Chapter 2 Methods for Describing Sets of Data

Transcription

1 Statitic for Buie ad Ecoomic 1th Editio McClave Solutio Maual Full Dowload: Chapter Method for Decribig Set of Data.1 Firt, we fid the frequecy of the grade A. The um of the frequecie for all five grade mut be. Therefore, ubtract the um of the frequecie of the other four grade from. The frequecy for grade A i: ( ) = 184 = 16 To fid the relative frequecy for each grade, divide the frequecy by the total ample ize,. The relative frequecy for the grade B i 36/ =.18. The ret of the relative frequecie are foud i a imilar maer ad appear i the table: Grade o Statitic Exam Frequecy Relative Frequecy A: B: C: D: F: Below Total 1.. a. To fid the frequecy for each cla, cout the umber of time each letter occur. The frequecie for the three clae are: Cla Frequecy X 8 Y 9 Z 3 Total b. The relative frequecy for each cla i foud by dividig the frequecy by the total ample ize. The relative frequecy for the cla X i 8/ =.4. The relative frequecy for the cla Y i 9/ =.45. The relative frequecy for the cla Z i 3/ =.15. Cla Frequecy Relative Frequecy X 8.4 Y 9.45 Z 3.15 Total 1. 1 Copyright 14 Pearo Educatio, Ic. Full dowload all chapter itatly pleae go to Solutio Maual, Tet Bak ite: tetbaklive.com

2 Method for Decribig Set of Data 11 c. The frequecy bar chart i: Frequecy X Y Cla Z d. The pie chart for the frequecy ditributio i: Pie Chart of Cla Z 15.% Category X Y Z X 4.% Y 45.%.3 a. The type of graph i a bar graph. b. The variable meaured for each of the robot i type of robotic limb. c. From the graph, the deig ued the mot i the leg oly deig. d. The relative frequecie are computed by dividig the frequecie by the total ample ize. The total ample ize i = 16. The relative frequecie for each of the categorie are: Type of Limb Frequecy Relative Frequecy Noe 15 15/16 =.14 Both 8 8 / 16 =.75 Leg ONLY 63 63/16 =.594 Wheel ONLY /16 =.189 Total Copyright 14 Pearo Educatio, Ic.

3 1 Chapter e. Uig MINITAB, the Pareto diagram i:.6.5 Relative Frequecy Leg Wheel Type Noe Both Percet withi all data..4 a. From the pie chart, 5.4% or.54 of the ampled adult livig i the U.S. ue the iteret ad pay to dowload muic. From the data, 56 out of 1,3 adult or 56/1,3 =.54 of ampled adult i the U.S. ue the iteret ad pay to dowload muic. Thee two reult agree. b. Uig MINITAB, a pie chart of the data i: Pie Chart of Dowload-Muic Category Pay No Pay No Pay 33.% Pay 67.% Copyright 14 Pearo Educatio, Ic.

4 Method for Decribig Set of Data 13.5 Uig MINITAB, the Pareto diagram for the data i: 5 Chart of Teat 4 Percet 3 1 Small SmallStadard Large Teat Major Achor Percet withi all data. Mot of the teat i UK hoppig mall are mall or mall tadard. They accout for approximately 84% of all teat ([ ]/1,81 =.84). Very few (le tha 1%) of the teat are achor..6 a. The relative frequecy for each repoe category i foud by dividig the frequecy by the total ample ize. The relative frequecy for the category Iurace Compaie i 869/119 =.41. The ret of the relative frequecie are foud i a imilar maer ad are reported i the table. Mot repoible for riig health-care cot Number repodig Relative Frequecie Iurace compaie /119 =.41 Pharmaceutical compaie /119 =.16 Govermet /119 =.16 Hopital 17 17/119 =.6 Phyicia 85 85/119 =.4 Other 18 18/119 =.6 Not at all ure 33 33/119 =.11 TOTAL, Copyright 14 Pearo Educatio, Ic.

5 14 Chapter b. Uig MINITAB, the relative frequecy bar chart i: Chart of Category 4 3 Cout 1 Iurace Co Pharm Govermet Hopital Category Phyicia Other Not ure c. Uig MINITAB, the Pareto diagram i: Chart of Category.4 Relative Frequecy.3..1 Iurace Co. Govermet Pharm Not ure Category Hopital Other Phyicia Mot America adult i the ample (41%) believe that the Iurace compaie are the mot repoible for the riig cot of health care. The ext highet categorie are Govermet ad Pharmaceutical compaie with about 16% each. Oly 4% of America adult i the ample believe phyicia are the mot repoible for the riig health care cot..7 a. Sice the variable meaured i maufacturer, the data type i qualitative. Copyright 14 Pearo Educatio, Ic.

6 Method for Decribig Set of Data 15 b. Uig MINITAB, a frequecy bar chart for the data i: 1 Number Shipped 1 Frequecy Bitel CyberNet Fujia Ladi Glitt EItelliget KwagWoo Omro Pax Tech Proveco SZZT Tohiba Urmet Maufacturer c. Uig MINITAB, the Pareto diagram i: 1 Number Shipped 1 Frequecy Fujia Ladi SZZT KwagWoo Omro Proveco CyberNet Bitel Tohiba Pax Tech Glitt EItelliget Urmet Maufacturer Mot PIN pad hipped i 7 were maufactured by either Fujia Ladi or SZZT Electroic. Thee two categorie make up (119, + 67,3)/334,39= 186,3/334,39 =.558 of all PIN pad hipped i 7. Urmet hipped the fewet umber of PIN pad amog thee 1 maufacturer. Copyright 14 Pearo Educatio, Ic.

7 16 Chapter.8 Uig MINITAB, the bar graph of the wave i: Chart of Job Statu WorkNoMBA WorkMBA NoWorkBuSch Percet NoWorkGrad Sch WorkNoMBA WorkMBA NoWorkBuSch NoWorkGrad Sch Pael variable: Wave; Percet withi all data. Job Statu I wave 1, mot of thoe takig the GMAT were workig (657/344 =.819) ad oe had MBA. About % were ot workig but were i either a 4-year ititutio or other graduate chool ([ ]/344 =.181). I wave, almot all were ow workig ([ ]/344 =.974). Of thoe workig, more tha half had MBA (1787/[ ] =.566). Of thoe ot workig, mot were i aother graduate chool..9 Uig MINITAB, the pie chart i: Pie Chart of Percet v Blog/Forum Not Idetified 15.4% Category Compay Employee Third Party Not Idetified Third Party 11.5% Compay 38.5% Employee 34.6% Compaie ad Employee repreet ( = 73.1) lightly more tha 73% of the etitie creatig blog/forum. Third partie are the leat commo etity. Copyright 14 Pearo Educatio, Ic.

8 Method for Decribig Set of Data 17.1 Uig MINITAB, a bar chart of the data i: Chart of INDUSTRY Cout Aeropace & Defee Bakig Buie Service & Capital Good Chemical Coglomerate Cotructio Coumer Durable Diverified Fiacia Drug & biotecholog Food Drik & tobacco Health care equipme Hotel, Retaurat Houehold & peroal Iurace Material Media Oil & Ga Operatio Retailig Semicoductor Software & Service Techology Hardware Telecommuicatio Traportatio Utilitie INDUSTRY Idutrie with the highet frequecie iclude Oil & Ga Operatio, Retailig, Drug & biotechologie, ad Health care equipmet. Idutrie with the mallet frequecie iclude Buie Service, Cotructio, Bakig, ad Coumer Durable..11 a. Uig MINITAB, a pie chart of the data i: Pie Chart of PREVUSE Category NEVER USED USED 8.8% NEVER 71.% From the chart, 71.% or.71 of the ampled phyicia have ever ued ethic coultatio. Copyright 14 Pearo Educatio, Ic.

9 18 Chapter b. Uig MINITAB, a pie chart of the data i: Pie Chart of FUTUREUSE NO 19.5% Category NO YES YES 8.5% From the chart, 19.5% or.195 of the ampled phyicia tate that they will ot ue the ervice i the future. c. Uig MINITAB, the ide-by-ide pie chart are: Pie Chart of PREVUSE MED SURG Category NEVER USED USED 9.3% USED 7.9% NEVER 7.7% NEVER 7.1% Pael variable: SPEC The proportio of medical practitioer who have ever ued ethic coultatio i.77. The proportio of urgical practitioer who have ever ued ethic coultatio i.71. Thee two proportio are almot the ame. Copyright 14 Pearo Educatio, Ic.

10 Method for Decribig Set of Data 19 d. Uig MINITAB, the ide-by-ide pie chart are: Pie Chart of FUTUREUSE MED SURG Category NO YES NO 17.3% NO 3.3% YES 8.7% YES 76.7% Pael variable: SPEC The proportio of medical practitioer who will ot ue ethic coultatio i the future i.173. The proportio of urgical practitioer who will ot ue ethic coultatio i the future i.33. The proportio of urgical practitioer who will ot ue ethic coultatio i the future i greater tha that of the medical practitioer..1 Uig MINITAB, the ide-by-ide bar graph are: Chart of Acquiitio No Ye Percet No Ye Acquiitio Pael variable: Year; Percet withi all data. I 198, very few firm had acquiitio 18 /1, By 199, the proportio of firm havig acquiitio icreaed to 35 /, By, the proportio of firm havig acquiitio icreaed to 748 /, Copyright 14 Pearo Educatio, Ic.

11 Chapter.13 Uig MINITAB, the ide-by-ide bar graph are: Chart of Dive Left Middle Right Ahead Behid Percet 8 Tied 6 4 Left Middle Right Dive Pael variable: Situatio; Percet withi all data. From the graph, it appear that if the team i either tied or ahead, the goal-keeper ted to dive either right or left with equal probability, with very few divig i the middle. However, if the team i behid, the the majority of goal-keeper ted to dive right (71%)..14 Uig MINITAB, a pie chart of the data i: Pie Chart of Meaure Total viitor 6.7% Big Show.% Category Big Show Fud Raied Member Payig viitor Total viitor Payig viitor 16.7% Fud Raied 3.3% Member 13.3% Sice the ize of the lice are cloe to each other, it appear that the reearcher i correct. There i a large amout of variatio withi the mueum commuity with regard to performace meauremet ad evaluatio. Copyright 14 Pearo Educatio, Ic.

12 Method for Decribig Set of Data 1.15 a. The variable meaured by Performark i the legth of time it took for each advertier to repod back. b. The pie chart i: Pie Chart of Repoe Time Never repoded 1.% > 1 day 1.% Category > 1 day day 6-1 day Never repoded day 33.% 6-1 day 34.% c. Twety-oe percet or.117, 3,57 of the advertier ever repod to the ale lead. d. The iformatio from the pie chart doe ot idicate how effective the "bigo card" are. It jut idicate how log it take advertier to repod, if at all..16 a. Uig MINITAB, the ide-by-ide graph are: Chart of Frequecy v Star Cotet Expoure Frequecy 16 Faculty Opportuity Star Pael variable: Criteria From thee graph, oe ca ee that very few of the top 3 MBA program got 5-tar i ay criteria. I additio, about the ame umber of program got 4 tar i each of the 4 criteria. The bigget differece i ratig amog the 4 criteria wa i the umber of program receivig 3-tar. More program received 3-tar i Coure Cotet tha i ay of the other criteria. Coequetly, fewer program received -tar i Coure Cotet tha i ay of the other criteria. Copyright 14 Pearo Educatio, Ic.

13 Chapter b. Sice thi chart lit the rakig of oly the top 3 MBA program i the world, it i reaoable that oe of thee bet program would be rated a 1-tar o ay criteria..17 a. Uig MINITAB, bar chart for the 3 variable are: 1 Chart of Well Cla 1 8 Cout 6 4 Private Well Cla Public Chart of Aquifer 15 Cout 1 5 Bedrock Aquifer Ucoolidated Chart of Detectio Cout Below Limit Detectio Detect Copyright 14 Pearo Educatio, Ic.

14 Method for Decribig Set of Data 3 b. Uig MINITAB, the ide-by-ide bar chart i: Chart of Detectio 8 Private Below Limit Public Detect 7 6 Percet Below Limit Detect Detectio Pael variable: Well Cla; Percet withi all data. c. Uig MINITAB, the ide-by-ide bar chart i: Chart of Detectio 7 Bedrock Below Limit Ucooli Detect 6 5 Percet Below Limit Detect Detectio Pael variable: Aquifer; Percet withi all data. d. From the bar chart i part a-c, oe ca ifer that mot aquifer are bedrock ad mot level of MTBE were below the limit ( /3). Alo the percetage of public well vere private well are relatively cloe. Approximately 8% of private well are ot cotamiated, while oly about 6% of public well are ot cotamiated. The percetage of cotamiated well i about the ame for both type of aquifer ( 3%). Copyright 14 Pearo Educatio, Ic.

15 4 Chapter.18 Uig MINITAB, the relative frequecy hitogram i:.5. Relative Frequecy Cla To fid the umber of meauremet for each meauremet cla, multiply the relative frequecy by the total umber of obervatio, = 5. The frequecy table i: Meauremet Cla Relative Frequecy Frequecy (.1) = (.15) = (.5) = (.) = (.5) = (.1) = (.1) = (.5) = 5 5 Uig MINITAB, the frequecy hitogram i: Frequecy Cla Copyright 14 Pearo Educatio, Ic.

16 . a. The origial data et ha = 3 obervatio. b. For the bottom row of the tem-ad-leaf diplay: Method for Decribig Set of Data 5 The tem i. The leave are, 1,. Aumig that the data are up to two digit, rouded off to the earet whole umber, the umber i the origial data et are, 1, ad. c. Agai, aumig that the data are up to two digit, rouded off to the earet whole umber, the dot plot correpodig to all the data poit i:.1 a. Thi i a frequecy hitogram becaue the umber of obervatio i graphed for each iterval rather tha the relative frequecy. b. There are 14 meauremet clae. c. There are 49 meauremet i the data et.. a. The meauremet cla 1 ha the highet proportio of repodet. b. The approximate proportio of the 144 orgaizatio that reported a percetage moetary lo from maliciou iider actio le tha % i =.68. c. The approximate proportio of the 144 orgaizatio that reported a percetage moetary lo from maliciou iider actio greater tha 6% i =.19. d. The approximate proportio of the 144 orgaizatio that reported a percetage moetary lo from maliciou iider actio betwee % ad 3% i.11. Therefore about.11(144) = or 16 of the 144 orgaizatio reported a percetage moetary lo from maliciou iider actio betwee % ad 3%..3 a. Sice the label o the vertical axi i Percet, thi i a relative frequecy hitogram. We ca divide the percet by 1% to get the relative frequecie. b. Summig the percet repreeted by all of the bar above 1, we get approximately 1%. Copyright 14 Pearo Educatio, Ic.

17 6 Chapter.4 a. Uig MINITAB, the tem-ad-leaf diplay ad hitogram are: Stem-ad-Leaf Diplay: Score Stem-ad-leaf of Score N = 186 Leaf Uit = (37) Hitogram of Score 5 Frequecy Score b. From the tem-ad-leaf diplay, there are oly 7 obervatio with aitatio core le tha 86. The proportio of hip with accepted aitatio tadard i (186 7) / / c. The core of 69 i highlighted i the tem-ad-leaf diplay. Copyright 14 Pearo Educatio, Ic.

18 Method for Decribig Set of Data 7.5 a. Uig MINITAB, a dot plot of the data i: Dotplot of Acquiitio Acquiitio b. By lookig at the dot plot, oe ca coclude that the year had the highet umber of firm with at leat oe acquiitio. The lowet umber of acquiitio i that time frame (748) i almot 1 higher tha the highet value from the remaiig year..6 a. Uig MINITAB, a hitogram of the curret value of the 3 NFL team i: 14 Hitogram of Value ($mil) 1 1 Frequecy Value ($mil) Copyright 14 Pearo Educatio, Ic.

19 8 Chapter b. Uig MINITAB, a hitogram of the 1-year chage i curret value for the 3 NFL team i: 1 Hitogram of Chag1Yr (%) 8 Frequecy Chag1Yr (%) c. Uig MINITAB, a hitogram of the debt-to-value ratio for the 3 NFL team i: Hitogram of Debt/Value (%) 15 Frequecy Debt/Value (%) Copyright 14 Pearo Educatio, Ic.

20 d. Uig MINITAB, a hitogram of the aual reveue for the 3 NFL team i: Method for Decribig Set of Data 9 16 Hitogram of Reveue ($mil) 14 1 Frequecy Reveue ($mil) e. Uig MINITAB, a hitogram of the operatig icome for the 3 NFL team i: 1 Hitogram of Icome ($mil) 8 Frequecy Icome ($mil) f. For all of the hitogram, there i 1 team that ha a very high core. The Dalla Cowboy have the larget value for curret value, aual reveue, ad operatig icome. However, the New York Giat have the highet 1-year chage, while the New York Jet have the highet debt-to-value ratio. All of the graph except the oe howig the 1-Yr Value Chage are kewed to the right. Copyright 14 Pearo Educatio, Ic.

21 3 Chapter.7 a. Uig MINITAB, the frequecy hitogram for 11 ad 1 SAT mathematic core are: Hitogram of MATH11, MATH1 14 MATH MATH Frequecy It appear that the core have ot chaged very much at all. The graph are very imilar. b. Uig MINITAB, the frequecy hitogram for 11 ad 1 SAT mathematic core are: Hitogram of MATH11, MATH1 14 MATH MATH Frequecy It appear that the core have hifted to the right. The core i 11 appear to be omewhat better tha the core i 11. Copyright 14 Pearo Educatio, Ic.

22 c. Uig MINITAB, the frequecy hitogram of the differece i: Method for Decribig Set of Data Hitogram of DiffMath 14 1 Frequecy DiffMath 16 3 From thi graph of the differece, we ca ee that there are more obervatio to the right of tha to the left of. Thi idicate that, i geeral, the core have improved ice 1. d. From the graph, the larget improvemet core i i the eighborhood of 3. The actual larget core i 3 ad it i aociated with Michiga..8 Uig MINITAB, the two dot plot are: Dotplot of Arrive, Depart Arrive Depart Data Ye. Mot of the umber of item arrivig at the work ceter per hour are i the 135 to 165 area. Mot of the umber of item departig the work ceter per hour are i the 11 to 14 area. Becaue the umber of item arrivig i larger tha the umber of item departig, there will probably be ome ort of bottleeck. Copyright 14 Pearo Educatio, Ic.

23 3 Chapter.9 Uig MINITAB, the tem-ad-leaf diplay i: Stem-ad-Leaf Diplay: Dioxide Stem-ad-leaf of Dioxide N = 16 Leaf Uit = () The highlighted value are value that correpod to water pecime that cotai oil. There i a tedecy for crude oil to be preet i water with lower level of dioxide a 6 of the lowet 8 pecime with the lowet level of dioxide cotai oil..3 Ye, we would agree with the tatemet that hoey may be the preferable treatmet for the cough ad leep difficulty aociated with childhood upper repiratory tract ifectio. For thoe receivig the hoey doage, 14 of the 35 childre (or 4%) had improvemet core of 1 or higher. For thoe receivig the DM doage, oly 9 of the 33 (or 4%) childre had improvemet core of 1 or higher. For thoe receivig o doage, oly of the 37 childre (or 5%) had improvemet core of 1 or higher. I additio, the media improvemet core for thoe receivig the hoey doage wa 11, the media for thoe receivig the DM doage wa 9 ad the media for thoe receivig o doage wa Uig MINITAB, the relative frequecy hitogram of the year i practice for the two group of doctor are: Hitogram of YRSPRAC 5 NO YES Percet YRSPRAC Pael variable: FUTUREUSE The reearcher hypotheized that older, more experieced phyicia will be le likely to ue ethic coultatio i the future. From the hitogram, approximately 38% of the doctor that aid o have more tha year of experiece. Oly about 19% of the doctor that aid ye had more tha year of experiece. Thi upport the reearcher aertio. Copyright 14 Pearo Educatio, Ic.

24 Method for Decribig Set of Data 33.3 a. Uig MINITAB, the tem-ad-leaf diplay i a follow, where the tem are the uit place ad the leave are the decimal place: Stem-ad-Leaf Diplay: Time Stem-ad-leaf of Time N = 49 Leaf Uit =.1 (6) b. A little more tha half (6/49 =.53) of all compaie pet le tha moth i bakruptcy. Oly two of the 49 compaie pet more tha 6 moth i bakruptcy. It appear that, i geeral, the legth of time i bakruptcy for firm uig "prepack" i le tha that of firm ot uig prepack." c. A dot diagram will be ued to compare the time i bakruptcy for the three type of "prepack" firm: Dotplot of Time v Vote Vote Joit Noe Prepack Time d. The highlighted time i part a correpod to compaie that were reorgaized through a leverage buyout. There doe ot appear to be ay patter to thee poit. They appear to be cattered about evely throughout the ditributio of all time. Copyright 14 Pearo Educatio, Ic.

25 34 Chapter.33 Uig MINITAB, the hitogram of the data i: 6 Hitogram of INTTIME 5 4 Frequecy INTTIME Thi hitogram look very imilar to the oe how i the problem. Thu, there appear that there wa miimal or o collaboratio or colluio from withi the compay. We could coclude that the phihig attack agait the orgaizatio wa ot a iide job..34 Uig MINITAB, the tem-ad-leaf diplay for the data i: Stem-ad-Leaf Diplay: Time Stem-ad-leaf of Time N = 5 Leaf Uit = (7) The umber i bold repreet delivery time aociated with cutomer who ubequetly did ot place additioal order with the firm. Sice there were oly cutomer with delivery time of 68 day or loger that placed additioal order, I would ay the maximum tolerable delivery time i about 65 to 67 day. Everyoe with delivery time le tha 67 day placed additioal order..35 Aume the data are a ample. The ample mea i: x x The media i the average of the middle two umber whe the data are arraged i order (ice = 6 i eve). The data arraged i order are:.,.1,.5,.8, 3., 3.7. The middle two umber are.5 ad.8. The media i: Copyright 14 Pearo Educatio, Ic.

26 Method for Decribig Set of Data a. b. c. d. x 85 x x x x The mea ad media of a ymmetric data et are equal to each other. The mea i larger tha the media whe the data et i kewed to the right. The mea i le tha the media whe the data et i kewed to the left. Thu, by comparig the mea ad media, oe ca determie whether the data et i ymmetric, kewed right, or kewed left..38 The media i the middle umber oce the data have bee arraged i order. If i eve, there i ot a igle middle umber. Thu, to compute the media, we take the average of the middle two umber. If i odd, there i a igle middle umber. The media i thi middle umber. A data et with five meauremet arraged i order i 1, 3, 5, 6, 8. The media i the middle umber, which i 5. A data et with ix meauremet arraged i order i 1, 3, 5, 5, 6, 8. The media i the average of the middle two umber which i Aume the data are a ample. The mode i the obervatio that occur mot frequetly. For thi ample, the mode i 15, which occur three time. The ample mea i: x x The media i the middle umber whe the data are arraged i order. The data arraged i order are: 1, 11, 1, 13, 15, 15, 15, 16, 17, 18, 18. The middle umber i the 6th umber, which i a. x x Media = (mea of 3rd ad 4th umber, after orderig) Mode = 3 b. x 4 4 x Media = 3 (7th umber, after orderig) Mode = 3 Copyright 14 Pearo Educatio, Ic.

27 36 Chapter c. x x Media = (mea of 5th ad 6th umber, after orderig) Mode = 5.41 a. For a ditributio that i kewed to the left, the mea i le tha the media. b. For a ditributio that i kewed to the right, the mea i greater tha the media. c. For a ymmetric ditributio, the mea ad media are equal..4 a. The mea i x 9 (.1) ( 1.6) x The average aualized percetage retur o ivetmet for 13 radomly elected tock creeer i 1.8. b. Sice the umber of obervatio i odd, the media i the middle umber oce the data have bee arraged i order. The data arraged i order are: The middle umber i 9.8 which i the media. Half of the aualized percetage retur o ivetmet are below 9.8 ad half are above a. The mea amout exported o the pritout i 653. Thi mea that the average amout of moey per market from exportig parklig wie wa $653,. b. The media amout exported o the pritout i 31. Sice the media i the middle value, thi mea that half of the 3 parklig wie export value were above $31, ad half of the parklig wie export value were below $31,. c. The mea 3-year percetage chage o the pritout i 481. Thi mea that i the lat three year, the average chage i 481%, which idicate a large icreae. d. The media 3-year percetage chage o the pritout i 156. Sice the media i the middle value, thi mea that half, or 15 of the 3 coutrie 3-year percetage chage value were above 156% ad half, or 15 of the 3 coutrie 3-year percetage chage value were below 156%..44 a. The ample mea i: xi i x The ample average urface roughe of the obervatio i Copyright 14 Pearo Educatio, Ic.

28 Method for Decribig Set of Data 37 b. The media i foud a the average of the 1 th ad 11 th obervatio, oce the data have bee ordered. The ordered data are: The 1 th ad 11 th obervatio are.3 ad.5. The media i: The middle urface roughe meauremet i.4. Half of the ample meauremet were le tha.4 ad half were greater tha.4. c. The data are omewhat kewed to the left. Thu, the media might be a better meaure of cetral tedecy tha the mea. The few mall value i the data ted to make the mea maller tha the media. x 1,68,97 885,18 881, ,967 15,19,1.45 a. The mea i x 759,61.5. The average reearch expediture for the top raked uiveritie i 759,61.5 thouad dollar. b. Sice the umber of obervatio i eve, the media i the average of the middle umber oce the data have bee arraged i order. Sice the data are already arraged i order, the media i 7,59 688, 5 695, Half of the ititutio have a reearch expediture le tha 695,48.5 thouad dollar ad half have reearch expediture greater tha 695,48.5 thouad dollar. c. No, the mea from part a would ot be a good meaure for the ceter of the ditributio for all America uiveritie. The data i part a come from oly the top uiveritie. Thee uiveritie would ot be repreetative of all America uiveritie..46 a. The mea i The tatemet i accurate. b. The media i 68.. The tatemet i accurate. c. The mode i 64. The tatemet i ot accurate. A better tatemet would be: The mot commo reported level of upport for corporate utaiability for the 99 eior maager wa 64. d. Sice the mea ad media are almot the ame, the ditributio of the 99 upport level hould be fairly ymmetric. The hitogram i Exercie.3 i almot ymmetric..47 a. The media i the middle umber (18 th ) oce the data have bee arraged i order becaue = 35 i odd. The hoey doage data arraged i order are: 4,5,6,8,8,8,8,9,9,9,9,1,1,1,1,1,1,11,11,11,11,1,1,1,1,1,1,13,13,14,15,15,15,15,16 The 18 th umber i the media = 11. Copyright 14 Pearo Educatio, Ic.

29 38 Chapter b. The media i the middle umber (17 th ) oce the data have bee arraged i order becaue = 33 i odd. The DM doage data arraged i order are: 3,4,4,4,4,4,4,6,6,6,7,7,7,7,7,8,9,9,9,9,9,1,1,1,11,1,1,1,1,1,13,13,15 The 17 th umber i the media = 9. c. The media i the middle umber (19 th ) oce the data have bee arraged i order becaue = 37 i odd. The No doage data arraged i order are:,1,1,1,3,3,4,4,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,1,11,1,1 The 19 th umber i the media = 7. d. Sice the media for the Hoey doage i larger tha the other two, it appear that the hoey doage lead to more improvemet tha the other two treatmet..48 a The mea dioxide level i x The average dioxide amout i b. Sice the umber of obervatio i eve, the media i the average of the middle umber oce the data are arraged i order. The data arraged i order are: The media i Half of the dioxide level are below 1.35 ad half are above c. The mode i the umber that occur the mot. For thi data et the mode i 4.. The mot frequet level of dioxide i 4.. d. Sice the umber of obervatio i eve, the media i the average of the middle umber oce the data are arraged i order. The data arraged i order are: The media i e. Sice the umber of obervatio i eve, the media i the average of the middle umber oce the data are arraged i order. The data arraged i order are: The media i f. The media level of dioxide whe crude oil i preet i.45. The media level of dioxide whe crude oil i ot preet i.85. It i apparet that the level of dioxide i much higher whe crude oil i ot preet. Copyright 14 Pearo Educatio, Ic.

30 Method for Decribig Set of Data a. Skewed to the right. There will be a few people with very high alarie uch a the preidet ad football coach. b. Skewed to the left. O a eay tet, mot tudet will have high core with oly a few low core. c. Skewed to the right. O a difficult tet, mot tudet will have low core with oly a few high core. d. Skewed to the right. Mot tudet will have a moderate amout of time tudyig while a few tudet might tudy a log time. e. Skewed to the left. Mot car will be relatively ew with a few much older. f. Skewed to the left. Mot tudet will take the etire time to take the exam while a few might leave early..5 a. The ample mea i: x x The media i foud a the th ad 1 t obervatio, oce the data have bee ordered. The th ad 1 t obervatio are 1.75 ad The media i: The mode i the umber that occur the mot ad i 1.4, which occur 3 time. b. The ample average drivig performace idex i The media drivig performace idex i Half of all drivig performace idexe are le tha ad half are higher. The mot commo drivig performace idex value i 1.4. c. Sice the mea i larger tha the media, the data are kewed to the right. Uig MINITAB, a hitogram of the drivig performace idex value i: 1 Hitogram of INDEX 8 Frequecy INDEX The mea i hour. Thi mea that the average umber of emeter hour per cadidate for the CPA exam i hour. The media i 14 hour. Thi mea that 5% of the cadidate had more tha 14 emeter hour of credit ad 5% had le tha 14 emeter hour of credit. Sice the mea ad media are o cloe i value, the data are probably ot kewed, but cloe to ymmetric. Copyright 14 Pearo Educatio, Ic.

31 4 Chapter.5 a. Uig MINITAB, the output i: Decriptive Statitic: YRSPRAC N for Variable N N* Mea Miimum Media Maximum Mode Mode YRSPRAC ,, 5 9 The mea i The average legth of time i practice for thi ample i year. The media i 14. Half of the phyicia have bee i practice le tha 14 year ad half have bee i practice loger tha 14 year. There are 3 mode: 14,, ad 5. The mot frequet year i practice are 14,, ad 5 year. b. Uig MINITAB, the reult are: Decriptive Statitic: YRSPRAC N for Variable FUTUREUSE N N* Mea Miimum Media Maximum Mode Mode YRSPRAC NO YES , 8 The mea for the phyicia who would refue to ue ethic coultatio i the future i The average time i practice for thee phyicia i year. The media i 18. Half of the phyicia who would refue ethic coultatio i the future have bee i practice le tha 18 year ad half have bee i practice more tha 18 year. The mode i 5. The mot frequet year i practice for thee phyicia i 5 year. c. From the reult i part b, the mea for the phyicia who would ue ethic coultatio i the future i The average time i practice for thee phyicia i year. The media i 14. Half of the phyicia who would ue ethic coultatio i the future have bee i practice le tha 14 year ad half have bee i practice more tha 14 year. There are mode: 14 ad. The mot frequet year i practice for thee phyicia are 14 ad year. d. The reult i part b ad c cofirm the reearcher theory. The mea, media ad mode of year i practice are larger for the phyicia who would refue to ue ethic coultatio i the future tha thoe who would ue ethic coultatio i the future..53 For the "Joit exchage offer with prepack" firm, the mea time i.6545 moth, ad the media i 1.5 moth. Thu, the average time pet i bakruptcy for "Joit" firm i.6545 moth, while half of the firm ped 1.5 moth or le i bakruptcy. For the "No prefilig vote held" firm, the mea time i moth, ad the media i 3. moth. Thu, the average time pet i bakruptcy for "No prefilig vote held" firm i moth, while half of the firm ped 3. moth or le i bakruptcy. For the "Prepack olicitatio oly" firm, the mea time i moth, ad the media i 1.4 moth. Thu, the average time pet i bakruptcy for "Prepack olicitatio oly" firm i moth, while half of the firm ped 1.4 moth or le i bakruptcy. Sice the mea ad media for the three group of firm differ quite a bit, it would be ureaoable to ue a igle umber to locate the ceter of the time i bakruptcy. Three differet "ceter" hould be ued..54 a. The ample mea i: xi i x 3.9 Copyright 14 Pearo Educatio, Ic.

32 Copyright 14 Pearo Educatio, Ic. Method for Decribig Set of Data 41 The ample media i foud by fidig the average of the 1 th ad 11 th obervatio oce the data are arraged i order. The data arraged i order are: The 1 th ad 11 th obervatio are 3 ad 4. The average of thee two umber (media) i: media 3.5 The mode i the obervatio appearig the mot. For thi data et, the mode i 1, which appear 5 time. b. Elimiatig the larget umber which i 11 reult i the followig: The ample mea i: xi i x The ample media i foud by fidig the middle obervatio oce the data are arraged i order. The data arraged i order are: The 1 th obervatio i 3. The media i 3 The mode i the obervatio appearig the mot. For thi data et, the mode i 1, which appear 5 time. By droppig the larget umber, the mea i reduced from 4.5 to The media i reduced from 3.5 to 3. There i o effect o the mode. c. The data arraged i order are: If we drop the lowet ad larget obervatio we are left with: The ample 1% trimmed mea i: xi i x The advatage of the trimmed mea over the regular mea i that very large ad very mall umber that could greatly affect the mea have bee elimiated..55 a. Due to the "elite" upertar, the alary ditributio i kewed to the right. Sice thi implie that the media i le tha the mea, the player' aociatio would wat to ue the media. b. The ower, by the logic of part a, would wat to ue the mea..56 a. The primary diadvatage of uig the rage to compare variability of data et i that the two data et ca have the ame rage ad be vatly differet with repect to data variatio. Alo, the rage i greatly affected by extreme meaure.

33 4 Chapter b. The ample variace i the um of the quared deviatio of the obervatio from the ample mea divided by the ample ize miu 1. The populatio variace i the um of the quared deviatio of the value from the populatio mea divided by the populatio ize. c. The variace of a data et ca ever be egative. The variace of a ample i the um of the quared deviatio from the mea divided by 1. The quare of ay umber, poitive or egative, i alway poitive. Thu, the variace will be poitive. The variace i uually greater tha the tadard deviatio. However, it i poible for the variace to be maller tha the tadard deviatio. If the data are betwee ad 1, the variace will be maller tha the tadard deviatio. For example, uppoe the data et i.8,.7,.9,.5, ad.3. The ample mea i: x x The ample variace i: x 3. x The tadard deviatio i a. Rage = 4 = 4 x 8 x b. Rage = 6 = 6 x 17 x c. Rage = 8 () = 1 x 3 x d. Rage = 1 (3) = 4 x ( 6.8) x a. b. x x x 1 x Copyright 14 Pearo Educatio, Ic.

34 Method for Decribig Set of Data 43 c. x 17 x a. x x x 8 x x 8 x b. x x 55 x feet 4 x x x quare feet feet c. x 1 ( 4) ( 3) 1 ( 4) ( 4) 15 x ( 1) ( 4) ( 3) 1 ( 4) ( 4) 59 x 15 x.5 6 x ( 15) x d x x 1 x.33 ouce x x 4 x quare ouce ouce Copyright 14 Pearo Educatio, Ic.

35 44 Chapter.6 a. Rage = 4 37 = 5 x 199 x b. Rage = 1 1 = 99 x 33 x 5, , c. Rage = 1 = 98 x 95 x,33 8 1, , , Thi i oe poibility for the two data et. Data Set 1:, 1,, 3, 4, 5, 6, 7, 8, 9 Data Set :,, 1, 1,,, 3, 3, 9, 9 The two et of data above have the ame rage = larget meauremet mallet meauremet = 9 = 9. The mea for the two data et are: x x x x The dot diagram for the two data et are how below. Dotplot of x1, x x1 4 x x 6 8 x Copyright 14 Pearo Educatio, Ic.

36 Method for Decribig Set of Data 45.6 Thi i oe poibility for the two data et. Data Set 1: 1, 1,,, 3, 3, 4, 4, 5, 5 Data Set : 1, 1, 1, 1, 1, 5, 5, 5, 5, 5 x x x x Therefore, the two data et have the ame mea. The variace for the two data et are: 1 x 3 x x 3 x The dot diagram for the two data et are how below. Dotplot of x1, x x1 x 1 3 x 4 5 x a. Rage = 3 = 3 x 7 x b. After addig 3 to each of the data poit, Rage = 6 3 = 3 Copyright 14 Pearo Educatio, Ic.

37 46 Chapter x x c. After ubtractig 4 from each of the data poit, Rage = 1 (4) = 3 x ( 13) x d. The rage, variace, ad tadard deviatio remai the ame whe ay umber i added to or ubtracted from each meauremet i the data et..64 a. The rage i the differece betwee the maximum ad miimum value. The rage The uit of meauremet are percet. b. The variace i x 14.7 x The uit are quare percet. c. The tadard deviatio i The uit are percet..65 a. The rage i the differece betwee the larget obervatio ad the mallet obervatio. From the pritout, the larget obervatio i $4,85 thouad ad the mallet obervatio i $7 thouad. The rage i: R $4,85 $7 $4,88 thouad b. From the pritout, the tadard deviatio i = $1,113 thouad. c. The variace i the tadard deviatio quared. The variace i: 1,113 1,38,769 millio dollar quared.66 a. The ample variace of the hoey doage group i: x 375 x The tadard deviatio i: Copyright 14 Pearo Educatio, Ic.

38 Method for Decribig Set of Data 47 b. The ample variace of the DM doage group i: x 75 The tadard deviatio i: c. The ample variace of the cotrol group i: x 41 x x The tadard deviatio i: d. The group with the mot variability i the group with the larget tadard deviatio, which i the DM group. The group with the leat variability i the group with the mallet tadard deviatio, which i the hoey group..67 a. The rage i 155. The tatemet i accurate. b. The variace i The tatemet i ot accurate. A more accurate tatemet would be: The variace of the level of upport for corporate utaiability for the 99 eior maager i c. The tadard deviatio i If the uit of meaure for the two ditributio are the ame, the the ditributio of upport level for the 99 eior maager ha le variatio tha a ditributio with a tadard deviatio of 5. If the uit of meaure for the ecod ditributio i ot kow, the we caot compare the variatio i the two ditributio by lookig at the tadard deviatio aloe. d. The tadard deviatio bet decribe the variatio i the ditributio. The rage ca be greatly affected by extreme meaure. The variace i meaured i quare uit, which i hard to iterpret. Thu, the tadard deviatio i the bet meaure to decribe the variatio..68 a. Uig MINITAB, the reult are: Decriptive Statitic: YRSPRAC Variable N N* Mea StDev Variace Rage YRSPRAC The rage i 39. The differece betwee the larget year i practice ad the mallet year i practice i 39 year. The variace i quare year. The tadard deviatio i year. b. Uig MINITAB, the reult are: Decriptive Statitic: YRSPRAC Variable FUTUREUSE N N* Mea StDev Variace Rage YRSPRAC NO YES For the phyicia who would refue to ue ethic coultatio i the future, the tadard deviatio i 1.5 year. Copyright 14 Pearo Educatio, Ic.

39 48 Chapter c. For the phyicia who would ue ethic coultatio i the future, the tadard deviatio i 8.95 year. d. The variatio i the legth of time i practice for the phyicia who would refue to ue ethic coultatio i the future i greater tha that for the phyicia who would ue ethic coultatio i the future..69 a. The rage i the larget obervatio miu the mallet obervatio or 11 1 = 1. The variace i: xi i x 78 i 45 i The tadard deviatio i: b. The larget obervatio i 11. It i deleted from the data et. The ew rage i: 9 1 = 8. The variace i: xi i x 67 i 39 i The tadard deviatio i: Whe the larget obervatio i deleted, the rage, variace ad tadard deviatio decreae. c. The larget obervatio i 11 ad the mallet i 1. Whe thee two obervatio are deleted from the data et, the ew rage i: 9 1 = 8. The variace i: xi i x 66 i 38 i The tadard deviatio i: Whe the larget ad mallet obervatio are deleted, the rage, variace ad tadard deviatio decreae..7 a. A worker' overall time to complete the operatio uder tudy i determied by addig the ubtaktime average. Worker A x 11 The average for ubtak 1 i: x x 1 The average for ubtak i: x 3 7 Worker A' overall time i = Copyright 14 Pearo Educatio, Ic.

40 Method for Decribig Set of Data 49 Worker B The average for ubtak 1 i: x 13 x x 9 The average for ubtak i: x Worker B' overall time i = b. Worker A x 11 x Worker B x 13 x c. The tadard deviatio repreet the amout of variability i the time it take the worker to complete ubtak 1. d. Worker A x 1 x Worker B x 9 x e. I would chooe worker imilar to worker B to perform ubtak 1. Worker B ha a lightly higher average time o ubtak 1 (A: x 3.14, B: x 3.43 ). However, Worker B ha a maller variability i the time it take to complete ubtak 1 (part b). He or he i more coitet i the time eeded to complete the tak. I would chooe worker imilar to Worker A to perform ubtak. Worker A ha a maller average time o ubtak (A: x 3, B: x 4.14 ). Worker A alo ha a maller variability i the time eeded to complete ubtak (part d)..71 a. The uit of meauremet of the variable of iteret i dollar (the ame a the mea ad tadard deviatio). Baed o thi, the data are quatitative. b. Sice o iformatio i give about the hape of the data et, we ca oly ue Chebyhev' Rule. $9 i tadard deviatio below the mea, ad $1 i tadard deviatio above the mea. Uig Chebyhev' Rule, at leat 3/4 of the meauremet (or 3/4 = 15 meauremet) will fall betwee $9 ad $1. Copyright 14 Pearo Educatio, Ic.

41 5 Chapter $6 i 3 tadard deviatio below the mea ad $4 i 3 tadard deviatio above the mea. Uig Chebyhev' Rule, at leat 8/9 of the meauremet (or 8/9 178 meauremet) will fall betwee $6 ad $4. $1 i 1 tadard deviatio below the mea ad $18 i 1 tadard deviatio above the mea. Uig Chebyhev' Rule, othig ca be aid about the umber of meauremet that will fall betwee $1 ad $18. $15 i equal to the mea ad $1 i tadard deviatio above the mea. Uig Chebyhev' Rule, at leat 3/4 of the meauremet (or 3/4 = 15 meauremet) will fall betwee $9 ad $1. It i poible that all of the 15 meauremet will be betwee $9 ad $15. Thu, othig ca be aid about the umber of meauremet betwee $15 ad $1..7 Sice o iformatio i give about the data et, we ca oly ue Chebyhev' Rule. a. Nothig ca be aid about the percetage of meauremet which will fall betwee x ad x. b. At leat 3/4 or 75% of the meauremet will fall betwee x ad x. c. At leat 8/9 or 89% of the meauremet will fall betwee x 3 ad x Accordig to the Empirical Rule: a. Approximately 68% of the meauremet will be cotaied i the iterval x to x. b. Approximately 95% of the meauremet will be cotaied i the iterval x to x. c. Eetially all the meauremet will be cotaied i the iterval x 3 to x a. x 6 x x 6 x b. Iterval x Number of Meauremet i Iterval Percetage, or (6.41, 1.7) / 5.7 or 7% x, or (4.58, 11.9) 4 4 / 5.96 or 96% x 3, or (.75, 13.73) 5 5 / 5 1. or 1% c. The percetage i part b are i agreemet with Chebyhev' Rule ad agree fairly well with the percetage give by the Empirical Rule. Copyright 14 Pearo Educatio, Ic.

42 Method for Decribig Set of Data 51 d. Rage ad Rage The rage approximatio provide a atifactory etimate of 1.83 from part a..75 Uig Chebyhev' Rule, at leat 8/9 of the meauremet will fall withi 3 tadard deviatio of the mea. Thu, the rage of the data would be aroud 6 tadard deviatio. Uig the Empirical Rule, approximately 95% of the obervatio are withi tadard deviatio of the mea. Thu, the rage of the data would be aroud 4 tadard deviatio. We would expect the tadard deviatio to be omewhere betwee Rage/6 ad Rage/4. For our data, the rage The Rage ad Rage Therefore, I would etimate that the tadard deviatio of the data et i betwee ad It would ot be feaible to have a tadard deviatio of 5. If the tadard deviatio were 5, the data would pa 65/5 = 5 tadard deviatio. Thi would be extremely ulikely..76 a. Uig MINITAB, the hitogram of the data i: 1 Hitogram of Wheel 1 8 Frequecy Wheel Sice the ditributio i kewed to the right, it i ot moud-haped ad it i ot ymmetric. b. Uig MINITAB, the reult are: Decriptive Statitic: Wheel Variable N Mea StDev Miimum Q1 Media Q3 Maximum Wheel The mea i 3.14 ad the tadard deviatio i c. The iterval i: x 3.14 (1.371) (.47, 5.956). d. Accordig to Chebyhev rule, at leat 75% of the obervatio will fall withi tadard deviatio of the mea. Copyright 14 Pearo Educatio, Ic.

43 5 Chapter e. Accordig to the Empirical Rule, approximately 95% of the obervatio will fall withi tadard deviatio of the mea. f. Actually, 6 of the 8 or 6/8 =.99 of the obervatio fall withi the iterval. Thi value i cloe to the 95% that we would expect with the Empirical Rule..77 a. The iterval x will cotai at leat 75% of the obervatio. Thi iterval i x 3.11 (.66) (1.79, 4.43). b. No. The value 1.5 doe ot fall i the iterval x. We kow that at leat 75% of all obervatio will fall withi tadard deviatio of the mea. Sice 1.5 fall more tha tadard deviatio from the mea, it would ot be a likely value to oberve..78 a. Uig Chebyhev Rule, at leat 75% of the obervatio will fall withi tadard deviatio of the mea. x 4.5 (1.) ( 19.79, 8.9) or (, 8.9) ice we caot have a egative umber blog. b. We would expect the ditributio to be kewed to the right. We kow that we caot have a egative umber of blog/forum. Eve 1 tadard deviatio below the mea i a egative umber. We would aume that there are a few very large obervatio becaue the tadard deviatio i o big compared to the mea..79 a. The tadard deviatio iterval aroud the mea i: x (17.77) (15.77, ) b. Uig Chebyhev Theorem, at leat ¾ of the obervatio will fall withi tadard deviatio of the mea. Thu, at leat ¾ of firt-time cadidate for the CPA exam have total credit hour betwee ad c. I order for the above tatemet to be true, othig eed to be kow about the hape of the ditributio of total emeter hour..8 a. Sice the data are moud-haped ad ymmetric, we kow from the Empirical Rule that approximately 95% of the obervatio will fall withi tadard deviatio of the mea. Thi iterval will be: x 39 (6) 39 1 (7, 51). b. We kow that approximately.5 of the obervatio will fall outide the rage 7 to 51. Sice the ditributio of core i ymmetric, we kow that half of the.5 or.5 will fall above 51. c. We kow from the Empirical Rule that approximately 99.7% (eetially all) of the obervatio will fall withi 3 tadard deviatio of the mea. Thi iterval i: x (6) 3918 (1, 57). xi i1 17,8.81 a. The ample mea i: x Copyright 14 Pearo Educatio, Ic.

44 Method for Decribig Set of Data 53 The ample variace i: i1 xi i 1 x 17,8 1,77, The tadard deviatio i: b. x (9.736, 1.66) x (4.963) (85.773, 15.65) x (4.963) (8.81, ) c. There are 166 out of 186 obervatio i the firt iterval. Thi i (166 /186) 1% 89.%. There are 179 out of 186 obervatio i the ecod iterval. Thi i (179 /186) 1% 96.%. There are 18 out of 186 obervatio i the ecod iterval. Thi i (18 /186) 1% 97.8%. The percetage for the firt iterval are much larger tha we would expect uig the Empirical Rule. The Empirical Rule idicate that approximately 68% of the obervatio will fall withi 1 tadard deviatio of the mea. It alo idicate that approximately 95% of the obervatio will fall withi tadard deviatio of the mea. Chebyhev Theorem ay that at leat ¾ or 75% of the obervatio will fall withi tadard deviatio of the mea ad at leat 8/9 or 88.9% of the obervatio will fall withi 3 tadard deviatio of the mea. It appear that our oberved percetage agree with Chebyhev Theorem better tha the Empirical Rule..8 a. The iterval i: x 13. (19.5) ( 5.8, 5.) or (, 5.) ice we caot have egative umber of miute. b. Sice thi iterval cotai egative umber, we kow that the ditributio caot be ymmetric. Oe caot have egative value for time pet o a laptop computer. c. Sice we kow the data are ot ymmetric, we mut ue Chebyhev Rule. At leat ¾ or 75% of the obervatio will fall betwee -5.8 ad 5. or betwee ad 5. miute..83 The ample mea i: xi i x The ample variace deviatio i: xi i 1 x i 551, 91.1 i The ample tadard deviatio i: The data are fairly ymmetric, o we ca ue the Empirical Rule. We kow from the Empirical Rule that almot all of the obervatio will fall withi 3 tadard deviatio of the mea. Thi iterval would be: x (9.91) (5.1, 64.47) Copyright 14 Pearo Educatio, Ic.

45 54 Chapter.84 a. Uig MINITAB, the frequecy hitogram for the time i bakruptcy i: Hitogram of TIME 15 Frequecy Time i Bakruptcy 8 1 The Empirical Rule i ot applicable becaue the data are ot moud haped. b. Uig MINITAB, the decriptive meaure are: Decriptive Statitic: TIME Variable N Mea StDev Miimum Q1 Media Q3 Maximum TIME From Chebyhev Theorem, we kow that at leat 75% of the obervatio will fall withi tadard deviatio of the mea. Thi iterval i: x.549 (1.88) ( 1.17, 6.5) or (, 6.5) ice we caot have egative moth. c. There are 47 of the 49 obervatio withi thi iterval. The percetage would be (47 / 49) 1% 95.9%. Thi agree with Chebyhev Theorem (at leat 75%). It alo agree with the Empirical Rule (approximately 95%). d. From the above iterval we kow that about 95% of all firm filig for prepackaged bakruptcy will be i bakruptcy betwee ad 6. moth. Thu, we would etimate that a firm coiderig filig for bakruptcy will be i bakruptcy up to 6. moth..85 a. The iterval x for the flexed arm group i x 59 3(4) 591 (47, 71). The iterval for the exteded are group i x 43 3() 43 6 (37, 49). We kow that at leat 8/9 or 88.9% of the obervatio will fall withi 3 tadard deviatio of the mea uig Chebyhev Rule. Sice thee iterval barely overlap, the iformatio upport the reearcher theory. The hopper from the flexed arm group are more likely to elect vice optio tha the exteded arm group. b. The iterval x for the flexed arm group i x 59 (1) 59 (39, 79). The iterval for the exteded are group i x 43 (15) 43 3 (13, 73). Sice thee two iterval overlap almot completely, the iformatio doe ot upport the reearcher theory. There doe ot appear to be ay differece betwee the two group. Copyright 14 Pearo Educatio, Ic.