Chapter 1: Overview and Descriptive Statistics CHAPTER 1. a. Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post

Transcription

1 Chapter : Overvew ad Descrptve Statstcs CHAPTER Secto.. a. Housto Chrocle, Des Moes Regster, Chcago Trbue, Washgto Post b. Captal Oe, Campbell Soup, Merrll Lych, Pultzer c. Bll Jasper, Kay Reke, Hele Ford, Davd Meedez d..78,.44,.5,.4. a. 9. yd., 8. yd., 4.7 yd.,. yd. b. 4, 96, 84, c.., 4.,., 6. d..7 g,.58 g, 7. g, 7. g. a. I a sample of VCRs, what are the chaces that more tha eed servce whle uder warratee? What are the chaces tha oe eed servce whle stll uder warratee? b. What proporto of all VCRs of ths brad ad model wll eed servce wth the warratee perod?

2 Chapter : Overvew ad Descrptve Statstcs 4. a. Cocrete: All lvg U.S. Ctzes, all mutual fuds marketed the U.S., all books publshed 98. Hypothetcal: All grade pot averages for Uversty of Calfora udergraduates durg the et academc year. Page legths for all books publshed durg the et caledar year. Battg averages for all major league players durg the et baseball seaso. b. Cocrete: Probablty: I a sample of 5 mutual fuds, what s the chace that all 5 have rates of retur whch eceeded % last year? Statstcs: If prevous year rates-of-retur for 5 mutual fuds were 9.6, 4.5, 8., 9.9 ad., ca we coclude that the average rate for all fuds was below %? Coceptual: Probablty: I a sample of books to be publshed et year, how lkely s t that the average umber of pages for the s betwee ad 5? Statstcs: If the sample average umber of pages for books s 7, ca we be hghly cofdet that the average for all books s betwee ad 45? 5. a. No, the relevat coceptual populato s all scores of all studets who partcpate the SI cojucto wth ths partcular statstcs course. b. The advatage to radomly choosg studets to partcpate the two groups s that we are more lkely to get a sample represetatve of the populato at large. If t were left to studets to choose, there may be a dvso of abltes the two groups whch could uecessarly affect the outcome of the epermet. c. If all studets were put the treatmet group there would be o results wth whch to compare the treatmets. 6. Oe could take a smple radom sample of studets from all studets the Calfora State Uversty system ad ask each studet the sample to report the dstace form ther hometow to campus. Alteratvely, the sample could be geerated by takg a stratfed radom sample by takg a smple radom sample from each of the campuses ad aga askg each studet the sample to report the dstace from ther hometow to campus. Certa problems mght arse wth self reportg of dstaces, such as recordg error or poor recall. Ths study s eumeratve because there ests a fte, detfable populato of objects from whch to sample. 7. Oe could geerate a smple radom sample of all sgle famly homes the cty or a stratfed radom sample by takg a smple radom sample from each of the dstrct eghborhoods. From each of the homes the sample the ecessary varables would be collected. Ths would be a eumeratve study because there ests a fte, detfable populato of objects from whch to sample.

3 Chapter : Overvew ad Descrptve Statstcs 8. a. Number observatos equal 8 b. Ths could be called a aalytc study because the data would be collected o a estg process. There s o samplg frame. 9. a. There could be several eplaatos for the varablty of the measuremets. Amog them could be measurg error, (due to mechacal or techcal chages across measuremets), recordg error, dffereces weather codtos at tme of measuremets, etc. b. Ths could be called a aalytc study because there s o samplg frame. Secto.. a. Mtab geerates the followg stem-ad-leaf dsplay of ths data: stem: oes 7 leaf: teths 68 What costtutes large or small varato usually depeds o the applcato at had, but a ofte-used rule of thumb s: the varato teds to be large wheever the spread of the data (the dfferece betwee the largest ad smallest observatos) s large compared to a represetatve value. Here, 'large' meas that the percetage s closer to % tha t s to %. For ths data, the spread s - 5 6, whch costtutes 6/8.75, or, 75%, of the typcal data value of 8. Most researchers would call ths a large amout of varato. b. The data dsplay s ot perfectly symmetrc aroud some mddle/represetatve value. There teds to be some postve skewess ths data. c. I Chapter, outlers are data pots that appear to be very dfferet from the pack. Lookg at the stem-ad-leaf dsplay part (a), there appear to be o outlers ths data. (Chapter gves a more precse defto of what costtutes a outler). d. From the stem-ad-leaf dsplay part (a), there are 4 values greater tha. Therefore, the proporto of data values that eceed s 4/7.48, or, about 5%.

4 Chapter : Overvew ad Descrptve Statstcs. 6l 4 6h l 44 7h StemTes 8l 44 LeafOes 8h l 9h 58 Ths dsplay brgs out the gap the data: There are o scores the hgh 7's.. Oe method of deotg the pars of stems havg equal values s to deote the frst stem by L, for 'low', ad the secod stem by H, for 'hgh'. Usg ths otato, the stem-ad-leaf dsplay would appear as follows: L H L 4 4H L 44 5H 58 stem: teths 6L leaf: hudredths 6H L 7H 5 The stem-ad-leaf dsplay o the prevous page shows that.45 s a good represetatve value for the data. I addto, the dsplay s ot symmetrc ad appears to be postvely skewed. The spread of the data s , whch s.44/ , or about 98% of the typcal value of.45. Ths costtutes a reasoably large amout of varato the data. The data value.75 s a possble outler 4

5 Chapter : Overvew ad Descrptve Statstcs. a. Leaf oes 445 Stem tes The observatos are hghly cocetrated at 4 5, where the dsplay suggests the typcal value falls. b. 4 Frequecy stregth The hstogram s symmetrc ad umodal, wth the pot of symmetry at appromately 5. 5

6 Chapter : Overvew ad Descrptve Statstcs 4. a. stem uts: leaf uts: b. A represetatve value could be the meda, 7.. c. The data appear to be hghly cocetrated, ecept for a few values o the postve sde. d. No, the data s skewed to the rght, or postvely skewed. e. The value 8.9 appears to be a outler, beg more tha two stem uts from the prevous value. 5. Cruchy Creamy Both sets of scores are reasoably spread out. There appear to be o outlers. The three hghest scores are for the cruchy peaut butter, the three lowest for the creamy peaut butter. 6

7 Chapter : Overvew ad Descrptve Statstcs 6. a. beams cylders The data appears to be slghtly skewed to the rght, or postvely skewed. The value of 4. appears to be a outler. Three out of the twety, / or.5 of the observatos eceed Mpa. b. The majorty of observatos are betwee 5 ad 9 Mpa for both beams ad cylders, wth the modal class the 7 Mpa rage. The observatos for cylders are more varable, or spread out, ad the mamum value of the cylder observatos s hgher. c. Dot Plot cylder... :.. :.:... : a. Number Nocoformg Frequecy RelatveFrequecy(Freq/6) does't add eactly to because relatve frequeces have bee rouded. b. The umber of batches wth at most 5 ocoformg tems s , whch s a proporto of 55/6.97. The proporto of batches wth (strctly) fewer tha 5 ocoformg tems s 5/ Notce that these proportos could also have bee computed by usg the relatve frequeces: e.g., proporto of batches wth 5 or fewer ocoformg tems - (.5.7.7).96; proporto of batches wth fewer tha 5 ocoformg tems - ( )

8 Chapter : Overvew ad Descrptve Statstcs c. The followg s a Mtab hstogram of ths data. The ceter of the hstogram s somewhere aroud or ad t shows that there s some postve skewess the data. Usg the rule of thumb Eercse, the hstogram also shows that there s a lot of spread/varato ths data. Relatve Frequecy Number 8. a. The followg hstogram was costructed usg Mtab: 8 7 Frequecy Number of papers The most terestg feature of the hstogram s the heavy postve skewess of the data. Note: Oe way to have Mtab automatcally costruct a hstogram from grouped data such as ths s to use Mtab's ablty to eter multple copes of the same umber by typg, for eample, 784() to eter 784 copes of the umber. The frequecy data ths eercse was etered usg the followg Mtab commads: MTB > set c DATA> 784() 4() 7() 5(4) (5) 8(6) 9(7) 9(8) DATA> 6(9) 7() 6() 7() 4() 4(4) 5(5) (6) (7) DATA> ed 8

9 Chapter : Overvew ad Descrptve Statstcs b. From the frequecy dstrbuto (or from the hstogram), the umber of authors who publshed at least 5 papers s , so the proporto who publshed 5 or more papers s 44/9., or %. Smlarly, by addg frequeces ad dvdg by 9, the proporto who publshed or more papers s 9/9.98, or about %. The proporto who publshed more tha papers (.e., or more) s /9.45, or about.5%. c. No. Strctly speakg, the class descrbed by ' 5 ' has o upper boudary, so t s mpossble to draw a rectagle above t havg fte area (.e., frequecy). d. The category 5-7 does have a fte wdth of, so the cumulated frequecy of ca be plotted as a rectagle of heght 6.5 over ths terval. The basc rule s to make the area of the bar equal to the class frequecy, so area (wdth)(heght) (heght) yelds a heght of a. From ths frequecy dstrbuto, the proporto of wafers that cotaed at least oe partcle s (-)/.99, or 99%. Note that t s much easer to subtract (whch s the umber of wafers that cota partcles) from tha t would be to add all the frequeces for,,, partcles. I a smlar fasho, the proporto cotag at least 5 partcles s ( )/ 7/.7, or, 7%. b. The proporto cotag betwee 5 ad partcles s (5845)/ 64/.64, or 64%. The proporto that cota strctly betwee 5 ad (meag strctly more tha 5 ad strctly less tha ) s (84)/ 44/.44, or 44%. c. The followg hstogram was costructed usg Mtab. The data was etered usg the same techque metoed the aswer to eercse 8(a). The hstogram s almost symmetrc ad umodal; however, t has a few relatve mama (.e., modes) ad has a very slght postve skew. Relatve frequecy... 5 Number of partcles 5 9

10 Chapter : Overvew ad Descrptve Statstcs. a. The followg stem-ad-leaf dsplay was costructed: stem: thousads leaf: hudreds A typcal data value s somewhere the low 's. The dsplay s almost umodal (the stem at 5 would be cosdered a mode, the stem at aother) ad has a postve skew. b. A hstogram of ths data, usg classes of wdth cetered at,,, 6 s show below. The proporto of subdvs os wth total legth less tha s ()/47.489, or 48.9%. Betwee ad 4, the proporto s (7 )/47.9, or 9.%. The hstogram shows the same geeral shape as depcted by the stem-ad-leaf part (a). Frequecy 5 legth 4 5 6

11 Chapter : Overvew ad Descrptve Statstcs. a. A hstogram of the y data appears below. From ths hstogram, the umber of subdvsos havg o cul-de-sacs (.e., y ) s 7/47.6, or 6.%. The proporto havg at least oe cul-de-sac (y ) s (47-7)/47 /47.68, or 6.8%. Note that subtractg the umber of cul-de-sacs wth y from the total, 47, s a easy way to fd the umber of subdvsos wth y. Frequecy 4 5 y b. A hstogram of the z data appears below. From ths hstogram, the umber of subdvsos wth at most 5 tersectos (.e., z 5) s 4/47.894, or 89.4%. The proporto havg fewer tha 5 tersectos (z < 5) s 9/47.8, or 8.%. Frequecy 5 4 z

12 Chapter : Overvew ad Descrptve Statstcs. A very large percetage of the data values are greater tha, whch dcates that most, but ot all, ruers do slow dow at the ed of the race. The hstogram s also postvely skewed, whch meas that some ruers slow dow a lot compared to the others. A typcal value for ths data would be the eghborhood of secods. The proporto of the ruers who ra the last 5 km faster tha they dd the frst 5 km s very small, about % or so.. a. Percet 4 5 brkstgth The hstogram s skewed rght, wth a majorty of observatos betwee ad cycles. The class holdg the most observatos s betwee ad cycles.

13 Chapter : Overvew ad Descrptve Statstcs b..4. Desty brkstgth 6 9 c [proporto ] [proporto < ] Percet weldstr 58 6

14 Chapter : Overvew ad Descrptve Statstcs 5. Hstogram of orgal data: 5 Frequecy 5 4 IDT Hstogram of trasformed data: 9 8 Frequecy log(idt) The trasformato creates a much more symmetrc, moud-shaped hstogram. 4

15 Chapter : Overvew ad Descrptve Statstcs 6. a. Class Itervals Frequecy Rel. Freq..5 -< < < < < < < < < Desty clearess b. The proporto of days wth a clearess de smaller tha.5 s ( ) c. The proporto of days wth a clearess de of at least.65 s ( ). 6, or 6%., or 6%. 5

16 Chapter : Overvew ad Descrptve Statstcs 7. a. The edpots of the class tervals overlap. For eample, the value 5 falls both of the tervals 5 ad 5. b. Class Iterval Frequecy Relatve Frequecy - < < < < < <.4 - < < 4. > Frequecy lfetme The dstrbuto s skewed to the rght, or postvely skewed. There s a gap the hstogram, ad what appears to be a outler the 5 55 terval. 6

17 Chapter : Overvew ad Descrptve Statstcs c. Class Iterval Frequecy Relatve Frequecy.5 - < < < < < < < < Frequecy l lfetme The dstrbuto of the atural logs of the orgal data s much more symmetrc tha the orgal. d. The proporto of lfetme observatos ths sample that are less tha s , ad the proporto that s at least s There are seasoal treds wth lows ad hghs moths apart. radt Ide 4 7

18 Chapter : Overvew ad Descrptve Statstcs 9. Complat Frequecy Relatve Frequecy B 7.67 C.5 F 9.5 J.667 M N 6. O.5 6. Cout of complat B C F J M N O complat. Cout of prodprob 4 5 prodprob. correct comp oet. mssg compoet. faled compoet 4. suffcet solder 5. ecess solder 8

19 Chapter : Overvew ad Descrptve Statstcs. Relatve Cumulatve Relatve Class Frequecy Frequecy Frequecy. - uder uder uder uder uder uder uder a. The frequecy dstrbuto s: Relatve Relatve Class Frequecy Class Frequecy -< <5.9 5-<.8 5-<.9 -< < < <5.4 6-< < < <8. 8-<95. The relatve frequecy dstrbuto s almost umodal ad ehbts a large postve skew. The typcal mddle value s somewhere betwee 4 ad 45, although the skewess makes t dffcult to ppot more eactly tha ths. b. The proporto of the fre loads less tha 6 s The proporto of loads that are at least s c. The proporto of loads betwee 6 ad s

20 Chapter : Overvew ad Descrptve Statstcs Secto.. ~ a , 89. The mea s larger tha the meda, but they are stll farly close together., ~ 89 b. Chagg the oe value, meda stays the same. c. 9.. The mea s lowered, the tr or 7% trmmed from each tal. d. For, Σ (9.769),557 For 4, Σ,557 59,76 or a. The sum of the data pots s 54.9, so 54.9/ b. The sample sze ( ) s odd, so there wll be a mddle value. Sortg from smallest to largest: The sth value, 6.6 s the mddle, or meda, value. The mea dffers from the meda because the largest sample observatos are much further from the meda tha are the smallest values. c. Deletg the smallest ( 4.4) ad largest ( 9.9) values, the sum of the remag 9 observatos s 4.6. The trmmed mea tr s 4.6/ The trmmg percetage s (/) 9.%. tr les betwee the mea ad meda. 5. a. The sample mea s (.4/8).55. The sample sze ( 8) s eve. Therefore, the sample meda s the average of the (/) ad (/) values. By sortg the 8 values order, from smallest to largest: , the forth ad ffth values are ad. The sample meda s (..)/.5. The.5% trmmed mea requres that we frst trm (.5)() or value from the eds of the ordered data set. The we average the remag 6 values. The.5% trmmed mea s 74.4/6.4. tr(.5) All three measures of ceter are smlar, dcatg lttle skewess to the data set. b. The smallest value (8.) could be creased to ay umber below. (a chage of less tha 4.) wthout affectg the value of the sample meda.

21 Chapter : Overvew ad Descrptve Statstcs c. The values obtaed part (a) ca be used drectly. For eample, the sample mea of.55 ps could be re-epressed as ks. ps (.55 ps) 5.7ks. 6. a. A stem-ad leaf dsplay of ths data appears below: 55 stem: oes 49 leaf: teths The dsplay s reasoably symmetrc, so the mea ad meda wll be close. b. The sample mea s 968/ The sample meda s ~ (697)/ c. The largest value (curretly 44) could be creased by ay amout. Dog so wll ot chage the fact that the mddle two observatos are 69 ad 7, ad hece, the meda wll ot chage. However, the value 44 ca ot be chaged to a umber less tha 7 (a chage of ) sce that wll lower the values(s) of the two mddle observatos. d. Epressed mutes, the mea s (7.7 sec)/(6 sec) 6.8 m; the meda s 6.6 m. 7.., ~. 5,. 46 tr( ) choces because of the outler.9.. The meda or the trmmed mea would be good 8. a. The reported values are ( creasg order), 5,,, 5,,, 5, ad 4. Thus the meda of the reported values s 5. b. 7.6 s reported as, so the meda s ow, a very substatal chage. Whe there s roudg or groupg, the meda ca be hghly sestve to small chage.

22 Chapter : Overvew ad Descrptve Statstcs a. Σ l so ~ (.7.).9 b..94 ca be decreased utl t reaches.(the largest of the mddle values).e. by.94..8, If t s decreased by more tha.8, the meda wll chage. 4. ~ tr( 5) tr( ) All four measures of ceter have about the same value a.. 7 b.. 7 proporto of successes s 5. c. 8 so s (.8)(5) total of successes 7 of the ew cars would have to be successes 4. Σy Σ( c) Σ c y~ the meda of ( c, c,..., c ),,..., ) c ~ c a. y c ( meda of Σy Σ( c) cσ b. y c y~ c, c,..., c ) c meda ( (,,..., ) c~ (57 79) 4. meda 68., % trmmed mea 66., % trmmed mea 67.5.

23 Chapter : Overvew ad Descrptve Statstcs Secto a. rage b. ( ) ( ) Σ. Σ( ) Σ( ) Σ( ),7. 4 s. Σ( ) c. s s 7. Σ ( Σ) /,7.4 (.) / d. s a / Devatos from the mea: , , , , ad b. s [(.8) (.) (-.98) (-.8) (.) ]/(5-).98/4.48, so s.694. c. 66,795.6, so s [66, (577.9) /5]/4.98/4.48. d. Subtractg from all values gves 5. 58, all devatos are the same as part b, ad the trasformed varace s detcal to that of part b.

24 Chapter : Overvew ad Descrptve Statstcs 46. a. 448/ The sorted data s: , so the sample meda s ~ 888. b. Subtractg a costat from each observato shfts the data, but does ot chage ts sample varace (Eercse 6). For eample, by subtractg 7 from each observato we get the values 8,,, 56, ad 88, whch are smaller (fewer dgts) ad easer to work wth. The sum of squares of ths trasformed data s 4 ad ts sum s 98, so the computatoal formula for the varace gves s [4-(98) /5]/(5-) The sample mea, (,6) 6. The sample stadard devato,. s ( ) (,6) 4, O average, we would epect a fracture stregth of 6.. I geeral, the sze of a typcal devato from the sample mea (6.) s about Some observatos may devate from 6. by more tha ths ad some by less. 48. Usg the computatoal formula, s [,587,566-(968) /6]/(6-) 59.45, so s 4.6. I geeral, the sze of a typcal devato from the sample mea (7.7) s about 4.4. Some observatos may devate from 7.7 by a lttle more tha ths, some by less. 49. a Σ, Σ (.75)... (.) (56.8) /7 8.5 s s b..56, 4

25 Chapter : Overvew ad Descrptve Statstcs 5. Frst, we eed (,79) devato 7 (,79) 4,657,5 s s (66.89) 96.6 bt less tha the $.5 mllo that was awarded orgally.. The we eed the sample stadard. The mamum award should be, or dollar uts, $,96,6. Ths s qute a 5. a. Σ 56 ad Σ 68, 5, so [68,5 (56) /9] s ad s c b. If y tme mutes, the y c where, so s y c s.5 ad s y cs Let d deote the ffth devato. The..9.. d or.5 d, so d.5. Oe sample for whch these are the devatos s.8, 4.4, 4.5,, (obtaed by addg.5 to each devato; addg ay other umber wll produce a dfferet sample wth the desred property) 5. a. lower half: upper half: Thus the lower fourth s.74 ad the upper fourth s.88. b. f s c. f s would t chage, sce creasg the two largest values does ot affect the upper fourth. d. By at most.4 (that s, to aythg ot eceedg.74), sce the t wll ot chage the lower fourth. e. Sce s ow eve, the lower half cossts of the smallest 9 observatos ad the upper half cossts of the largest 9. Wth the lower fourth.74 ad the upper fourth.9, f.9. s 5

26 Chapter : Overvew ad Descrptve Statstcs 54. a. The lower half of the data set: , whose meda, ad.. ( ) therefore, the lower quartle, s 6.. The top half of the data set: , whose meda, ad ( ) therefore, the upper quartle, s 7. So, the IQR (7. 6.) 44.. b. A boplot (created Mtab) of ths data appears below: 5 sheer stregth There s a slght postve skew to the data. The varato seems qute large. There are o outlers. c. A observato would eed to be further tha.5(44.) 66.5 uts below the lower quartle [( ) 4.5 uts] or above the upper quartle [( ) 6.5 uts] to be classfed as a mld outler. Notce that, ths case, a outler o the lower sde would ot be possble sce the sheer stregth varable caot have a egatve value. A etreme outler would fall ()44.). or more uts below the lower, or above the upper quartle. Sce the mmum ad mamum observatos the data are 4.4 ad 9.9 respectvely, we coclude that there are o outlers, of ether type, ths data set. d. Not utl the value 9.9 s lowered below 7.7 would there be ay chage the value of the upper quartle. That s, the value 9.9 could ot be decreased by more tha ( ) 6. uts. 6

27 Chapter : Overvew ad Descrptve Statstcs 55. a. Lower half of the data set: , whose meda, ad therefore the lower quartle, s 59 (the 7 th observato the sorted lst). The top half of the data s , whose meda, ad therefore the upper quartle s 9. So, the IQR b..5(iqr).5() 49.5 ad (IQR) () 99. Observatos that are further tha 49.5 below the lower quartle (.e., or less) or more tha 49.5 uts above the upper quartle (greater tha ) are classfed as 'mld' outlers. 'Etreme' outlers would fall 99 or more uts below the lower, or above the upper, quartle. Sce the mmum ad mamum observatos the data are 5 ad 44, we coclude that there are o mld outlers ths data (ad therefore, o 'etreme' outlers ether). c. A boplot (created by Mtab) of ths data appears below. There s a slght postve skew to the data, but t s ot far from beg symmetrc. The varato, however, seems large (the spread s a large percetage of the meda/typcal value) 7 Escape tme 4 d. Not utl the value 44 s lowered below the upper quartle value of 9 would there be ay chage the value of the upper quartle. That s, the value 44 could ot be decreased by more tha 44-9 uts. 7

28 Chapter : Overvew ad Descrptve Statstcs 56. A boplot (created Mtab) of ths data appears below. alumum 4 5 There s a slght postve skew to ths data. There s oe etreme outler (5). Eve whe removg the outler, the varato s stll moderately large. 57. a..5(iqr).5( ). ad (IQR) ( ) 6.4. Mld outlers: observatos below or above Etreme outlers: observatos below or above Of the observatos gve, 5.8 s a etreme outler ad 5. s a mld outler. b. A boplot of ths data appears below. There s a bt of postve skew to the data but, ecept for the two outlers detfed part (a), the varato the data s relatvely small. * * The most otceable feature of the comparatve boplots s that mache s sample values have cosderably more varato tha does mache s sample values. However, a typcal value, as measured by the meda, seems to be about the same for the two maches. The oly outler that ests s from mache. 8

29 Chapter : Overvew ad Descrptve Statstcs 59. a. ED: meda.4 (the 4 th value the sorted lst of data). The lower quartle (meda of the lower half of the data, cludg the meda, sce s odd) s (.. )/.. The upper quartle s (.7.8)/.75. Therefore, IQR No-ED: meda (.5.7)/.6. The lower quartle (meda of the lower 5 observatos) s.; the upper quartle (meda of the upper half of the data) s 7.9. Therefore, IQR b. ED: mld outlers are less tha. -.5(.65) or greater tha.75.5(.65) Etreme outlers are less tha. - (.65) or greater tha.75 (.65).7. So, the two largest observatos (.7,.) are etreme outlers ad the et two largest values (8.9, 9.) are mld outlers. There are o outlers at the lower ed of the data. No-ED: mld outlers are less tha. -.5(7.6) -. or greater tha 7.9.5(7.6) 9.. Note that there are o mld outlers the data, hece there ca ot be ay etreme outlers ether. c. A comparatve boplot appears below. The outlers the ED data are clearly vsble. There s otceable postve skewess both samples; the No-Ed data has more varablty the the Ed data; the typcal values of the ED data ted to be smaller tha those for the No-ED data. No-ED ED Cocetrato (mg/l) 9

30 Chapter : Overvew ad Descrptve Statstcs 6. A comparatve boplot (created Mtab) of ths data appears below. test type caster burst stregth 8 The burst stregths for the test ozzle closure welds are qute dfferet from the burst stregths of the producto caster ozzle welds. The test welds have much hgher burst stregths ad the burst stregths are much more varable. The producto welds have more cosstet burst stregth ad are cosstetly lower tha the test welds. The producto welds data does cota outlers. 6. Outlers occur the 6 a.m. data. The dstrbutos at the other tmes are farly symmetrc. Varablty ad the 'typcal' values the data crease a lttle at the oo ad p.m. tmes.

31 Chapter : Overvew ad Descrptve Statstcs Supplemetary Eercses 6. To somewhat smplfy the algebra, beg by subtractg 76, from the orgal data. Ths trasformato wll affect each date value ad the mea. It wll ot affect the stadard devato. 68,,48, y 8 ( 4)(8),4 so,, 4 4,4 4, ad (, ) ad 59 Net, s ( 8) So,,859, 444 ( 4) 4 4,859,444 4,94,65 59,,859, 444 By substtutg ( 59 ) (,59 ),94,65,59 6,499 ad we obta the equato. we obta ad, Evaluatg for Thus,,68 76, 9.

32 Chapter : Overvew ad Descrptve Statstcs 6. Flow Lower Upper rate Meda quartle quartle IQR.5(IQR) (IQR) There are o outlers the three data sets. However, as the comparatve boplot below shows, the three data sets dffer wth respect to ther cetral values (the medas are dfferet) ad the data for flow rate 6 s somewhat less varable tha the other data sets. Flow rates 5 ad also ehbt a small degree of postve skewess. Flow rate Uformty (%) 5

33 Chapter : Overvew ad Descrptve Statstcs stemoes 7 7 leafteths , ~.6 s f s (.5)(.) (.5)(.) 4.6 o outlers lower fourth 8.85, upper fourth Radato There are o outlers. The dstrbuto s skewed to the left.

34 Chapter : Overvew ad Descrptve Statstcs 65. a. HC data: 68.4 ad 96.8, so s [ (96.8) /4]/ 9.95 ad the sample stadard devato s s CO data: ad 75, so s [ (75) /4]/ ad the sample stadard devato s s b. The mea of the HC data s 96.8/4 4.; the mea of the CO data s 75/ Therefore, the coeffcet of varato of the HC data s 9.59/4..96, or 9.6%. The coeffcet of varato of the CO data s 59.4/8.75., or.%. Thus, eve though the CO data has a larger stadard devato tha does the HC data, t actually ehbts less varablty ( percetage terms) aroud ts average tha does the HC data. 66. a. The hstogram appears below. A represetatve value for ths data would be 9. The hstogram s reasoably symmetrc, umodal, ad somewhat bell-shaped. The varato the data s ot small sce the spread of the data (99-8 8) costtutes about % of the typcal value of 9. Relatve frequecy Fracture stregth (MPa) b. The proporto of the observatos that are at least 85 s - (67)/69.9. The proporto less tha 95 s - ()/ c. 9 s the mdpot of the class 89-<9, whch cotas 4 observatos (a relatve frequecy of 4/ Therefore about half of ths frequecy,.7, should be added to the relatve frequeces for the classes to the left of 9. That s, the appromate proporto of observatos that are less tha 9 s

35 Chapter : Overvew ad Descrptve Statstcs % trmmedmea % trmmedmea.6 % trmmedmea 5 5 ( ) ( ) (.7) (.6) a. dc { ( c) } d c dc( d c) ( c c c) ( c c). b. ( ) ssmallert ha ( µ ). 69. a. b. y s y a 9 y 5 y ( a b) ( y y) ( a b ( a b) ) ( a a) ( ) ο ο C, y F ( 87.) s y s y 9 5 a s. a 89.4 b a b. (.4)

36 Chapter : Overvew ad Descrptve Statstcs 7. a. 5 Oyge Cosumpto 5 5 Treadmll Weght Eercse Type There s a sgfcat dfferece the varablty of the two samples. The weght trag produced much hgher oyge cosumpto, o average, tha the treadmll eercse, wth the meda cosumptos beg appromately ad lters, respectvely. b. Subtractg the y from the for each subject, the dffereces are., 9.,.4, 9., 6.,.5,., 8.4, 8.7, 4.4,.5, -.8, -.4, 5., ad Dfferece The majorty of the dffereces are postve, whch suggests that the weght trag produced hgher oyge cosumpto for most subjects. The meda dfferece s about 6 lters. 6

37 Chapter : Overvew ad Descrptve Statstcs 7. a. The mea, meda, ad trmmed mea are vrtually detcal, whch suggests symmetry. If there are outlers, they are balaced. The rage of values s oly 5.5, but half of the values are betwee.95 ad 8.5. b. 5 4 stregth The boplot also dsplays the symmetry, ad adds a vsual of the outlers, two o the lower ed, ad oe o the upper. 7

38 Chapter : Overvew ad Descrptve Statstcs 7. A table of summary statstcs, a stem ad leaf dsplay, ad a comparatve boplot are below. The healthy dvduals have hgher receptor bdg measure o average tha the dvduals wth PTSD. There s also more varato the healthy dvduals values. The dstrbuto of values for the healthy s reasoably symmetrc, whle the dstrbuto for the PTSD dvduals s egatvely skewed. The bo plot dcates that there are o outlers, ad cofrms the above commets regardg symmetry ad skewess. PTSD Healthy Mea.9 5. Meda 7 5 Std Dev M Ma 46 7 stem tes 58 leaf oes PTSD Idvduals Healthy Receptor Bdg 8

39 Chapter : Overvew ad Descrptve Statstcs stemteths leafhudredths , s.89, ~.9 lowerfourth.855, upperfourth Cadece The data appears to be a bt skewed toward smaller values (egatvely skewed). There are o outlers. The mea ad the meda are close value. 74. a. Mode.9. It occurs four tmes the data set. b. The Modal Category s the oe whch the most observatos occur. 9

40 Chapter : Overvew ad Descrptve Statstcs 75. a. The meda s the same (7) each plot ad all three data sets are very symmetrc. I addto, all three have the same mmum value (5) ad same mamum value (9). Moreover, all three data sets have the same lower (64) ad upper quartles (78). So, all three boplots wll be detcal. b. A comparatve dotplot s show below. These graphs show that there are dffereces the varablty of the three data sets. They also show dffereces the way the values are dstrbuted the three data sets... :. :::. : Type Type.... :... :.: Type c. The boplot (a) s ot capable of detectg the dffereces amog the data sets. The prmary reaso s that boplots gve up some detal descrbg data because they use oly 5 summary umbers for comparg data sets. Note: The defto of lower ad upper quartle used ths tet s slghtly dfferet tha the oe used by some other authors (ad software packages). Techcally speakg, the meda of the lower half of the data s ot really the frst quartle, although t s geerally very close. Istead, the medas of the lower ad upper halves of the data are ofte called the lower ad upper hges. Our boplots use the lower ad upper hges to defe the spread of the mddle 5% of the data, but other authors sometmes use the actual quartles for ths purpose. The dfferece s usually very slght, usually uotceable, but ot always. For eample the data sets of ths eercse, a comparatve boplot based o the actual quartles (as computed by Mtab) s show below. The graph shows substatally the same type of formato as those descrbed (a) ecept the graphs based o quartles are able to detect the slght dffereces varato betwee the three data sets. Type of wre MPa 8 9 4

41 Chapter : Overvew ad Descrptve Statstcs 76. The measures that are sestve to outlers are: the mea ad the mdrage. The mea s sestve because all values are used computg t. The mdrage s sestve because t uses oly the most etreme values ts computato. The meda, the trmmed mea, ad the mdhge are ot sestve to outlers. The meda s the most resstat to outlers because t uses oly the mddle value (or values) ts computato. The trmmed mea s somewhat resstat to outlers. The larger the trmmg percetage, the more resstat the trmmed mea becomes. The mdhge, whch uses the quartles, s reasoably resstat to outlers because both quartles are resstat to outlers. 77. a stem: oes 7 5 leaf: teths HI

42 Chapter : Overvew ad Descrptve Statstcs b. Iterval Frequecy Rel. Freq. Desty -<.5.5 -< < < <.. -< Desty Repar Tme 78. a. Sce the costat s subtracted from each value to obta each y value, ad addto or subtracto of a costat does t affect varablty, s ad s y s y s b. Let c /s, where s s the sample stadard devato of the s ad also (by a ) of the y s. The s z cs y (/s)s, ad s z. That s, the stadardzed quattes z,, z have a sample varace ad stadard devato of. 4

43 Chapter : Overvew ad Descrptve Statstcs 79. a. b. [, so ( ) s ( ) s ( { ( ) } ) ( ) ( ) ] Whe the epresso for from a s substtuted, the epresso braces smplfes to the followg, as desred: ( ) ( ) 5(.58).8.5 c.. 5 s 6 6 ( ) 4 s.5 ( ) 5 ( ) ( ) (.8.58) (6) So the stadard devato s

44 Chapter : Overvew ad Descrptve Statstcs 8. a. Bus Route Legth Desty legth b. Proporto less tha Proporto at least. c. Frst compute (.9)(9 ) 5.8. Thus, the 9 th percetle should be about the 5 d ordered value. The 5 st ordered value les the terval 8 - <. The 5 d ordered value les the terval - < 5. There are 7 values the terval - < 5. We do ot kow how these values are dstrbuted, however, the smallest value (.e., the 5 d value the data set) caot be smaller tha. So, the 9 th percetle s roughly. d. Frst compute (.5)(9 ) 96. Thus the meda (5 th percetle) should be the 96 ordered value. The 74 th ordered value les the terval 6 -< 8. The et 4 observato le the terval 8 - <. So, ordered observato 75 to 6 le the tervals 8 - <. The 96 th observato s about the mddle of these. Thus, we would say, the meda s roughly Assumg that the hstogram s umodal, the there s evdece of postve skewess the data sce the meda les to the left of the mea (for a symmetrc dstrbuto, the mea ad meda would cocde). For more evdece of skewess, compare the dstaces of the 5th ad 95th percetles from the meda: meda - 5th percetle 5-4 whle 95th percetle -meda 7-5. Thus, the largest 5% of the values (above the 95th percetle) are further from the meda tha are the lowest 5%. The same skewess s evdet whe comparg the th ad 9th percetles to the meda: meda - th percetle whle 9th percetle -meda Fally, ote that the largest value (95) s much further from the meda ( ) tha s the smallest value (5-8), aga a dcato of postve skewess. 44

45 Chapter : Overvew ad Descrptve Statstcs 8. a. There s some evdece of a cyclcal patter. 6 Temperature 5 4 Ide 5 b (.)(54) (.9)(47) 47.7 (.)(5) (.9)(47.7) , etc. t for. α. t forα α. gves a smoother seres. t α ( α) t α ( α)[ α ( α) c. t t α α( α)... α t t t t t α( α) t t ] ( α) [ α t α( α) ( α) t t ]... α ( α) t ( α) t Thus, ( bar) t depeds o t ad all prevous values. As k creases, the coeffcet o t- k decreases (further back tme mples less weght). d. Not very sestve, sce (-α) t- wll be very small. 45

46 Chapter : Overvew ad Descrptve Statstcs 8. a. Whe there s perfect symmetry, the smallest observato y ad the largest observato y wll be equdstat from the meda, so y y. Smlarly, the secod smallest ad secod largest wll be equdstat from y y the meda, so ad so o. Thus, the frst ad secod umbers each par wll be equal, so that each pot the plot wll fall eactly o the 45 degree le. Whe the data s postvely skewed, y wll be much further from the meda tha s y, so ~ y ad the pot ~, ~ ) y ~ wll cosderably eceed ( y y wll fall cosderably below the 45 degree le. A smlar commet aples to other pots the plot. b. The frst pot the plot s ( ,.6-4.) (54., 7.5). The others are: (476.,.9), (44.4, 4.), ( 756.4, 9.), ( 48.8, 88.9), ( 67.5, 8.), ( 8.4, 9.), (.5, 6.), ( 8.,.), ( 5.,.6), ( 5., 9.), (.4,.), ad (.9,.9). The frst umber each of the frst seve pars greatly eceed the secod umber, so each pot falls well below the 45 degree le. A substatal postve skew (stretched upper tal) s dcated. 46

47 CHAPTER Secto.. a. S { 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 } b. Evet A cotas the outcomes where s frst the lst: A { 4, 4, 4, 4 } c. Evet B cotas the outcomes where s frst or secod: B { 4, 4, 4, 4, 4, 4, 4, 4 } d. The compoud evet A B cotas the outcomes A or B or both: A B {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 }. a. Evet A { RRR, LLL, SSS } b. Evet B { RLS, RSL, LRS, LSR, SRL, SLR } c. Evet C { RRL, RRS, RLR, RSR, LRR, SRR } d. Evet D { RRL, RRS, RLR, RSR, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS, LSS } e. Evet D cotas outcomes where all cars go the same drecto, or they all go dfferet drectos: D { RRR, LLL, SSS, RLS, RSL, LRS, LSR, SRL, SLR } Because Evet D totally ecloses Evet C, the compoud evet C D D: C D { RRL, RRS, RLR, RSR, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS, LSS } Usg smlar reasog, we see that the compoud evet C D C: C D { RRL, RRS, RLR, RSR, LRR, SRR } 47

48 Chapter : Probablty. a. Evet A { SSF, SFS, FSS } b. Evet B { SSS, SSF, SFS, FSS } c. For Evet C, the system must have compoet workg ( S the frst posto), the at least oe of the other two compoets must work (at least oe S the d ad rd postos: Evet C { SSS, SSF, SFS } d. Evet C { SFF, FSS, FSF, FFS, FFF } Evet A C { SSS, SSF, SFS, FSS } Evet A C { SSF, SFS } Evet B C { SSS, SSF, SFS, FSS } Evet B C { SSS SSF, SFS } 4. a. Home Mortgage Number Outcome 4 F F F F F F F V F F V F 4 F F V V 5 F V F F 6 F V F V 7 F V V F 8 F V V V 9 V F F F V F F V V F V F V F V V V V F F 4 V V F V 5 V V V F 6 V V V V b. Outcome umbers,, 5,9 c. Outcome umbers, 6 d. Outcome umbers,,, 5, 9 e. I words, the UNION descrbed s the evet that ether all of the mortgages are varable, or that at most all of them are varable: outcomes,,,5,9,6. The INTERSECTION descrbed s the evet that all of the mortgages are fed: outcome. f. The UNION descrbed s the evet that ether eactly three are fed, or that all four are the same: outcomes,,, 5, 9, 6. The INTERSECTION words s the evet that eactly three are fed AND that all four are the same. Ths caot happe. (There are o outcomes commo) : b c. 48

49 Chapter : Probablty 5. a. Outcome Number Outcome b. Outcome Numbers, 4, 7 c. Outcome Numbers 6, 8,, 6,, d. Outcome Numbers,, 7, 9, 9,, 5, 7 49

50 Chapter : Probablty 6. a. Outcome Number Outcome b. Outcomes, 4, 5 c. Outcomes, 6, 9,, 5 d. Outcomes,,,, 4, 5 7. a. S {BBBAAAA, BBABAAA, BBAABAA, BBAAABA, BBAAAAB, BABBAAA, BABABAA, BABAABA, BABAAAB, BAABBAA, BAABABA, BAABAAB, BAAABBA, BAAABAB, BAAAABB, ABBBAAA, ABBABAA, ABBAABA, ABBAAAB, ABABBAA, ABABABA, ABABAAB, ABAABBA, ABAABAB, ABAAABB, AABBBAA, AABBABA, AABBAAB, AABABBA, AABABAB, AABAABB, AAABBBA, AAABBAB, AAABABB, AAAABBB} b. {AAAABBB, AAABABB, AAABBAB, AABAABB, AABABAB} 5

51 Chapter : Probablty 8. a. A A A b. A A A c. A A A 5

52 Chapter : Probablty d. (A A A ) (A A A ) (A A A ) e. A (A A ) 5

53 Chapter : Probablty 9. a. I the dagram o the left, the shaded area s (A B). O the rght, the shaded area s A, the strped area s B, ad the tersecto A B occurs where there s BOTH shadg ad strpes. These two dagrams dsplay the same area. b. I the dagram below, the shaded area represets (A B). Usg the dagram o the rght above, the uo of A ad B s represeted by the areas that have ether shadg or strpes or both. Both of the dagrams dsplay the same area.. a. A {Chev, Pot, Buck}, B {Ford, Merc}, C {Plym, Chrys} are three mutually eclusve evets. b. No, let E {Chev, Pot}, F {Pot, Buck}, G {Buck, Ford}. These evets are ot mutually eclusve (e.g. E ad F have a outcome commo), yet there s o outcome commo to all three evets. 5

54 Chapter : Probablty Secto.. a..7 b c. Let evet A selected customer ows stocks. The the probablty that a selected customer does ot ow a stock ca be represeted by P(A ) - P(A) (.8.5) Ths could also have bee doe easly by addg the probabltes of the fuds that are ot stocks.. a. P(A B) b. P(A B) c. A B ; P(A B ) P(A) P(A B) a. awarded ether # or # (or both): P(A A ) P(A ) P(A ) - P(A A ) b. awarded ether # or #: P(A A ) P[(A A ) ] - P(A A ) c. awarded at least oe of #, #, #: P(A A A ) P(A ) P(A ) P(A ) - P(A A ) - P(A A ) P(A A ) P(A A A ) d. awarded oe of the three projects: P( A A A ) P(awarded at least oe) e. awarded # but ether # or #: P( A A A ) P(A ) - P(A A ) P(A A ) P(A A A )

55 Chapter : Probablty f. ether (ether # or #) or #: P[( A A ) A ] P(shaded rego) P(awarded oe) P(A ) Alteratvely, aswers to a f ca be obtaed from probabltes o the accompayg Ve dagram 55

56 Chapter : Probablty 4. a. P(A B) P(A) P(B) - P(A B), so P(A B) P(A) P(B) - P(A B) b. P(shaded rego) P(A B) - P(A B) Shaded rego evet of terest (A B ) (A B) 5. a. Let evet E be the evet that at most oe purchases a electrc dryer. The E s the evet that at least two purchase electrc dryers. P(E ) P(E) b. Let evet A be the evet that all fve purchase gas. Let evet B be the evet that all fve purchase electrc. All other possble outcomes are those whch at least oe of each type s purchased. Thus, the desred probablty P(A) P(B) a. There are s smple evets, correspodg to the outcomes CDP, CPD, DCP, DPC, PCD, ad PDC. The probablty assged to each s 6. b. P( C raked frst) P( {CPD, CDP} ). c. P( C raked frst ad D last) P({CPD})

57 Chapter : Probablty 7. a. The probabltes do ot add to because there are other software packages besdes SPSS ad SAS for whch requests could be made. b. P(A ) P(A) -..7 c. P(A B) P(A) P(B)..5.8 (sce A ad B are mutually eclusve evets) d. P(A B ) P[(A B) ] (De Morga s law) - P(A B) Ths stuato requres the complemet cocept. The oly way for the desred evet NOT to happe s f a 75 W bulb s selected frst. Let evet A be that a 75 W bulb s selected frst, ad P(A) 5 6. The the desred evet s evet A. 6 9 So P(A ) P(A) Let evet A be that the selected jot was foud defectve by spector A. P(A),. Let evet B be aalogous for spector B. P(B) 75,. Compoud evet A B s the evet that the selected jot was foud defectve by at least oe of the two spectors. P(A B) 59,. a. The desred evet s (A B), so we use the complemet rule: 59 P(A B) - P(A B) -, 884,.884 b. The desred evet s B A. P(B A ) P(B) - P(A B). P(A B) P(A) P(B) - P(A B), So P(B A ) P(B) - P(A B) Let S, S ad S represet the swg ad ght shfts, respectvely. Let C ad C represet the usafe codtos ad urelated to codtos, respectvely. a. The smple evets are {S,C}, {S,C}, {S,C}, {S,C},{S,C}, {S,C}. b. P({C}) P({S,C},{S,C},{S,C}) c. P({S} ) - P({S,C}, {S,C}) (..5).55 57

58 Chapter : Probablty. a. P({M,H}). b. P(low auto) P[{(L,N}, (L,L), (L,M), (L,H)}] Followg a smlar patter, P(low homeower s) c. P(same deductble for both) P[{ LL, MM, HH }] d. P(deductbles are dfferet) P(same deductbles) e. P(at least oe low deductble) P[{ LN, LL, LM, LH, ML, HL }] f. P(ether low) P(at least oe low) a. P(A A ) P(A ) P(A ) - P(A A ) b. P(A A ) P(A ) - P(A A ) c. P(eactly oe) P(A A ) - P(A A ) Assume that the computers are umbered 6 as descrbed. Also assume that computers ad are the laptops. Possble outcomes are (,) (,) (,4) (,5) (,6) (,) (,4) (,5) (,6) (,4) (,5) (,6) (4,5) (4,6) ad (5,6). a. P(both are laptops) P[{ (,)}] 5.67 b. P(both are desktops) P[{(,4) (,5) (,6) (4,5) (4,6) (5,6)}] c. P(at least oe desktop) P(o desktops) P(both are laptops).67.9 d. P(at least oe of each type) P(both are the same) P(both laptops) P(both desktops)

59 Chapter : Probablty 4. Sce A s cotaed B, the B ca be wrtte as the uo of A ad (B A ), two mutually eclusve evets. (See dagram). From Aom, P[A (B A )] P(A) P(B A ). Substtutg P(B), P(B) P(A) P(B A ) or P(B) - P(A) P(B A ). From Aom, P(B A ), so P(B) P(A) or P(A) P(B). For geeral evets A ad B, P(A B) P(A), ad P(A B) P(A). 5. P(A B) P(A) P(B) - P(A B).65 P(A C).55, P(B C).6 P(A B C) P(A B C) P(A) P(B) P(C) P(A B) P(A C) P(B C) a. P(A B C).98, as gve. b. P(oe selected) - P(A B C) c. P(oly automatc trasmsso selected). from the Ve Dagram d. P(eactly oe of the three)

60 Chapter : Probablty 6. a. P(A ) P(A ) b. P(A A ) P(A ) P(A ) - P(A A ) c. P(A A A ) P(A A ) - P(A A A ) d. P(at most two errors) P(all three types) - P(A A A ) Outcomes: (A,B) (A,C ) (A,C ) (A,F) (B,A) (B,C ) (B,C ) (B,F) (C,A) (C,B) (C,C ) (C,F) (C,A) (C,B) (C,C ) (C,F) (F,A) (F,B) (F,C ) (F,C ) a. P[(A,B) or (B,A)]. 4 7 b. P(at least oe C). 7 c. P(at least 5 years) P(at most 4 years) P[(,6) or (6,) or (,7) or (7,) or (,) or (,) or (6,7) or (7,6)] There are 7 equally lkely outcomes. a. P(all the same) P[(,,) or (,,) or (,,)] 7 9 b. P(at most are assged to the same stato) P(all are the same) c. P(all dfferet) [{(,,) (,,) (,,) (,,) (,,) (,,)}]

61 Chapter : Probablty Secto. 9. a. (5)(4) (5 choces for presdet, 4 rema for vce presdet) b. (5)(4)() 6 5 5!!! c. (No orderg s mpled the choce). a. Because order s mportat, we ll use P 8, 8(7)(6) 6. b. Order does t matter here, so we use C,6 59, c. From each group we choose : 8, 6 8,6 59,775 d. The umerator comes from part c ad the deomator from part b:. 4 e. We use the same deomator as part d. We ca have all zfadel, all merlot, or all caberet, so P(all same) P(all z) P(all m) P(all c) ,775.. a. ( )( ) (9)(7) 4 b. ( )( )( ) (9)(7)(5) 645, so such a polcy could be carred out for 645 successve ghts, or appromately years, wthout repeatg eactly the same program. 6

62 Chapter : Probablty. a b. 4 c. 4 8 d. # wth at least o Soy total # - # wth o Soy 4 8 e. P(at least oe Soy) P(eactly oe Soy) P(oly Soy s recever) P(oly Soy s CD player) P(oly Soy s deck) ! a. 5, 5 5!! 8 7 b , c. P(eactly 4 have cracks). d. P(at least 4) P(eactly 4) P(eactly 5)

63 Chapter : Probablty 6 4. a.. 8,76 6 P(all from day shft) 48. 8,45,6 8, b. P(all from same shft) c. P(at least two shfts represeted) P(all from same shft) d. Let A day shft urepreseted, A swg shft urepreseted, ad A graveyard shft urepreseted. The we wsh P(A A A ). P(A ) P(day urepreseted) P(all from swg ad graveyard) P(A ) , P(A ) , P(A ) , P(A A ) P(all from graveyard) P(A A ) , P(A A ) , P(A A A ), So P(A A A )

64 Chapter : Probablty 5. There are possble outcomes -- 5 ways to select the postos for B s votes: BBAAA, BABAA, BAABA, BAAAB, ABBAA, ABABA, ABAAB, AABBA, AABAB, ad AAABB. Oly the last two have A ahead of B throughout the vote cout. Sce the outcomes are equally lkely, the desred probablty s.. 6. a., 4, 5, so 6 rus b., (just oe temperature),, 5 mples that there are such rus. 7. There are 5 6 ways to select the 5 rus. Each catalyst s used dfferet rus, so the umber of ways of selectg oe ru from each of these 5 groups s 5. Thus the desred probablty s a. P(selectg - 75 watt bulbs) b. P(all three are the same) c

65 Chapter : Probablty d. To eame eactly oe, a 75 watt bulb must be chose frst. (6 ways to accomplsh ths). To eame eactly two, we must choose aother wattage frst, the a 75 watt. ( 9 6 ways). Followg the patter, for eactly three, ways; for four, ; for fve, P(eame at least 6 bulbs) P(eame 5 or less) P( eame eactly or or or 4 or 5) [P(oe) P(two) P(fve)] [ ] a. We wat to choose all of the 5 cordless, ad 5 of the others, to be amog the frst servced, so the desred probablty s. 89 b. Isolatg oe group, say the cordless phoes, we wat the other two groups represeted the last 5 servced. So we choose 5 of the others, ecept that we do t wat to clude the outcomes where the last fve are all the same. 5 So we have But we have three groups of phoes, so the desred probablty s (5).498. c. We wat to choose of the 5 cordless, of the 5 cellular, ad of the corded phoes:

66 Chapter : Probablty 4. a. If the A s are dstgushable from oe aother, ad smlarly for the B s, C s ad D s, the there are! Possble cha molecules. S of these are: A A A B C C D C D D B B, A A A B C C D C D D B B A A A B C C D C D D B B, A A A B C C D C D D B B A A A B C C D C D D B B, A A A B C C D C D D B B These 6 (!) dffer oly wth respect to orderg of the A s. I geeral, groups of 6 cha molecules ca be created such that wth each group oly the orderg of the A s s dfferet. Whe the A subscrpts are suppressed, each group of 6 collapses to a sgle molecule (B s, C s ad D s are stll dstgushable). At ths pot there are! molecules. Now suppressg subscrpts o the B s, C s ad D s tur gves!! ultmately 69, 6 cha molecules. 4 (!) b. Thk of the group of A s as a sgle etty, ad smlarly for the B s, C s, ad D s. The there are 4! Ways to order these ettes, ad thus 4! Molecules whch the A s are cotguous, the B s, C s, ad D s are also. Thus, P(all together)! a. P(at least oe F amog st ) P(o F s amog st ) A alteratve method to calculate P(o F s amog st ) would be to choose oe of the females ad of the 4 males, as follows: , obvously producg the same result b. P(all F s amog st 5) c. P(ordergs are dfferet) P(ordergs are the same for both semesters) (# ordergs such that the orders are the same each semester)/(total # of possble ordergs for semesters) ( ) ( ) 66