CHAPTER 8 ANSWERS. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley

Transcription

1 CHAPTER 8 ANSWERS Sectio 8.1 Statistical Literacy ad Critical Thikig 1 The distributio of radomly selected digits from to 9 is uiform. The distributio of sample meas of 5 such digits is approximately ormal. 2 The sample mea is the best estimate of a populatio mea, ad the sample proportio is the best estimate of a populatio proportio. 3 The symbol x deotes the mea of a sample while μ deotes the meas of a populatio. 4 The symbol p deotes a sample proportio while p deotes the populatio proportio. 5 The statemet is ot sesible. The large size of the sample caot compesate for the biased sample that is likely to occur with the coveiece sample. A sample of college studets, o matter how it is obtaied, caot be take as a represetative sample of all adult Americas. College studets have ot lived as log as the average adult America ad have had less opportuity to be ivolved i car crashes, so it is likely that Ted will uderestimate the proportio of adults who have bee ivolved i car crashes. 6 The statemet is ot sesible. A samplig error is the result of radom variatio, but the pollster's mistakes are i a differet category. 7 The statemet is sesible. Commo sese suggests that a very small sample with oly a few observatios ca vary widely, but larger samples ted to be more cosistet ad they ted to better reflect the true ature of the populatio. 8 The statemet is ot sesible. The umbers 91 ad 112 will icrease with a larger sample, but the sample proportio is equally likely to icrease or decrease. Cocepts ad Applicatios 9 The best estimate of the populatio proportio for the Northeast is the same as the sample proportio: p =.19. The give sample proportio is ot likely to be a good estimate of the proportio for the populatio of all childre i the Uited States simply because the sample was ot take from the etire U.S. There are regioal factors that could have a strog ifluece o that proportio i differet parts of the coutry. 1 The best estimate for the mea cholesterol level for all wome is the sample mea x = We would be more cofidet of the estimate if the sample icluded measuremets from 5 wome sice as the sample size icreases, the sample mea becomes a more reliable estimate of the populatio mea. 11 a) The sample mea is ( )/.2 = 9.5 stadard deviatios away from the mea of the distributio of sample meas. b) Based o Table 5.1, the probability is very small, less tha.1 or.1%. c) No. It appears that the cas are beig filled with a amout that is greater tha 12. oz, so cosumers are gettig more tha the amout stated o the label. 12 a) The sample mea x = 2.55 is ( )/.6 = -1.5 stadard deviatios from the mea of the distributio of sample meas μ. That is, x is 1.5 stadard deviatios below the mea of the distributio. b) From Table 5.1, the probability of a sample mea less tha 2.55 is 6.68% or a) The populatio proportio is p = 6523/12345 =.528. b) The sample proportio is p = 245/5 = Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

2 SECTION 8.1, SAMPLING DISTRIBUTIONS 137 c) The sample proportio is a little too low. 14 a) The populatio proportio is p = 137/1348 =.12. b) The sample proportio is p =.11 c) The sample does appear to be represetative of the populatio. The sample proportio is p = 73/ = This is the best estimate of the populatio proportio, so we estimate that.4867 x 168 = 783 people traveled from abroad. We would be more cofidet of the estimate if we sampled 3 people istead of sice the estimate p becomes more reliable as the sample size icreases. 16 The sample proportio is p = 4/75 =.56. This is the best estimate of 17 a) 18 a) the populatio proportio, so we estimate that.56 x 74,512 = 41,727 people at the game support the Chicago Bears.. We would be more cofidet of the estimate if we sampled 2- people istead of 75 sice the estimate p becomes more reliable as the sample size icreases. sample proportio - populatio proportio z.7,so the stadard deviatio.3 sample proportio is.7 stadard deviatios below the mea of the samplig distributio. b) From Table 5.1, z = -.7 correspods to the 24. percetile, so the probability of a sample proportio less tha.32 is.24. sample proportio - populatio proportio z.1, so the sample stadard deviatio.21 proportio is.1 stadard deviatios above the mea of the samplig distributio. b) From Table 5.1, z =.1 correspods to the percetile, so the probability of a sample proportio greater tha.45 is = a) Sample Sample Mea 1,1 1. 1, ,5 3. 2, ,. 2, ,1 3. 5, ,5 5. b) The mea of the sample meas i part (a) is ( )/9 = 24/9 = 8/3 = 2.7. c) The mea of the populatio is ( )/3 = 8/3 = 2.7, which is the same as the mea of the sample meas i part (a). Yes, these meas will always be equal. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

3 138 CHAPTER 8, FROM SAMPLES TO POPULATIONS a) Sample Mea 62, , , , , , , , , , , , , , , , , , , , , , , , , b) The mea of the sample meas is ( )/25 = 1485/25 = c) The mea of the populatio is ( )/5 = 297/5 = 59.4, which is equal to the mea of the sample meas i part (b). These meas will always be equal. 21 O the right side of each of the followig tables, we show the sample meas i icreasig order. Followig the tables are histograms of the distributios of the sample meas. Samples of size = 1 Sample Mea Sample Mea A 52 C 5 C 5 M 33 G 6 A 52 M 33 G 6 O 97 O 97 Mea 49.4 Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

4 Samples of size = 2 Sample Mea Sample Mea AC 28.5 CM 19. AG 56. AC 28.5 AM 42.5 CG 32.5 AO 74.5 AM 42.5 CG 32.5 GM 46.5 CM 19. CO 51. CO 51. AG 56. GM 46.5 MO 65. GO 78.5 AO 74.5 MO 65. GO 78.5 Mea 49.4 SECTION 8.1, SAMPLING DISTRIBUTIONS 139 Samples of size = 3. Sample Mea Sample Mea ACG 39. ACM 3. ACM 3. CGM ACO ACG 39. AGM CMO 45. AGO AGM AMO 6.67 ACO CGM CGO 54. CGO 54. AMO 6.67 CMO 45. GMO GMO AGO Mea 49.4 Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

5 14 CHAPTER 8, FROM SAMPLES TO POPULATIONS Samples of size = 4. Sample Mea Sample Mea ACGM 37.5 ACGM 37.5 ACGO 53.5 ACMO ACMO CGMO AGMO 6.5 ACGO 53.5 CGMO AGMO 6.5 Mea 49.4 Samples of size = 5. Sample Mea Sample Mea ACGMO 49.4 ACGMO 49.4 Mea 49.4 Mea 49.4 The mea of the samplig distributio of the meas is 49.4 for each sample size, the same as the mea of the populatio. Histogram of Mea1 Histogram of Mea2 Per cet 1 Pe rce t Mea Mea Histogram of Mea3 Histogram of Mea4 Percet 1 Percet Mea Mea The graph of the distributio for the sample of size 5 is a sigle spike at Note that the distributios become more cocetrated as the sample Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

6 SECTION 8.1, SAMPLING DISTRIBUTIONS 141 size icreases. 22 The samplig is to be doe without replacemet. O the right side of each of the followig tables, we show the sample meas i icreasig order. Followig the tables are histograms of the distributios of the sample meas. Samples of size = 1. Sample Mea Sample Mea A 11 E 62 B 87 D 66 C 75 C 75 D 66 B 87 E 62 A 11 Mea 78.2 Samples of size = 2. Sample Mea Sample Mea AB 94. DE 64. AC 88. CE 68.5 AD 83.5 CD 7.5 AE 81.5 BE 74.5 BC 81. BD 76.5 BD 76.5 BC 81. BE 74.5 AE 81.5 CD 7.5 AD 83.5 CE 68.5 AC 88. DE 64. AB 94. Mea 78.2 Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

7 142 CHAPTER 8, FROM SAMPLES TO POPULATIONS Samples of size = 3. Sample Mea Sample Mea ABC CDE ABD BDE ABE BCE ACD 8.67 BCD 76. ACE ADE ADE ACE BCD 76. ACD 8.67 BCE ABE BDE ABD CDE ABC Mea 78.2 Samples of size = 4. Sample Mea Sample Mea ABCD BCDE 72.5 ABCE ACDE 76. ABDE 79. ABDE 79. ACDE 76. ABCE BCDE 72.5 ABCD Mea 78.2 Samples of size = 5. Sample Mea Sample Mea ABCDE 78.2 ABCDE 78.2 Mea 78.2 Mea 78.2 The mea of the samplig distributio of the meas is 78.2 for each sample size, the same as the mea of the populatio. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

8 SECTION 8.2, ESTIMATING POPULATION MEANS 143 Histogram of Mea1 Histogram of Mea2 Per cet 1 Percet Mea Me a Histogram of Mea3 Histogram of Mea4 Perce t 1 Per cet Mea Mea The graph of the distributio for the sample of size 5 is a sigle spike at Note that the distributios become more cocetrated as the sample size icreases. Sectio 8.2 Statistical Literacy ad Critical Thikig 1 We have 95% cofidece that the limits of g ad g actually do cotai the true populatio mea cholesterol level of all wome. We expect that 95% of such samples will result i cofidece iterval limits that do cotai the populatio mea. 2 The cofidece iterval is from to or from 7.4 to 76. mm Hg. 3 The media ofte omit referece to the cofidece level, which is typically 95%. The word mea should be used istead of the word average sice there are other kids of averages besides the mea. 4 No. A sample ca provide a good estimate of a populatio mea it its size is large eough based o the stadard deviatio ad desired margi or error. The size of the populatio is geerally ot a factor i determiig how good a estimate is. The oly exceptio is whe the populatio itself is small. 5 The statemet is ot sesible because the sigle value of $47, is ot a iterval. 6 The statemet is ot sesible because whe costructig a cofidece for the mea icome i dollars, the margi or error should also be a amout i dollars, ot a percetage. 7 The statemet is ot sesible. As the sample size icreases, the margi of error teds to decrease. 8 The statemet is sesible. Because the cofidece iterval limits are foud by takig the sample mea ad addig ad subtractig the margi of error, the sample mea is the value midway betwee the two cofidece iterval Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

9 144 CHAPTER 8, FROM SAMPLES TO POPULATIONS limits. Cocepts ad Applicatios 2 1. E 2.. The approximate 95% cofidece 1 iterval is x E to E.27. The approximate 95% 64 cofidece iterval is x E to E The approximate 95% 168 cofidece iterval is x E $46, $44,736 to $47, E 2kg. The approximate 95% cofidece 692 iterval is x E to7. 9 The margi of error is 1 The margi of error is 11 The margi of error is 12 The margi of error is E E E E E E E E 5 21 The populatio mea μ is estimated by the sample mea x = g. The 2.62 margi of error is E.. The approximate 95% cofidece 4 iterval is x E to5.659 g. 22 The populatio mea μ is estimated by the sample mea x = The margi 2 28 of error is E 35. The approximate 95% cofidece iterval is 25 x E to1757. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

10 SECTION 8.2, ESTIMATING POPULATION MEANS The populatio mea μ is estimated by the sample mea x = 5. years. The margi of error is E.5. The approximate 95% cofidece 44 iterval is x E to5. years. 24 The populatio mea μ is estimated by the sample mea x = 1.91 pouds. The margi of error is E.27. The approximate 95% cofidece 62 iterval is x E to 2.18 pouds. 25 Use the software of your choice to fid the mea ad stadard deviatio of the sample of = 36 weights. You should fid x = ad s = The margi of error is E ad the 95% cofidece 36 iterval is x E to 221 pouds or 146 < μ < 221 lb. 26 Use the software of your choice to fid the mea ad stadard deviatio of the sample of = 4 weights. You should fid x = ad s = The margi of error is E ad the 95% cofidece 4 iterval is x E to21 or 135 < μ < Use the software of your choice to fid the mea ad stadard deviatio of the sample of = 31 family sizes. a) x = 3.1 b) s = c) The best estimate for the mea family size for the populatio of all America families is the sample mea, E.51 ad the 95% cofidece 31 iterval is x E to3.61 or 2.59 < μ < 3.61 d) The margi of error is e) We ca be 95% cofidet that the iterval from 2.59 to 3.61 cotais the populatio mea μ. That is, we expect 95% of cofidece itervals computed from such samples to cotai the populatio mea. Sice the sample is take etirely from oe eighborhood, this should be a reliable estimate for homes i the eighborhood, but it may ot be very reliable if the populatio of iterest is larger tha the eighborhood. This problem ca also be doe by had although it is somewhat tedious. create a table of x values, x x values ad the squares of those differeces as show followig. We Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

11 146 CHAPTER 8, FROM SAMPLES TO POPULATIONS x x x (x x ) Dividig the total i the first colum by 31, we fid that x = 3.1. Dividig the total i the third colum by 1 = 3, ad the takig the square root of the result, we fid that s = Use the software of your choice to fid the mea ad stadard deviatio of the sample of = 31 household TV sets. a) x = 2.3 b) s = c) The best estimate for the mea family size for the populatio of all America families is the sample mea, E ad the 95% cofidece 31 iterval is x E to2.48 or 1.58 < μ < 2.48 d) The margi of error is e) We ca be 95% cofidet that the iterval from 1.58 to 2.48 cotais the populatio mea μ. That is, we expect 95% of cofidece itervals computed from such samples to cotai the populatio mea umber of household TV sets. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

12 SECTION 8.3, ESTIMATING POPULATION PROPORTIONS 147 Sectio 8.3 Statistical Literacy ad Critical Thikig 1 We have 95% cofidece that the limits of.457 ad.551 actually do cotai the true populatio proportio. We expect that 95% of such samples will result i cofidece iterval limits that do cotai the populatio proportio. 2 The 95% cofidece iterval for p is p + E = or.47 < p < The media ofte omit referece to the cofidece level, which is typically 95%. 4 The claim is ot valid. The coveiece sample (or eve a radom sample) of studets all from the same college is ot likely to represetative of all college studets ad will likely be biased. A large sample caot compesate for a poor samplig procedure. 5 The statemet is ot sesible because the sigle value of.45 is ot a iterval. 6 This statemet does make sese. The give cofidece iterval correspods to the origial statemet. 7 The statemet is sesible. We ca see from the expressio for E that larger values of result i smaller values of E. Also, commo sese suggests that larger samples are likely to result i better ad more accurate (that is, smaller errors) estimates of a populatio proportio. 8 This statemet does ot make sese. The margi of error does ot ivolve the size of the populatio, ad the reliability of the cofidece iterval depeds o the sample size ad the samplig method. Cocepts ad Applicatios p (1 p ).25(1.25) E The 95% cofidece iterval is p E to.3366 ; 9 The approximate margi of error is.1634 < p < p (1 p ).4(1.4) E The 4 95% cofidece iterval is p E to.449 ; 1 The approximate margi of error is.351 < p <.449. p (1 p ).228(1.228) E The 95% cofidece iterval is p E to.254 ; 11 The approximate margi of error is.2 < p <.254. p (1 p ).377(1.377) E The 95% cofidece iterval is p E to.42 ; 12 The approximate margi of error is < p < E.1 1, 1 1 E.3 1, ,112 Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

13 148 CHAPTER 8, FROM SAMPLES TO POPULATIONS E E.25 1,6 E 2 p (1 p ).(1.) The 95% cofidece iterval is p E to.16 ; or 17 The approximate margi of error is.14 < p <.16. p (1 p ).(1.) E The 1 95% cofidece iterval is p E to.7 ; or 18 The approximate margi of error is.143 < p <.7. Doublig the sample size produced oly a small chage i the cofidece iterval, so it does ot appear to be worth the extra cost ad effort. Percetage wise, the iterval was shorteed by 3% from. to.14, but eve that gai hardly seems worth the extra effort. p (1 p ).8(1.8) E The 95% 14 cofidece iterval is p E to.821 ; or.779 < p < The approximate margi of error is For legalized abortio, the margi of error is p (1 p ).51(1.51) E The 95% cofidece iterval is 276 p E to.512 ; or.58 < p <.512. p (1 p ).4(1.4) For casual sex, the margi of error is E The 95% cofidece iterval is p E to.42 ; or.398 < p <.42. p (1 p ).172(1.172) E The 95% cofidece iterval is p E to.191; or 21 a) The approximate margi of error is.3 < p <.191. p (1 p ).132(1.132) E The 95% cofidece iterval is p E to.146 ; or b) The approximate margi of error is.118 < p <.146 p (1 p ).229(1.229) E The 95% cofidece iterval is p E to.243 ; or c) The approximate margi of error is.2 < p <.243. p (1 p ).79(1.79) E ad 26 p (1 p ).74(1.74) E Both are closer to two percetage poits The approximate margis of error are Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

14 SECTION 8.3, ESTIMATING POPULATION PROPORTIONS 149 tha they are to three. However, to claim that the margi of error is 2 percetage poits would be sayig that the survey was more accurate tha it was, so it is safer to claim a margi of error of 3 percetage poits. 23 a) The sample proportio is.98. p (1 p ).98(1.98) E The 95% cofidece iterval is p E to.994 ; or b) From this, the margi of error is.966 < p <.994. c) No. To be a radom sample, all films must have a equal chace of beig chose. p (1 p ).44(1.44) E The 95% cofidece 1 iterval is p E to.471 ; or.419 < p < The margi of error is 25 The sample proportio is p = 9/16 = The margi of error is p (1 p ).5625(1.5625) E The 95% cofidece iterval is 16 p E to.587 ; or.538 < p <.587. Sice we ca be 95% cofidet this iterval cotais the populatio proportio supportig the curret mayor ad this iterval lies etirely above 5%, we should be comfortable i claimig that the mayor will wi a majority of the votes. 26 The sample proportio is p = 13/25 =.52. The margi of error is p (1 p ).52(1.52) E The 95% cofidece iterval is 25 p E to ; or.52 < p < Sice we ca be 95% cofidet this iterval cotais the populatio proportio supportig the curret mayor ad this iterval lies etirely above 5%, we should be comfortable i claimig that the mayor will wi a majority of the votes (just barely). If the sample size were oly 25, with 13 supportig the mayor, the sample proportio would still be 13/25 =.52, but the margi of error would p (1 p ).52(1.52) E The 95% cofidece iterval is 25 p E to.583 ; or.457 < p <.583. Sice it icludes be values that are less tha.5, we would ot wat to make the claim that the mayor will wi a majority of the votes. 27 I the first poll, p = 78/ =.52. The margi of error is p (1 p ).52(1.52) E ad the cofidece iterval is p E to.546. I the secod poll, p = 1285/25 =.514. The margi of error is p (1 p ).514(1.514) E 2 2. ad the cofidece iterval is 25 p E to.534. I the third poll, p = 182/35 =.5. The margi of error is Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

15 CHAPTER 8, FROM SAMPLES TO POPULATIONS p (1 p ).5(1.5) E ad the cofidece iterval is 35 p E to.532. Sice all of these cofidece itervals iclude values that are less tha.5, Martiez caot yet be cofidet of wiig a majority of the votes. 28 a) The margi of error is p (1 p ).34(1.34) E , or.1% 6, to the earest teth of a percet. This level of precisio eables the Bureau to detect very small chages i the uemploymet rate ad is reasoable for that purpose. b) Multiplyig the sample size by 4 will reduce E by 1/2 (E will be halved) sice appears i the deomiator of the square root. c) Multiplyig the sample size by 1/4 will icrease E by a factor of two (E will be doubled) sice appears i the deomiator of the square root. 29 a) The 95% cofidece iterval for the true populatio proportio approvig the Presidet s performace is to.58. b) Sice p (1 p ).54(1.54) E 2 2.4, we coclude that.54(1.54) (.54)(.46).2 or.4. This implies that (.54)(.46) p (1 p ).666(1.666) E a) The actual margi of error is b) This is cosistet with the stated margi of error of 4 percetage poits ,5 E.2 Chapter 8 Review Exercises p (1 p ).32(1.32) E The 95% 221 cofidece iterval is p E to a) The margi of error is b) We are 95% cofidet that the limits of.8 ad.56 actually cotai the populatio proportio p. c) E =.24 from part (a). d) The distributio of the sample proportios will be approximately a ormal distributio. e) Sice appears i the deomiator of the expressio for the margi or error, E, the limits would get closer together as icreases. f) The populatio proportio is a fixed value, ot a radom variable; either it is cotaied withi the limits or it is ot, ad there is o associated probability. The process used to costruct the cofidece iterval has a 95% chace of producig a iterval cotaiig the populatio proportio. That is why we are 95% cofidet that the particular cofidece iterval produced cotais the populatio proportio. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley

16 2 a) E E CHAPTER 8 QUIZ 1 b) If the sample size is larger tha ecessary, the estimate will be better, i.e., the cofidece iterval will be shorter tha plaed. If the sample size is smaller tha ecessary, the estimate will be worse, i.e., the cofidece iterval will be loger tha plaed. c) Sice college studets are more homogeeous tha the populatio as a whole, the stadard deviatio of their IQs will be less tha 16. is proportioal to the square of s, a smaller value of s will lead to a smaller value of. 3 a) The margi of error is E.77. The 95% cofidece 5 Sice iterval is x + E = = 6.4 to 7.8. b) We ca be 95% cofidet that the iterval from 6.4 to 7.8 cotais the populatio mea white blood cell cout. If such samples of size 5 were radomly selected may times, the resultig cofidece iterval limits would cotai the true populatio mea i 95% of those samples. c) The margi of error, from part (a), is E =.7. d) 4 a) E ,5 E.2 b) No. You will ed up with a sample about the right size, but it will be self-selected ad likely to be biased. Chapter 8 Quiz 1 The distributio of the sample meas is approximately a ormal distributio. 2 The distributio of the sample proportios is approximately a ormal distributio. 3 The symbol p represets the sample proportio < p <.68 5 The best estimate of the populatio mea is the mea of the sample (b). 6 The margi of error is oe-half of the legth of the cofidece iterval. Sice the legth of the iterval is =.4, the margi of error is.2. 7 The margi of error is oe-half of the legth of the cofidece iterval. Sice the legth of the iterval is = 4, the margi of error is. 8 The 95% cofidece iterval is x + E = = 4.45 to The 95% cofidece iterval is p + E =.6 +. =.4 to.8. 1 A cofidece iterval for a proportio must have limits that are both betwee ad 1, but these limits are ot. Copyright 12 Pearso Educatio, Ic. Publishig as Addiso-Wesley