CHAPTER 6. Confidence Intervals. 6.1 (a) y = 1269; s = 145; n = 8. The standard error of the mean is = s n = = 51.3 ng/gm.

Transcription

1 } CHAPTER 6 Cofidece Iterval 6.1 (a) y = 1269; = 145; = 8. The tadard error of the mea i SE ȳ = = = 51.3 g/gm. (b) y = 1269; = 145; = 30. The tadard error of the mea i ȳ = 145 = 26.5 g/gm (a) 15/ 25 = 3.0 cm (b) 15/ 100 = 1.5 cm 6.3 y = 9.520; = 1.429; 1.429/ 5 = gm/kg. 6.4 (a) 3.06/ 86 =.33 mm (b) 25 Frequecy Tail legth (mm) SE } 6.5 (a) We would predict the SD of the ew meauremet to be about 3 mm becaue thi i our etimate (baed o Exercie 6.4) of the populatio SD. (b) We would expect the SE of the ew meauremet to be 3/ mm. 6.6 To covey the homogeeity of the group of rat, the 10 gm hould be the SD, ice the SD decribe variability amog the rat. (The SE decribe the preciio of the ample mea, but thi deped o the ample ize.)

2 (a) the SE (b) the SD (c) the SE 6.8 ad 6.9 See Sectio III of thi Maual (a) y = mg; = mg; = 5. The tadard error of the mea i ȳ = = mg. 5 (b) The degree of freedom are - 1 = 5-1 = 4. The critical value i t.05 = The 90% cofidece iterval for µ i y ± t ± 2.132( ) (23.4,40.0) or 23.4 < µ < 40.0 mg (a) The degree of freedom are - 1 = 5-1 = 4. The critical value i t.025 = The 95% cofidece iterval for µ i y ± t ± 2.776( ) (20.9,42.5) or 20.9 < µ < 42.5 mg. (b) We are 95% cofidet that the mea thymu glad weight i the populatio of chick embryo i betwee 20.9 ad 42.5 mg (a) y = 28.7; = ; / 6 = µg/ml ± (2.571)(1.9) (23.8,33.6) or 23.8 < µ < 33.6 µg/ml. (b) µ = mea blood erum cocetratio of Getamici (1.5 hour after ijectio of 10 mg/kg body weight) i healthy three-year-old female Suffolk heep. (c) No. The "95%" refer to the percetage (i a meta-experimet) of cofidece iterval that would cotai µ. Sice the width of a cofidece iterval deped o, the percetage of obervatio cotaied i the cofidece iterval alo deped o, ad would be very mall if were large (a) Thi tatemet i fale. The cofidece iterval allow u to make a iferece cocerig the mea of the etire populatio. We kow that < y < (b) Thi tatemet i true. (See part (a).) 6.14 Thi tatemet i fale. The cofidece iterval cocer the mea of the populatio. It doe ot tell u where idividual data poit lie The higher the cofidece level, the greater the legth of the iterval. Thu, 90%, 85%, ad 80% cofidece iterval correpod to (.822,.858), (.824,.856), ad (.826,.854), repectively.

3 (a) y = 13.0; = 12.4; = 10. The degree of freedom are - 1 = 10-1 = 9. The critical value i t.05 = The 95% cofidece iterval for µ i y ± t ± 2.262( ) (4.1,21.9) or 4.1 < µ < 21.9 pg/ml. (b) We are 95% cofidet that the average drop i HBE level from Jauary to May i the populatio of all participat i phyical fite program like the oe i the tudy i betwee 4.1 ad 21.9 pg/ml A hitogram ad ormal probability plot of the data, how here, upport the ue of a ormal curve model for thee data core 6.18 (a) 5,111 ± (2.306)(818/ 9 ) (4482,5740) or 4,482 < µ < 5,740 uit (b) We are 95% cofidet that the average ivertae activity of all fugal tiue icubated at 95% relative humidity for 24 hour i betwee 4,482 uit ad 5,740 uit. (c) To check that the data are from a ormal populatio we ca make a ormal probability plot ± (2.042)(1.84/ 36 ) (5.58,6.84) or 5.58 < µ < 6.84 µg/dl y = 1.20; =.14; = 50. The degree of freedom are 50-1 = 49. From Table 4 with df = 50 (the df value cloet to 49) we fid that t.05 = The 90% cofidece iterval for µ i y ± t ± 1.676( ) (1.17,1.23) or 1.17 < µ < 1.23 mm We are 95% cofidet that the mea Bayley Idex of prematurely bor ifat who receive itraveoufeedig olutio i betwee 93.8 ad (Although the ceter of the iterval i 97.95, which i le

4 87 tha the geeral populatio average of 100, the iterval exted above 100, o we caot be ure that µ i le tha 100.) 6.22 y = 10.3; = 0.9; = 101. There are 100 degree of freedom, o t.025 = The 95% cofidece iterval for µ i 10.3 ± 1.984( ) (10.12,10.48) or < µ < g/dli =.975. I Table 3, z = 1.96 correpod to a area of.975. (A t ditributio with df = i a ormal ditributio.) = I Table 3, a area of.9975 correpod to z = A t ditributio with df = i a ormal ditributio; thu, t.0025 = 2.81 whe df = The cofidece coefficiet i approximately 68%, becaue a t ditributio with df = i a ormal ditributio, ad the area uder a ormal curve betwee z = ad z = 1.00 i approximately 68% (a) Smaller. The area icluded betwee t = ad t = 1.00 i maller for a Studet' t ditributio tha for a ormal ditributio. (b) It i ot affected, becaue of the Cetral Limit Theorem (a) Gueed SD = 20 kg; mut atify the iequality 20 5 o = 16. (b) mut atify the iequality 40 5 o = 64. The required ample ize doe ot double, but rather i four time a large We ue the iequality Gueed SD Deired SE. I thi cae, the deired SE i 3 mg/dl ad the gueed SD i 40 mg/dl. Thu, the iequality i 40 3 or 40 3 which mea that 177.8, o a ample of = 178 me i eeded Gueed SD = 80 g (a) The deired SE i 20 g, o mut atify which yield 16.

5 88 (b) The deired SE i 15 g, o mut atify which yield 28.4, o = Gueed SD = 1.5 i The SE hould be o more tha.25 i, o mut atify which yield The fact that the mea i le tha the SD cat doubt o the coditio that the populatio i ormal, for the followig reao. I a ormal populatio, about 15% of the obervatio fall more tha oe SD below the mea, wherea thi ample caot have ay obervatio that far below the mea becaue y - i egative ad the oberved variable (erum SGOT) caot be egative (a) There were 36 cell, but oly eve guiea pig, o there i a hierarchical tructure i the data, which ugget that the obervatio are ot idepedet. (b) Number of brach egmet Frequecy Total Frequecy Number of brach egmet The ditributio ha two or perhap three mode, which may reflect the hierarchical tructure i the data (that i, differet mode may repreet differet aimal or group of aimal.) 6.33 The outlier (1,060) ugget that the populatio ditributio i ot ormal but rather i kewed to the right or log-tailed. Becaue the ample ize i mall, Studet' t method i ot appropriate if the populatio i ot ormal.

6 6.34 (a) No. The oberved frequecy ditributio i highly kewed, which ugget that the populatio ditributio i kewed. (b) Eve if the populatio ditributio i ot ormal, Studet' t method i approximately valid if the ample ize i large. I thi cae, the ample ize i = 242, which i quite large (a) y = 5.68; = 1.54; = 9. The 90% cofidece iterval for µ i 5.68 ± 1.860( ) (4.73,6.63) or 4.73 < µ < 6.63 cm. (b) We are 90% cofidet that the average quare root of the diameter of all America Sycamore tree i the populatio i betwee 4.73 ad 6.63 cm (a) y = (50)(.84) = 42; ~ p = (42+2)/(50+4) = (1-.815) 50+4 =.053. (b) y = (200)(.84) = 168; ~ p = (168+2)/(200+4) = (1-.833) = (a) The umber of mutat i the ample i y = (100)(.20) = 20. Thu, p ~ = (20+2)/(100+4) =.212. The tadard error i p~ (1-p ~ ) =.212(1-.212) =.040. (b) The umber of mutat i the ample i y = (400)(.20) = 80. Thu, p ~ = (80+2)/(400+4) =.203. The tadard error i p~ (1-p ~ ) =.203(1-.203) (a) The 95% cofidece iterval i p~ ± 1.96SE p ~.212 ± (1.96)(.040).212 ±.078 (.134,.290) or.134 < p <.290. (b) The 95% cofidece iterval i p~ ± 1.96SE p ~.203 ± (1.96)(.020).203 ±.039 (.164,.242) or.164 < p <.242. = p ~ = (28 + 2)/( ) =.051;.051(1-.051) 584 =.009.

7 90 The 95% cofidece iterval i.051 ± (1.96)(.009) or (.033,.069) or.033 < p < (a) p ~ = (69+2)/(339+4) =.207. The tadard error i p~ (1-p ~ ) =.207(1-.207) The 95% cofidece iterval i p~ ± 1.96SE p ~.207 ± (1.96)(.022).207 ±.043 (.164,.250) or.164 < p <.250. =.022. (b) We are 95% cofidet that the probability of advere reactio i ifat who receive their firt ijectio of vaccie i betwee.164 ad (a) y = (959)(.157) = , o y mut be 151. Thu, ~ p = ( ).158(1-.158) ( =.158 ad ) = % cofidece iterval:.158 ± (1.645)(.012) or (.138,.178) or.138 < p <.178. (b) The cofidece iterval from part (a) i a cofidece iterval for the probability of iterferece with the pacemaker for that type of cellular telephoe ~.213(1-.213) p = (14+2)/(71+4) =.213; 71+4 = % cofidece iterval:.213 ± (1.96)(.047) or (.120,.306) or.120 < p < The required mut atify the iequality or (Gueed p ~ )(1 - Gueed p ~ ) (.6)(.4) It follow that.04. (.6)(.4).04 Deired SE or (.6)(.4).042 or 150, o 146.

8 6.44 ~ p (1-p ~ ) i larget whe ~ p =.5. Thu, the required mut atify the iequality.04. It follow that.04 or.042 or , o ; thu, Deired.01; gueed ~ p =.7. The required mut atify the iequality (.7)(.3).01. (.7)(.3) It follow that.01 or (.7)(.3).012 or 2100, o ~ p (1-p ~ ) i larget whe ~ p =.5. Thu, the required mut atify the iequality.01. It follow that.01 or.012 or 2500, o The required ample ize would be approximately reduced by a factor of 4 becaue i iverely proportioal to the quare of the SE requiremet (whe >> 4) ~.073(1.073) p = (103+2)/(1438+4) =.073; = % cofidece iterval:.073 ± (1.96)(.0069) or (.059,.087) or.059 < p < It i eceary to ue ~ p =.5 becaue the proportio of progey that are reitat i ukow i advace. Thu, the required mut atify the iequality.05. It follow that Ye. The two mechaim give p =.5 ad p =.75. The agroomit ca be ure that the cofidece iterval will ot cotai both.5 ad.75 becaue the cofidece iterval will be o larger tha p ~ ± (1.96)(.05), which i p ~ ±.098. The width of the iterval i o more tha.196, o the iterval caot cover both.5 ad.75.

9 (a) y = 51.0; = 3.195; 3.195/ 4 = (b) 51.0 ± (3.182)(1.6) (45.9,56.1) or 45.9 < µ < 56.1%. (c) Wider. The higher the cofidece level, the wider the iterval mut be (a) y = 2.275; =.238. The tadard error of the mea i ȳ =.238 =.084 mm. 8 (b) From Table 4 with df = - 1 = 7, we fid that t.025 = The 95% cofidece iterval for µ i y ± t ± (2.365)(.084) (2.08,2.47) or 2.08 < µ < 2.47 mm. (c) µ i the populatio mea tem diameter of plat of Tetraticho wheat three week after flowerig (a) The cofidece iterval formula i valid if (1) the data ca be regarded a a radom ample from a large populatio, (2) the obervatio are idepedet of oe aother, ad (3) the populatio i ormal. (b) We have o iformatio with which to check coditio (1) ad (2). A ormal probability plot, or a hitogram, how that the data look ormal. (c) The mot importat coditio i (1) We ue the iequality Gueed SD Deired SE. I thi cae the deired SE i.03 mm ad the gueed SD (from Exercie 6.46) i.238 mm. Thu, the iequality i or.03, o , which mea that Thu, the experimet hould iclude 63 plat (a) 4.3 ± (2.093)(2.03/ 20 ) (3.35,5.25) or 3.35 < µ < 5.25 puff (b) We are 95% cofidet that the average umber of puff for all fruit fly larva icubated at 37 C for 30 miute i betwee 3.35 ad (a) ± (2.576)(4.24/ 1353 ) (28.56,29.16) or < µ < day. (b) The cofidece iterval i ot coitet with the hypothei becaue 29.5 i ot i the iterval (a) The mea of all reported cycle i maller becaue the wome with horter cycle had more cycle durig the fixed time period, ad therefore cotributed more obervatio to the data. (b) It would ot be valid becaue the 5412 obervatio are ot idepedet -- there i a hierarchical tructure i the data.

10 } (a) ± (2.052)(.1237) (4.91,5.42) or 4.91 < µ < 5.42 kg. (b) ± (2.771)(.1237) (4.83,5.51) or 4.83 < µ < 5.51 kg. (c) We are 95% cofidet that the average birthweight of all lamb that were bor i April, were all the ame breed, ad were all igle birth, i betwee 4.91 kg ad 5.43 kg (a) We mut be able to view the data a a radom ample of obervatio from a large populatio, the obervatio i the ample mut be idepedet of each other, ad the populatio ditributio mut be approximately ormal. (Note, however, that becaue the ample ize ( = 28) i ot very mall, ome oormality of the populatio ditributio would be acceptable.) (b) The hape of the hitogram i a etimate of the hape of the populatio ditributio. Thu, the hitogram ca be ued to check the ormality coditio of the populatio (c) If twi birth were icluded, the idepedece of the obervatio would be quetioable, becaue birthweight of the member of a twi pair are likely to be depedet The cofidece iterval hould be (6.2,7.4). The cofidece iterval i a iterval etimate of the populatio mea. The data oly take o iteger value, but the mea of the populatio eed ot be a iteger (ad probably i ot) (a).42/ 84 = (b) 30 Frequecy Serum potaium SE } (c) 4.36 ± (1.984)(.04583) (4.269,4.451) or < µ < meq/l. (d) We are 95% cofidet that the average erum potaium cocetratio i the blood of all healthy wome i betwee meq/l ad meq/l No. The cofidece iterval would be much too arrow; oly a miority of healthy wome would fall withi the cofidece iterval. Itead, the iterval y ± 2SD would be a reaoable choice for referece limit.

11 (a) We would predict the SD of the ew meauremet to be about.42 meq/l, becaue thi i our bet etimate (baed o Exercie 6.54) of the populatio SD. (b).42/ 200 =.030 meq/l (a) y = ; = ; / 6 = (b) ± (2.015)(.41) (61.94,63.60) or < µ < 63.60% p ~ = (97+2)/(123+4) =.780. The tadard error i p~ (1-p ~ ) =.780(1-.780) The 95% cofidece iterval i p~ ± 1.96SE p ~.780 ± (1.96)(.037).780 ±.073 (.707,.853) or.707 < p <.853. = The data may ot be a radom ample becaue it i eaier to capture pregat female tha other (a) = 180; y = 23; p ~ = y+.5( ) =.133. p~ (1-p ~ ) =.025. The 90% cofidece iterval i.133 ± (1.645)(.025) or.133 ±.041 or (.092,.174). (b) The cofidece iterval formula i valid if (1) the data ca be regarded a a radom ample from a large populatio ad (2) the obervatio are idepedet (a) y = 145.3; = 12.87; 12.87/ 1139 =.381. The cofidece iterval i ± (1.96)(.381) or (144.55,146.05) or < µ < g/l. (b) No. The obtaied 95% cofidece iterval i a cofidece iterval for the populatio mea hemoglobi level. It doe ot give limit for idividual data poit. (c) No. See the awer to part (b) The required mut atify the iequality (.45)(.55).02 It follow that ~ p (1-p ~ ) i larget whe ~ p =.5. Thu, the required mut atify the iequality.02.

12 It follow that.02 or.022 or 625, o We ca etimate the SD a beig approximately 8. (Note that (110-80)/4 = ) It follow that SE 8/ The t multiplier i roughly 2. Thu, (iv) i a approximate 95% cofidece iterval for the populatio mea blood preure.