Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval Chapter 5 5. a. z α/ = 1.96, uig Table IV, Appedix B, P(0 z 1.96) =.4750. Thu, α/ =.5000.4750 =.05, α = (.05) =.05, ad 1 - α = 1 -.05 =.95. The cofidece level i 100%.95 = 95%. b. z α/ = 1.645, uig Table IV, Appedix B, P(0 z 1.645) =.45. Thu, α/ =.50.45 =.05, α = (.05) =.1, ad 1 α = 1.1 =.90. The cofidece level i 100%.90 = 90%. c. z α/ =.575, uig Table IV, Appedix B, P(0 z.575) =.495. Thu, α/ =.500.495 =.005, α = (.005) =.01, ad 1 α = 1.01 =.99. The cofidece level i 100%.99 = 99%. d. z α/ = 1.8, uig Table IV, Appedix B, P(0 z 1.8) =.4. Thu, α/ =.5.4 =.1, α = (.1) =., ad 1 α = 1. =.80. The cofidece level i 100%.80 = 80%. e. z α/ =.99, uig Table IV, Appedix B, P(0 z.99) =.3389. Thu, α/ =.5000.3389 =.1611, α = (.1611) =.3, ad 1 α = 1.3 =.6778. The cofidece level i 100%.6778 = 67.78%. 5.4 a. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: x ± z.05 5.9 ± 1.96.7 90 5.9 ±.56 (5.34, 6.46) b. For cofidece coefficiet.90, α =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. The cofidece iterval i: x ± z.05 5.9 ± 1.645.7 90 5.9 ±.47 (5.43, 6.37) c. For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.58. The cofidece iterval i: x ± z.005 5.9 ±.58.7 90 5.9 ±.73 (5.17, 6.63) 5.6 If we were to repeatedly draw ample from the populatio ad form the iterval x ± 1.96σ x each time, approximately 95% of the iterval would cotai μ. We have o way of kowig whether our iterval etimate i oe of the 95% that cotai μ or oe of the 5% that do ot. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 137
5.8 a. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: x ± z.05 33.9 ± 1.96 3.3 100 33.9 ±.647 (33.53, 34.547) b. x ± z.05 33.9 ± 1.96 3.3 400 33.9 ±.33 (33.577, 34.3) c. For part a, the width of the iterval i (.647) = 1.94. For part b, the width of the iterval i (.33) =.646. Whe the ample ize i quadrupled, the width of the cofidece iterval i halved. 5.10 a. A poit etimate for the average umber of latex glove ued per week by all healthcare worker with latex allergy i x =19. 3. b. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: 11.9 x ± zα / 19.3 ± 1.96 19.3 ± 3.44 (15.86,.74) 46 c. We are 95% cofidet that the true average umber of latex glove ued per week by all healthcare worker with a latex allergy i betwee 15.86 ad.74. d. The coditio required for the iterval to be valid are: a. The ample elected wa radomly elected from the target populatio. b. The ample ize i ufficietly large, i.e. > 30. 5.1 a. The poit etimate for the mea charitable commitmet of tax-exempt orgaizatio i x = 74.9667. b. From the pritout, the 95% cofidece iterval i (68.371, 81.696). c. The probability of etimatig the true mea charitable commitmet with a igle umber i 0. By etimatig the true mea charitable commitmet with a iterval, we ca be pretty cofidet that the true mea i i the iterval. 138 Chapter 5
5.14 Uig MINITAB, the decriptive tatitic are: Decriptive Statitic: r Variable N Mea Media TrMea StDev SE Mea r 34 0.44 0.4300 0.4310 0.1998 0.0343 Variable Miimum Maximum Q1 Q3 r -0.0800 0.7400 0.95 0.6000 For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i:.1998 x ± zα /.44 ± 1.96.44 ±.067 34 (.355,.4895) We are 95% cofidet that the mea value of r i betwee.355 ad.4895. 5.16 a. Uig MINITAB, the decriptive tatitic are: Decriptive Statitic: Rate Variable N Mea Media TrMea StDev SE Mea Rate 30 79.73 80.00 80.15 5.96 1.09 Variable Miimum Maximum Q1 Q3 Rate 60.00 90.00 76.75 84.00 For cofidece coefficiet.90, α =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. The cofidece iterval i: 5.96 x ± zα / 79.73 ± 1.645 79.73 ± 1.79 30 (77.94, 81.5) b. We are 90% cofidet that the mea participatio rate for all compaie that have 401(k) pla i betwee 77.94% ad 81.5%. c. We mut aume that the ample ize ( = 30) i ufficietly large o that the Cetral Limit Theorem applie. d. Ye. Sice 71% i ot icluded i the 90% cofidece iterval, it ca be cocluded that thi compay' participatio rate i lower tha the populatio mea. e. The ceter of the cofidece iterval i. If 60% i chaged to 80%, the value of will icreae, thu idicatig that the ceter poit will be larger. The value of will decreae if 60% i replaced by 80%, thu cauig the width of the iterval to decreae. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 139
5.18 a. Uig MINITAB, I geerated 30 radom umber uig the uiform ditributio from 1 to 308. The radom umber were: 9, 15, 19, 36, 46, 47, 63, 73, 90, 9, 108, 11, 117, 17, 144, 145, 150, 151, 17, 178, 18, 9, 30, 41, 4, 46, 5, 67, 74, 8 I umbered the 308 obervatio i the order that they appear i the file. Uig the radom umber geerated above, I elected the 9 th, 15 th, 19 th, etc. obervatio for the ample. The elected ample i:.31,.34,.34,.50,.5,.53,.64,.7,.70,.70,.75,.78, 1.00, 1.00, 1.03, 1.04, 1.07, 1.10,.1,.4,.58, 1.01,.50,.57,.58,.61,.70,.81,.85, 1.00 b. Uig MINITAB, the decriptive tatitic for the ample of 30 obervatio are: Decriptive Statitic: carat-amp Variable N Mea Media TrMea StDev SE Mea carat- 30 0.6910 0.7000 0.6965 0.60 0.0478 Variable Miimum Maximum Q1 Q3 carat- 0.100 1.1000 0.5150 1.0000 From above, x =.6910 ad =.60. c. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i:.6 x ± zα /.691± 1.96.691 ±.094 (.597,.785) 30 d. We are 95% cofidet that the mea umber of carat i betwee.597 ad.785. e. From Exercie.47, we computed the populatio mea to be.631. Thi mea doe fall i the 95% cofidece iterval we computed i part d. 5.0 11,98 x = =.6 5,000 For cofidece coefficiet,.95, α =.05 ad α/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: x ± z α/ 1.5.6 ± 1.96 5000.6 ±.04 (.,.30) We are 95% cofidet the mea umber of roache produced per roach per week i betwee. ad.30. 140 Chapter 5
5. a. If x i ormally ditributed, the amplig ditributio of x i ormal, regardle of the ample ize. b. If othig i kow about the ditributio of x, the amplig ditributio of x i approximately ormal if i ufficietly large. If i ot large, the ditributio of x i ukow if the ditributio of x i ot kow. 5.4 a. P(t t 0 ) =.05 where df = 11 t 0 =.01 b. P(t t 0 ) =.01 where df = 9 t 0 =.81 c. P(t t 0 ) =.005 where df = 6 Becaue of ymmetry, the tatemet ca be rewritte P(t t 0 ) =.005 where df = 6 t 0 = 3.707 d. P(t t 0 ) =.05 where df = 18 t 0 = 1.734 5.6 For thi ample, x 1567 x = = = 97.9375 16 = = ( x) 1567 x 155,867 = 16 1 16 1 = 1.6463 = 159.99 a. For cofidece coefficiet,.80, α = 1.80 =.0 ad α/ =.0/ =.10. From Table VI, Appedix B, with df = 1 = 16 1 = 15, t.10 = 1.341. The 80% cofidece iterval for μ i: 1.6463 x ± t.10 97.94 ± 1.341 97.94 ± 4.40 (93.700, 10.180) 16 b. For cofidece coefficiet,.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table VI, Appedix B, with df = 1 = 4 1 = 3, t.05 =.131. The 95% cofidece iterval for μ i: x ± t.05 97.94 ±.131 1.6463 16 97.94 ± 6.737 (91.03, 104.677) The 95% cofidece iterval for μ i wider tha the 80% cofidece iterval for μ foud i part a. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 141
c. For part a: We are 80% cofidet that the true populatio mea lie i the iterval 93.700 to 10.180. For part b: We are 95% cofidet that the true populatio mea lie i the iterval 91.03 to 104.677. The 95% cofidece iterval i wider tha the 80% cofidece iterval becaue the more cofidet you wat to be that μ lie i a iterval, the wider the rage of poible value. 5.8 a. Uig MINITAB, the decriptive tatitic are: Decriptive Statitic: MTBE Variable N N* Mea SE Mea StDev Miimum Q1 Media Q3 Maximum MTBE 1 0 97. 3.8 113.8 8.00 1.0 50.5 146.0 367.0 A poit etimate for the true mea MTBE level for all well ite located ear the New Jerey gaolie ervice tatio i x = 97.. b. For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table VI, Appedix B, with df = 1 = 1 1 = 11, t.005 = 3.106. The 99% cofidece iterval i: 113.8 x ± t.005 97. ± 3.106 97. ± 10.04 ( 4.84, 199.4) 1 We are 99% cofidet that the true mea MTBE level for all well ite located ear the New Jerey gaolie ervice tatio i betwee 4.84 ad 199.4. c. We mut aume that the data were ampled from a ormal ditributio. We will ue the four method to check for ormality. Firt, we will look at a hitogram of the data. Uig MINITAB, the hitogram of the data i: Hitogram of MTBE 5 4 Frequecy 3 1 0 0 50 100 150 00 MTBE 50 300 350 14 Chapter 5
From the hitogram, the data do ot appear to be moud-haped. Thi idicate that the data may ot be ormal. Next, we look at the iterval x ±, x ±, x ± 3. If the proportio of obervatio fallig i each iterval are approximately.68,.95, ad 1.00, the the data are approximately ormal. Uig MINITAB, the ummary tatitic are: x ± 97. ± 113.8 ( 16.6, 11.0) 10 of the 1 value fall i thi iterval. The proportio i.83. Thi i ot very cloe to the.68 we would expect if the data were ormal. x ± 97. ± (113.8) 97. ± 7.6 ( 130.4, 34.8) 11 of the 1 value fall i thi iterval. The proportio i.9. Thi i a omewhat maller tha the.95 we would expect if the data were ormal. x ± 97. ± 3(113.8) 97. ± 341.4 ( 44., 438.6) 1 of the 1 value fall i thi iterval. The proportio i 1.00. Thi i exactly the 1.00 we would expect if the data were ormal. From thi method, it appear that the data may ot be ormal. Next, we look at the ratio of the IQR to. IQR = Q U Q L = 146.0 1.0 = 134.0. IQR 134.0 = = 1.18 Thi i omewhat maller tha the 1.3 we would expect if the data 113.8 were ormal. Thi method idicate the data may ot be ormal. Fially, uig MINITAB, the ormal probability plot i: Probability Plot of MTBE Normal - 95% CI 99 95 90 Mea 97.17 StDev 113.8 N 1 AD 0.99 P-Value 0.01 80 Percet 70 60 50 40 30 0 10 5 1-300 -00-100 0 100 MTBE 00 300 400 500 Sice the data do ot form a fairly traight lie, the data may ot be ormal. From above, the all method idicate the data may ot be ormal. It appear that the data probably are ot ormal. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 143
5.30 We mut aume that the ditributio of the LOS' for all patiet i ormal. a. For cofidece coefficiet.90, α = 1.90 =.10 ad α/ =.10/ =.05. From Table VI, Appedix B, with df = 1 = 0 1 = 19, t.05 = 1.79. The 90% cofidece iterval i: x ± t.05 3.8 ± 1.79 1. 0 3.8 ±.464 (3.336, 4.64) b. We are 90% cofidet that the mea LOS i betwee 3.336 ad 4.64 day. c. 90% cofidece mea that if repeated ample of ize are elected from a populatio ad 90% cofidece iterval are cotructed, 90% of all iterval thu cotructed will cotai the populatio mea. 5.3 a. The 95% cofidece iterval for the mea urface roughe of coated iterior pipe i (1.63580,.160). b. No. Sice.5 doe ot fall i the 95% cofidece iterval, it would be very ulikely that the average urface roughe would be a high a.5 micrometer. 5.34 a. The populatio i the et of all DOT permaet cout tatio i the tate of Florida. b. Ye. There are everal type of route icluded i the ample. There are 3 recreatioal area, 7 rural area, 5 mall citie, ad 5 urba area. c. Uig MINITAB, the decriptive tatitic are: Decriptive Statitic: 30th hour, 100th hour Variable N Mea Media TrMea StDev SE Mea 30th hou 0 06 064 165 14 74 100th ho 0 096 1999 048 103 69 Variable Miimum Maximum Q1 Q3 30th hou 5 4905 149 3068 100th ho 9 4815 1318 877 For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table VI, Appedix B, with df = 1 = 0 1 = 19, t.05 =.093. The 95% cofidece iterval i: 1,4 x ± t.05,06 ±.093,06 ± 57.84 (1,633.16,,778.84) 0 We are 95% cofidet that the mea traffic cout at the 30 th highet hour i betwee 1,633.16 ad,778.84. d. We mut aume that the ditributio of the traffic cout at the 30th highet hour i ormal. From the tem-ad-leaf diplay, the data look fairly moud-haped. Thu, the aumptio of ormality i probably met. 144 Chapter 5
e. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table VI, Appedix B, with df = 1 = 0 1 = 19, t.05 =.093. The 95% cofidece iterval i: 1,03 x ± t.05,096 ±.093,096 ± 563.01 (1,53.99,,659.01) 0 We are 95% cofidet that the mea traffic cout at the 100 th highet hour i betwee 1,53.99 ad,659.01. We mut aume that the ditributio of the traffic cout at the 100th highet hour i ormal. From the tem-ad-leaf diplay, the data look fairly moud-haped. Thu, the aumptio of ormality i probably met. f. If μ =,700, it i very poible that it i the mea cout for the 30 th highet hour. It fall i the 95% cofidece iterval for the mea cout for the 30 th highet hour. It i ot very likely that the mea cout for the 100 th highet hour i,700. It doe ot fall i the 95% cofidece iterval for the mea cout for the 100 th highet hour. (See part c ad e above.) 5.36 By the Cetral Limit Theorem, the amplig ditributio of i approximately ormal with pq mea μ ˆp = p ad tadard deviatio σ ˆp =. 5.38 a. The ample ize i large eough if the iterval ˆp ± 3σ p ˆ doe ot iclude 0 or 1. ˆp ± 3σ p ˆ ˆp ± 3 pq ˆp ± ˆˆ 3 pq.88 ±.88(1.88) 11 (.791,.969).88 ±.089 Sice the iterval lie withi the iterval (0, 1), the ormal approximatio will be adequate. b. For cofidece coefficiet.90, α =.10 ad α/ =.05. From Table IV, Appedix B, z.05 = 1.645. The 90% cofidece iterval i: pq ± z ˆp ± 1.645 p ˆ.05 pq ˆˆ.88 ± 1.645.88(.1) 1.645 11.88 ±.049 (.831,.99) c. We mut aume that the ample i a radom ample from the populatio of iteret. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 145
5.40 a. 15 Of the 50 obervatio, 15 like the product p ˆ = =.30. 30 To ee if the ample ize i ufficietly large: ˆp ± 3σ ˆp ± 3 ˆp pq ˆˆ.3 ±.3(.7) 3 50.3 ±.194 (.106,.494) Sice thi iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. For the cofidece coefficiet.80, α =.0 ad α/ =.10. From Table IV, Appedix B, z.10 = 1.8. The cofidece iterval i: ˆp ± z.10 pq ˆˆ.3 ± 1.8.3(.7) 50.3 ±.083 (.17,.383) b. We are 80% cofidet the proportio of all coumer who like the ew ack food i betwee.17 ad.383. 5.4 a. The poit etimate of p i p ˆ =.11. b. To ee if the ample ize i ufficietly large: pq ˆˆ.11(.89) ± 3σ ± 3.11± 3.11 ±.077 (.033,.187) 150 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: pq ˆˆ.11(.89) ± z.05.11± 1.645.11 ±.05 (.06,.16) 150 c. We are 95% cofidet that the true proportio of MSDS that are atifactorily completed i betwee.06 ad.16. 5.44 a. The poit etimate of p i x 16 = = =.05. 308 To ee if the ample ize i ufficietly large: pq ˆˆ.05(.948) ± 3σ ± 3.05 ± 3.05 ±.038 (.014,.090) 308 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. 146 Chapter 5
For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.58. The cofidece iterval i: pq ˆˆ.05(.948) ± z.05.05 ±.58.05 ±.033 (.019,.085) 308 We are 99% cofidet that the true proportio of diamod for ale that are claified a D color i betwee.019 ad.085. x 81 b. The poit etimate of p i = = =.63. 308 To ee if the ample ize i ufficietly large: pq ˆˆ.63(.737) ± 3σ ± 3.63 ± 3.63 ±.075 (.188,.338) 308 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.58. The cofidece iterval i: pq ˆˆ.63(.737) ± z.05.63 ±.58.63 ±.065 (.198,.38) 308 We are 99% cofidet that the true proportio of diamod for ale that are claified a VS1 clarity, i betwee.198 ad.38. 5.46 a. The populatio i all eior huma reource executive at U.S. compaie. b. The populatio parameter of iteret i p, the proportio of all eior huma reource executive at U.S. compaie who believe that their hirig maager are iterviewig too may people to fid qualified cadidate for the job. c. The poit etimate of p i large: x 11 = = =.4. To ee if the ample ize i ufficietly 50 pq ˆˆ.4(.58) ± 3σ ± 3.4 ± 3.4 ±.066 (.354,.486) 50 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 147
d. For cofidece coefficiet.98, α =.0 ad α/ =.0/ =.01. From Table IV, Appedix B, z.01 =.33. The cofidece iterval i: pq ˆˆ.4(.58) ± z.01.4 ±.33.4 ±.051 (.369,.471) 50 We are 98% cofidet that the true proportio of all eior huma reource executive at U.S. compaie who believe that their hirig maager are iterviewig too may people to fid qualified cadidate for the job i betwee.369 ad.471. e. A 90% cofidece iterval would be arrower. If the iterval wa arrower, it would cotai fewer value, thu, we would be le cofidet. 5.48 a. The poit etimate of p i ˆp = x/ = 35/55 =.636. b. We mut check to ee if the ample ize i ufficietly large: ˆp ± 3σ p ˆ ˆp ± ˆˆ 3 pq.636 ±.636(.364) 3 55.636 ±.195 (.441,.831) Sice the iterval i wholly cotaied i the iterval (0, 1) we may aume that the ormal approximatio i reaoable. For cofidece coefficiet,.99, α =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.575. The cofidece iterval i: ˆp ± z.005 pq ˆˆ.636 ±.636(.364).575 55.636 ±.167 (.469,.803) c. We are 99% cofidet that the true proportio of fatal accidet ivolvig childre i betwee.469 ad.803. d. The ample proportio of childre killed by air bag who were ot wearig eat belt or were improperly retraied i 4/35 =.686. Thi i rather large proportio. Whether a child i killed by a airbag could be related to whether or ot he/he wa properly retraied. Thu, the umber of childre killed by air bag could poibly be reduced if the child were properly retraied. 5.50 The poit etimate of p i x 36 = = =.434. 83 To ee if the ample ize i ufficietly large: pq ˆˆ.434(.566) ± 3σ ± 3.434 ± 3.434 ±.163 (.71,.597) 83 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. 148 Chapter 5
For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: pq ˆˆ.434(.566) ± z.05.434 ± 1.96.434 ±.107 (.37,.541) 83 We are 95% cofidet that the true proportio of healthcare worker with latex allergie actually upect the he or he actually ha the allergy i betwee.37 ad.541. 5.5 To compute the eceary ample ize, ue = ( ) zα / σ where α = 1.95 =.05 ad α/ =.05/ =.05. SE From Table IV, Appedix B, z.05 = 1.96. Thu, = (1.96) (7.).3 = 307.38 308 You would eed to take 308 ample. 5.54 a. To compute the eeded ample ize, ue: = ( zα / ) pq SE where z.05 = 1.96 from Table IV, Appedix B. Thu, = (1.96) (.)(.8).08 = 96.04 97 You would eed to take a ample of ize 97. b. To compute the eeded ample ize, ue: = ( zα / ) pq SE = (1.96 (.5)(.5) ).08 = 150.065 151 You would eed to take a ample of ize 151. 5.56 a. For a width of 5 uit, SE = 5/ =.5. To compute the eeded ample ize, ue = ( ) zα / σ SE where α = 1.95 =.05 ad α/ =.05. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 149
From Table IV, Appedix B, z.05 = 1.96. Thu, = (1.96) (14).5 = 10.47 11 You would eed to take 11 ample at a cot of 11($10) = $110. Ye, you do have ufficiet fud. b. For cofidece coefficiet.90, α = 1.90 =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. = (1.645) (14).5 = 84.86 85 You would eed to take 85 ample at a cot of 85($10) = $850. You till have ufficiet fud but have a icreaed rik of error. 5.58 The ample ize will be larger tha eceary for ay p other tha.5. 5.60 a. The cofidece level deired by the reearcher i 90%. b. The amplig error deired by the reearcher i SE =.05. c. For cofidece coefficiet.90, α =.10 ad α/ =.10/ =.05. From Table IV, x 64 Appedix B, z.05 = 1.645. From the problem, we will ue = = =.604 106 to etimate p. Thu, ( zα /) pq 1.645.604(.396) = = = ( SE).05 58.9 59 Thu, we would eed a ample of ize 59. 5.6 For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. For thi tudy, = ( z ) 1.96 (5) SE 1 α / σ = 96.04 97 The ample ize eeded i 97. 150 Chapter 5
5.64 For cofidece coefficiet.90, α =.10 ad α/ =.05. From Table IV, Appedix B, z.05 = 1.645. For a width of.06, SE =.06/ =.03 The ample ize i = ( zα /) pq (.1645) (.17)(.83) = SE.03 = 44. 45 You would eed to take = 45 ample. 5.66 To compute the eceary ample ize, ue = ( z ) SE α / σ where α = 1.90 =.10 ad α/ =.05. From Table IV, Appedix B, z.05 = 1.645. Thu, = (1.645) (10) 1 = 70.6 71 5.68 a. To compute the eeded ample ize, ue = ( z ) SE α / σ where α = 1.90 =.10 ad α/ =.05. From Table IV, Appedix B, z.10 = 1.645. Thu, = (1.645) ().1 = 1,08.41 1,083 b. A the ample ize decreae, the width of the cofidece iterval icreae. Therefore, if we ample 100 part itead of 1,083, the cofidece iterval would be wider. c. To compute the maximum cofidece level that could be attaied meetig the maagemet' pecificatio, = ( z ) SE α / σ 100 = / ( z )() 100(.01) α ( z α /) = =.5 z α/ =.5.1 4 Uig Table IV, Appedix B, P(0 z.5) =.1915. Thu, α/ =.5000.1915 =.3085, α = (.3085) =.617, ad 1 α = 1.617 =.383. The maximum cofidece level would be 38.3%. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 151
5.70 σ x = σ N N a. b. σ x σ x 00 500 1000 = = 4.90 1000 500 00 5000 1000 = = 5.66 1000 5000 c. σ x = 00 10,000 1000 1000 10,000 = 6.00 d. σ x 00 100,000 1000 = = 6.93 1000 100,000 5.7 a. For = 36, with the fiite populatio correctio factor: N 4 5000 64 ˆ σ x = / = = 3.987 N 64 5000 =.9807 without the fiite populatio correctio factor: 4 ˆ σ = x / = 64 = 3 ˆ σ x without the fiite populatio correctio factor i lightly larger. b. For = 400, with the fiite populatio correctio factor: N 4 5000 400 ˆ σ x = / = = 1..9 N 400 5000 = 1.1510 without the fiite populatio correctio factor: 4 ˆ σ = x / = 400 = 1. c. I part a, i maller relative to N tha i part b. Therefore, the fiite populatio correctio factor did ot make a much differece i the awer i part a a i part b. 5.74 A approximate 95% cofidece iterval for μ i: x ± ˆ σ x x ± N 14 375 40 4 ± N 40 375 4 ± 4.184 (417.816, 46.184) 15 Chapter 5
5.76 a. For N =,193, = 3, x =116,754, ad = 39,185, the 95% cofidece iterval i: N 39,185,193 3 x ± ˆ σ x x ± 116,754 ± N 3,193 116,754 ± 4,974.06 (111,779.94, 11,78.06) b. We are 95% cofidet that the mea alary of all vice preidet who ubcribe to Quality Progre i betwee $111,777.94 ad $11,78.06. 5.78 a. The populatio of iteret i the et of all houehold headed by wome that have icome of $5,000 or more i the databae. b. Ye. Sice /N = 1,333/5,000 =.053 exceed.05, we eed to apply the fiite populatio correctio. c. The tadard error for ˆp hould be: ˆ σ p ˆ(1 ) N.708(1.708) 5,000 1,333 = = N 1333 5,000 =.01 d. For cofidece coefficiet.90, α = 1.90 =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. The approximate 90% cofidece iterval i: ˆp ± 1.645 ˆ σ p ˆ.708 ± 1.645(.01) (.688,.78) 5.80 For N = 1,500, = 35, x = 1, ad = 14, the 95% cofidece iterval i: x ± ˆ σ x x ± N N 1 ± 14 1,500 35 1 ± 41.43 35 1,500 ( 40.43, 4.43) We are 95% cofidet that the mea error of the ew ytem i betwee -$40.43 ad $4.43. 5.8 a. For a mall ample from a ormal ditributio with ukow tadard deviatio, we ue the t tatitic. For cofidece coefficiet.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table VI, Appedix B, with df = 1 = 3 1 =, t.05 =.074. b. For a large ample from a ditributio with a ukow tadard deviatio, we ca etimate the populatio tadard deviatio with ad ue the z tatitic. For cofidece coefficiet.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. c. For a mall ample from a ormal ditributio with kow tadard deviatio, we ue the z tatitic. For cofidece coefficiet.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 153
d. For a large ample from a ditributio about which othig i kow, we ca etimate the populatio tadard deviatio with ad ue the z tatitic. For cofidece coefficiet.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. e. For a mall ample from a ditributio about which othig i kow, we ca ue either z or t. 5.84 a. Of the 400 obervatio, 7 had the characteritic ˆp = 7/400 =.5675. To ee if the ample ize i ufficietly large: ˆp ± 3σ p ˆ ˆp ± 3 pq ˆp ± ˆˆ 3 pq.5675 ±.5675(.435) 3 400.5675 ±.0743 (.493,.6418) Sice the iterval lie withi the iterval (0, 1), the ormal approximatio will be adequate. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: ˆp ± z.05 pq ± 1.96 pq ˆˆ.5675 ± 1.96.5675(.435) 400.5675 ±.0486 (.5189,.6161) b. For thi problem, SE =.0. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. Thu, = ( z /) pq (1.96) (.5675)(.435) α = =,357.,358 SE.0 Thu, the ample ize wa,358. 5.86 a. The fiite populatio correctio factor i: ( N ) N = (,000 50),000 =.9874 b. The fiite populatio correctio factor i: ( N ) N = (100 0) 100 =.8944 c. The fiite populatio correctio factor i: ( N ) N = (1,500 300) 1,500 =.8944 154 Chapter 5
5.88 a. From the pritout, the 90% cofidece iterval i (4.77, 6.184). We are 90% cofidet that the mea umber of office operated by all Florida law firm i betwee 4.77 ad 6.184. b. From the hitogram, it appear that the data probably are ot from a ormal ditributio. The data appear to be kewed to the right. c. The iterval cotructed i part a deped o the aumptio that the data came from a ormal ditributio. From part b, it appear that thi aumptio i ot valid. Thu, the cofidece iterval i probably ot valid. 5.90 a. The poit etimate of p i x 67 = = =.638. 105 b. To ee if the ample ize i ufficietly large: pq ˆˆ.638(.36) ± 3σ ˆ ˆ p p± 3.638 ± 3.638 ±.141 (.497,.779) 105 Sice the iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: pq ˆˆ.638(.36) ± z.05.638 ± 1.96.638 ±.09 (.546,.730) 105 c. We are 95% cofidet that the true proportio of o-the-job homicide cae that occurred at ight i betwee.546 ad.730. 5.9 a. Uig MINITAB, the decriptive tatitic are: Decriptive Statitic: NJValue Variable N N* Mea SE Mea StDev Miimum Q1 Media Q3 Maximum NJValue 0 0 440.4 67.8 303.0 159.0 1.3 97.5 660.5 1190.0 For cofidece coefficiet.95, α =.05 ad α/ =.05/ =.05. From Table VI, Appedix B, with df = 1 = 0 1 = 19, t.05 =.093. The 95% cofidece iterval i: 303.0 x ± t.05 440.4 ±.093 440.4 ± 141.81 (98.59, 58.1) 0 b. We are 95% cofidet that the true mea ale price i betwee $98,590 ad $58,10. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 155
c. "95% cofidece" mea that i repeated amplig, 95% of all cofidece iterval cotructed will cotai the true mea ale price ad 5% will ot. d. Uig MINITAB, a hitogram of the data i: Hitogram of NJValue 9 8 7 Frequecy 6 5 4 3 1 0 00 400 600 800 NJValue 1000 100 Sice the ample ize i mall ( = 0), we mut aume that the ditributio of ale price i ormal. From the hitogram, it doe ot appear that the data come from a ormal ditributio. Thu, thi cofidece iterval i probably ot valid. 5.94 a. For cofidece coefficiet.90, α =.10 ad α/ =.05. From Table IV, Appedix B, z.05 = 1.645. The 90% cofidece iterval i: σ x ± z.05 x ± 1.645 1. ± 1.645 10 100 1. ± 1.645 (10.555, 13.845) We are 90% cofidet that the mea umber of day of ick leave take by all it employee i betwee 10.555 ad 13.845. b. For cofidece coefficiet.99, α =.01 ad α/ =.005. From Table IV, Appedix B, z.005 =.58. The ample ize i = ( ) zα / σ (.58) (10) = SE = 166.4 167 You would eed to take = 167 ample. 156 Chapter 5
5.96 a. For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.58. The cofidece iterval i:.1 x ± zα / 1.13 ±.58 1.13 ±.67 7 (.46, 1.80) We are 99% cofidet that the mea umber of peck at the blue trig i betwee.46 ad 1.80. b. Ye. The mea umber of peck at the white trig i 7.5. Thi value doe ot fall i the 99% cofidet iterval for the blue trig foud i part a. Thu, the chicke are more apt to peck at white trig. 5.98 a. Firt we mut compute ˆp : ˆp = x = 14 159 =.78 To ee if the ample ize i ufficietly large: pq ˆˆ.78() ˆp ± 3σ p ˆ ˆp ± 3.78 ± 3.78 ±.099 (.681,.879) 159 Sice thi iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. For cofidece coefficiet.90, α =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. The cofidece iterval i: ˆp ± z.05 pq ˆp ± 1.645 pq ˆˆ.78 ± 1.645.78(.) 159.78 ±.054 (.76,.834) We are 90% cofidet that the true proportio of all truck driver who uffer from leep apea i betwee.76 ad.834. b. Sleep reearcher believe that 5% of the populatio uffer from obtructive leep apea. Sice the 90% cofidece iterval for the proportio of truck driver who uffer from leep apea doe ot cotai.5, it appear that the true proportio of truck driver who uffer from leep apea i larger tha the proportio of the geeral populatio. 5.100 a. The populatio of iteret i the et of all debit cardholder i the U.S. c. Of the 15 obervatio, 180 had ued the debit card to purchae a product or ervice o the Iteret 180 p ˆ = =.144 15 Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 157
To ee if the ample ize i ufficietly large: pq ˆˆ.144(.856) ± 3σ ˆ ˆ p p± 3.144 ± 3.144 ±.030 (.114,.174) 15 Sice thi iterval i wholly cotaied i the iterval (0, 1), we may coclude that the ormal approximatio i reaoable. d. For cofidece coefficiet.98, α = 1.98 =.0 ad α/ =.0/ =.01. From Table IV, Appedix B, z.01 =.33. The cofidece iterval i: pq ˆˆ.144(.856) ± z.01.144 ±.33.144 ±.03 (.11,.167) 15 We are 98% cofidet that the proportio of debit cardholder who have ued their card i makig purchae over the Iteret i betwee.11 ad.167. e. Sice we would have le cofidece with a 90% cofidece iterval tha with a 98% cofidece iterval, the 90% iterval would be arrower. 5.10 a. Of the 100 cacer patiet, 7 were fired or laid off = 7/100 =.07. To ee if the ample ize i ufficietly large: ˆp ± 3σ p ˆ ˆp ± 3 pq ˆp ± ˆˆ 3 pq.07 ±.07(.93) 3 100.07 ±.077 (.007,.145) Sice the iterval doe ot lie withi the iterval (0, 1), the ormal approximatio will ot be adequate. We will go ahead ad cotruct the iterval ayway. For cofidece coefficiet.90, α =.10 ad α/ =.10/ =.05. From Table IV, Appedix B, z.05 = 1.645. The cofidece iterval i: ˆp ± z.05 pq ˆp ± 1.645 pq ˆˆ.07 ± 1.645.07(.93) 100.07 ±.04 (.08,.11) Covertig thee to percetage, we get (.8%, 11.%). b. We are 90% cofidet that the percetage of all cacer patiet who are fired or laid off due to their ille i betwee.8% ad 11.%. c. Sice the rate of beig fired or laid off for all America i 1.3% ad thi value fall outide the cofidece iterval i part b, there i evidece to idicate that employee with cacer are fired or laid off at a rate that i greater tha that of all America. 158 Chapter 5
5.104 a. x 996 ˆp = = 10,000 The approximate 95% cofidece iterval i: ˆp ± p ˆ(1 ) N N.996 ±.996(.0704) 500,000 10,000 10,000 500,000.996 ±.000006413.996 ±.0051 (.945,.9347) b. Oly 10,000 100% = % of the ubcriber retured the quetioaire. Ofte i mail 500,000 urvey, thoe that repod are thoe with trog view. Thu, the 10,000 that repoded may ot be repreetative. I would quetio the etimate i part a. 5.106 a. The poit etimate for the fractio of the etire market who refue to purchae bar i: x 3 p ˆ = = =.094 44 b. To ee if the ample ize i ufficiet: pq ˆˆ (.094)(.906) p ˆ ± 3.094 ± 3.094 ±.056 (.038,.150) 44 Sice the iterval above i cotaied i the iterval (0, 1), the ample ize i ufficietly large. c. For cofidece coefficiet.95, α = 1.95 =.05 ad α/ =.05/ =.05. From Table IV, Appedix B, z.05 = 1.96. The cofidece iterval i: pq ˆˆ (.094)(.906) p ˆ ± z.05.094 ± 1.96.094 ±.037 (.057,.131) 44 d. The bet etimate of the true fractio of the etire market who refue to purchae bar ix moth after the poioig i.094. We are 95% cofidet the true fractio of the etire market who refue to purchae bar ix moth after the poioig i betwee.057 ad.131. Iferece Baed o a Sigle Sample: Etimatio with Cofidece Iterval 159
5.108 The boud i SE =.1. For cofidece coefficiet.99, α = 1.99 =.01 ad α/ =.01/ =.005. From Table IV, Appedix B, z.005 =.575. We etimate p with from Exercie 7.48 which i =.636. Thu, = zα pq = 153.5 154 SE.1 ( /).575 (.636)(.364) The eceary ample ize would be 154. 5.110 Sice the maufacturer wat to be reaoably certai the proce i really out of cotrol before huttig dow the proce, we would wat to ue a high level of cofidece for our iferece. We will form a 99% cofidece iterval for the mea breakig tregth. For cofidece coefficiet.99, α =.01 ad α/ =.01/ =.005. From Table VI, Appedix B, with df = 1 = 9 1 = 8, t.005 = 3.355. The 99% cofidece iterval i:.9 x ± t.005 985.6 ± 3.355 985.6 ± 5.61 (959.99, 1,011.1) 9 We are 99% cofidet that the true mea breakig tregth i betwee 959.99 ad 1,011.1. Sice 1,000 i cotaied i thi iterval, it i ot a uuual value for the true mea breakig tregth. Thu, we would recommed that the proce i ot out of cotrol. 160 Chapter 5