. The test sttistic is Coditio Treted Utreted Totl Improved (95.5) (95.5) Not improved (4.5) (4.5) Totl 4 ( ) ( ) ( ) X = = To test oe-tiled ltertive of effectiveess, first check to see tht pˆ ˆ > p. The the rejectio regio with df hs right-til re of.5 =.or X > χ.5 =.76. is rejected d we coclude tht the serum is effective. b Cosider the treted d utreted ptiets s comprisig rdom smples of two hudred ech, drw from two popultios (i.e., smple of treted ptiets d smple of utreted ptiets). Let p be the probbility tht treted ptiet improves d let p be the probbility tht utreted ptiet improves. The the hypothesis to be tested is : p p = : p p > Usig the procedure described i Chpter 9 for testig hypothesis bout the differece betwee two biomil prmeters, the followig estimtors re clculted: x 7 x 74 x x 7 74 pˆ = = pˆ = = pˆ = = = The test sttistic is pˆ pˆ.5 z = = = 4.34 pq ˆˆ( ).4775(.55)(.) Ad the rejectio regio for α =.5 is z >.645. Agi, the test sttistic flls i the rejectio regio. We reject the ull hypothesis of o differece d coclude tht the serum is effective. Notice tht z = 4.34 = 8.5 = X (to withi roudig error) 4.47 Refer to Sectio 9.6. The two-tiled z test ws used to test the hypothesis : p p = : p p usig the test sttistic z = ( pˆ pˆ) ( pˆ ˆ p) pq ˆˆ( ) pˆ pˆ pq ˆˆ z = = pq ˆˆ Note tht x x pˆ ˆ p pˆ = = Now cosider the chi-squre test sttistic used i Exercise The hypothesis to be tested is : idepedece of clssifictio : depedece of clssifictio Tht is, the ull hypothesis sserts tht the percetge of ptiets who show improvemet is idepedet of whether or ot they hve bee treted with the serum. If the ull hypothesis is true, the p = p. ece, the two tests re desiged to test the sme hypothesis. I order to

35 show tht z is equivlet to X, it is ecessry to rewrite the chi-squre test sttistic i terms of the qutities, pˆ, pˆ ˆ, p, d. Cosider O, the observed umber of treted ptiets who hve improved. Sice ˆp = O, we hve O = ˆ p. Similrly, O = qˆ O = ˆ ˆ p O = q The estimted expected cell couts re clculted uder the ssumptio tht the ull hypothesis is true. Cosider rc ( O O )( O O ) ( x x )( O O ) E ˆ = = = = p Similrly, Eˆ = qˆ ˆ ˆ E ˆ ˆ = p E = q The tble of observed d estimted expected cell couts follows. Treted Utreted Totl Improved pˆ pˆ x x ( p ˆ ) ( pˆ ) Not improved qˆ qˆ ( x x) ( q ˆ ) ( qˆ ) Totl The ( O ) ij Eij X = Eij ( pˆ pˆ) ( qˆ qˆ) ( pˆ pˆ) ( qˆ qˆ) = pˆ qˆ pˆ qˆ ( pˆ pˆ) ( pˆ) ( pˆ) ( pˆ ˆ p) ( pˆ) ( pˆ) = pˆ qˆ pˆ qˆ ( pˆ) ( pˆ ˆ p) ˆ p( pˆ ˆ p) ( pˆ) ( pˆ ˆ p) ˆ p( pˆ ˆ p) = pq ˆˆ pˆˆ q pˆ ˆ ˆ ˆ p p p = pq ˆˆ pq ˆˆ Substitutig for ˆp, we obti X pˆ pˆ pˆ ˆ ˆ ˆ ˆ ˆ p p p p p = ˆˆ ˆˆ pq pq ˆ ˆ ˆ ˆ ˆ ˆ p p p p p p = = pq ˆˆ pq ˆˆ( ) ( ) Note tht X is ideticl to z, s defied t the begiig of the exercise The cotigecy tble with estimted expected cell couts i pretheses is show i the Miitb pritout below. Chi-Squre Test: Excellet, Good, Fir Expected couts re prited below observed couts Chi-Squre cotributios re prited below expected couts Excellet Good Fir Totl

36 Totl 87 Chi-Sq = 3.59, DF =, P-Vlue =.96 The test sttistic is The observed vlue of ( ) ( ) ( ) X = = X is less th χ. so tht p -vlue >. (the exct p -vlue =.96 from the pritout) d is ot rejected. There is o evidece of differece due to geder. b The cotigecy tble with estimted expected cell couts i pretheses is show i the Miitb pritout below. Chi-Squre Test: Excellet, Good, Fir, Poor Expected couts re prited below observed couts Chi-Squre cotributios re prited below expected couts Excellet Good Fir Poor Totl Totl Chi-Sq =.54, DF = 3, P-Vlue =.788 cells with expected couts less th 5. The test sttistic is The observed vlue of ( ) ( ) ( ) X = = X is less th χ. so tht p -vlue >. (the exct p -vlue =.788 from the pritout) d is ot rejected. There is o evidece of differece due to geder. c Notice tht the computer pritout i prt b wrs tht cells hve expected cell couts less th 5. This is violtio of the ssumptios ecessry for this test, d results should thus be viewed with cutio The 3cotigecy tble is lyzed s i previous exercises. The Miitb pritout below shows the observed d estimted expected cell couts, the test sttistic d its ssocited p-vlue. Chi-Squre Test: 3 or fewer, 4 or 5, 6 or more Expected couts re prited below observed couts Chi-Squre cotributios re prited below expected couts 3 or fewer 4 or 5 6 or more Totl Totl Chi-Sq =.69, DF =, P-Vlue =.3 The results re highly sigifict ( -vlue.3 p = ) d we coclude tht there is differece i the susceptibility to colds depedig o the umber of reltioships you hve. b The proportio of people with colds is clculted coditiolly for ech of the three groups, d is show i the tble below. Three or fewer Four or five Six or more

37 Cold = = = No cold = = = Totl.. As the resercher suspects, the susceptibility to cold seems to decrese s the umber of reltioships icreses! 4.6 The ull hypothesis to be tested is : p = ; p = ; p3 = ; p4 = ; p5 = ; p6 = gist the ltertive tht t lest oe of these probbilities is icorrect. A tble of observed d expected cell couts follows: Dy Mody Tuesdy Wedesdy Thursdy Fridy Sturdy O i E i The test sttistic is 95 4 X = = The umber of degrees of freedom is k = 5d the rejectio regio with α =.5 is X > χ.5 =.7 d is rejected. The mger s clim is refuted.