Applied Statistics and Probability for Engineers, 6 th edition December 31, 2013 CHAPTER 6. Section 6-1

Transcription

1 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 CHAPTER 6 Secto No, usually ot. For eample, f the sample s {, 3} the mea s.5 whch s ot a observato the sample No, usually ot. For eample, the mea of {1, 4, 4} s 3 whch s ot eve a observato the sample Yes. For eample, {5, 5, 5, 5, 5, 5, 5}, the sample mea = 5, sample stadard devato = Sample average: 1 Sample varace: mm s (mm) 7 Sample stadard devato: s mm The sample stadard devato could also be foud usg s Dot Dagram: 1 1 where : dameter There appears to be a possble outler the data set. 6-1

2 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample average: Sample varace: yards s yards 11 Sample stadard devato: s yards The sample stadard devato could also be foud usg s 1 1 where Dot Dagram: (roudg was used to create the dot dagram).. :.... : : C

3 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample average: Sample varace: s Sample stadard devato: s The sample stadard devato could also be foud usg s 1 1 where Dot Dagram: The value 5.44 s the populato mea because the actual physcal populato of all flght tmes durg the operato s avalable Sample average of eercse group: 6-3

4 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample varace of eercse group: 1 1 s ( ) Sample stadard devato of eercse group: s Dot Dagram of eercse group: Dotplot of 6 hours of Eercse hours of Eercse Sample average of o eercse group: Sample varace of o eercse group: 1 1 s (600.08)

5 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample stadard devato of o eercse group: s Dot Dagram of o eercse group: Dotplot of No Eercse No Eercse s s Eamples: repeatablty of the test equpmet, tme lag betwee samples, durg whch the ph of the soluto could chage, ad operator skll drawg the sample or usg the strumet a) F F s 1.16 Dot Dagram : : :.:..:..:.. :: temp b) Removg the smallest observato (31), the sample mea ad stadard devato become F s =.74 F 6-1. Sample mea: Sample varace:

6 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, s Sample stadard devato: s Dot Dagram: 6-3. Dot Dagram for Useeded clouds: Dot Dagram for Seeded clouds: The sample mea for useeded clouds s ad the data s ot cetered about the mea. Two large observatos crease the mea. The sample mea for seeded clouds s ad the data s ot cetered about the mea. The average rafall whe clouds are seeded s hgher whe compared to the rafall whe clouds are ot seeded. The amout of rafall of seeded clouds vares wdely whe compared to amout of rafall for useeded clouds Sample mea: 6-6

7 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample varace: 71 1 s Sample stadard devato: s Dot Dagram: There appears to be a outler the data. Secto a) N = 30 Leaf Ut = (7) Varable N N* Mea SE Mea StDev Mmum Q1 Meda Q3 CodNum Mamum

8 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 b) Oe partcular brdge has poor ratg of 3.4 c) Calculatg the mea after removg the brdge metoed above: Varable N Mea SE Mea StDev Mmum Q1 Meda Q3 CodNum Mamum There s a lttle dfferece betwee the two meas A back-to-back steam-ad-leaf dsplay s useful whe two data sets are to be compared. Therefore, for ths problem the comparso of the seeded versus useeded clouds ca be performed more easly wth the backto-back stem ad leaf dagram tha a dot dagram The meda wll equal the mode whe the sample s symmetrc wth a sgle mode. The symmetry mples the mode s at the meda of the sample Stem-ad-leaf dsplay for cycles to falure: ut = 0 1 represets 0 1 0T 3 1 0F 5 0S o * T 3333 (15) 1F S o * 011 T Meda = , Q 1 = 97.8, ad Q 3 = No, oly 5 out of 70 coupos survved beyod 000 cycles Stem-ad-leaf dsplay for yeld: ut = 1 1 represets 1 1 7o 8 1 8* 7 8T F S (11) 8o * T F S o 8 Meda = 89.50, Q 1 = 86.0, ad Q 3 = Sample meda s at = 35.5 th observato, the meda s Modes are 1, 1315, ad 1750 whch are the most frequet data. 6-8

9 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Sample mea: Do ot use the total as a observato. There are 3 observatos. Stem-ad-leaf of Bllo of klowatt hours N = 3 Leaf Ut = 0 (18) Sample meda s at 1 th = Sample mea: Sample varace: s = Sample stadard devato: s = Stem-ad-leaf dsplay. Stregth: ut = represets (13) th percetle s

10 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Stem-ad-leaf dsplay. Yard: ut = 1.0 Note: Mtab has dropped the value to the rght of the decmal to make ths dsplay (15) Sample Mea Sample Stadard Devato s s yards ad Sample Meda ad yards ( Varable N Meda yards yards th percetle s

11 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Stem-ad-leaf dsplay. Ratg: ut = 0. 1 represets (7) Sample Mea Sample Stadard Devato s s ad Sample Meda ad Varable N Meda ratg / = 55% of the taste testers cosdered ths partcular Pot Nor truly eceptoal Stem-ad-leaf dsplay. Heght: ut = 0. 1 represets 1. Female Studets Male Studets

12 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, (4) (15) The male egeerg studets are taller tha the female egeerg studets. Also there s a slghtly wder rage the heghts of the male studets. Secto

13 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Frequecy Tabulato for Eercse 6-3.Cycles Lower Upper Relatve Cumulatve Cum. Rel. Class Lmt Lmt Mdpot Frequecy Frequecy Frequecy Frequecy at or below above Mea = Stadard Devato =.385 Meda = Frequecy umber of cycles to falure 6-13

14 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Frequecy Tabulato for Eercse 6-5.Yeld Lower Upper Relatve Cumulatve Cum. Rel. Class Lmt Lmt Mdpot Frequecy Frequecy Frequecy Frequecy at or below above Mea = Stadard Devato = Meda = Hstogram 8 bs: 18 Hstogram of Cycles to falure (8 bs) Frequecy Cycles to falure of alumum test coupos 50 Hstogram 16 bs: 6-14

15 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Hstogram of Cycles to falure (16 bs) 1 Frequecy Cycles to falure of alumum test coupos 300 Yes, both of them gve the same smlar formato Hstogram 0 Hstogram of Eergy 15 Frequecy Eergy cosumpto 00 The data are skewed The hstogram for the spot weld shear stregth data shows that the data appear to be ormally dstrbuted (the same shape that appears the stem-leaf-dagram). 6-15

16 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Frequecy shear stregth Yes, the hstogram of the dstace data shows the same shape as the stem-ad-leaf dsplay eercse Frequecy dstace Yes, the hstogram of the we ratg data shows the same shape as the stem-ad-leaf dsplay Frequecy ratg 6-16

17 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Frequecy Tabulato for Brdge Codto B Frequecy <=3 0 (3,4] 3 (4,5] 9 (4,6] (6,7] 8 (7,8] 0 >8 0 Mea Meda 5. Stadard Devato Frequecy Tabulato for Cloud Seedg B Frequecy <00 3 (00, 0] 11 (0, 600] (600, 800] 1 (800, 00] (00, 0] 0 >

18 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Mea Meda Stadard Devato Secto a) Descrptve Statstcs: Brdge Codto Varable N N* Mea SE Mea StDev Mmum Q1 Meda Q3 BrgCd Mamum Meda = 5.0 Lower Quartle = Q1= Upper Quartle = Q3= 6.77 b) c) No obvous outlers. 6-18

19 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, a) Descrptve Statstcs for useeded cloud data: Varable Mea SE Mea StDev Mmum Q1 Meda Q3 Mamum Useeded Meda = 44. Lower Quartle = Q1 = 3.7 Upper Quartle = Q3 =.6 b) Descrptve Statstcs for seeded data: Varable Mea SE Mea StDev Mmum Q1 Meda Q3 Mamum Seeded Meda = Lower Quartle = Q1 = 79 Upper Quartle = Q3 = 445 c) d) The two data sets are plotted here. A greater mea ad greater dsperso the seeded data are see. Both plots show outlers Descrptve Statstcs Varable N Mea Meda TrMea StDev SE Mea tme Varable Mmum Mamum Q1 Q3 tme a)sample Mea:.415, Sample Stadard Devato: b) Bo Plot: There are o outlers the data. 6-19

20 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Boplot of Tme Tme Descrptve Statstcs Varable N Mea Meda Tr Mea StDev SE Mea Temperat Varable M Ma Q1 Q3 Temperat a) Sample Mea = 95.44, Sample Varace = 9.53, Sample Stadard Devato = 3.09 b) Meda = 953; A crease the largest temperature measuremet does ot affect the meda. c) Boplot of Temperature Temperature Descrptve Statstcs of O-rg jot temperature data Varable N Mea Meda TrMea StDev SE Mea Temp Varable Mmum Mamum Q1 Q3 Temp a) Meda = 67.50, Lower Quartle: Q 1 = 58.50, Upper Quartle: Q 3 = b) Data wth lowest pot removed Varable N Mea Meda TrMea StDev SE Mea Temp

21 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Varable Mmum Mamum Q1 Q3 Temp The mea ad meda have creased ad the stadard devato ad dfferece betwee the upper ad lower quartle have decreased. c)bo Plot: The bo plot dcates that there s a outler the data. 90 Boplot of Jot Temp Jot Temp The bo plot shows the same basc formato as the stem ad leaf plot but a dfferet format. The outlers that were separated from the ma porto of the stem ad leaf plot are show here separated from the whskers Boplot of Bllo of klowatt hours 1600 Bllo of klowatt hours Ths plot, as the stem ad leaf oe, dcates that the data fall mostly oe rego ad that the measuremets toward the eds of the rage are more rare. 6-1

22 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Boplot of Weld Stregth 5450 Weld Stregth We ca see that the two dstrbutos seem to be cetered at dfferet values. 76 Boplot of Female, Male Data Female Male All dstrbutos are cetered at about the same value, but have dfferet varaces. 6-

23 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Boplot of Hgh Dose, Cotrol, Cotrol_1, Cotrol_ Data Hgh Dose Cotrol Cotrol_1 Cotrol_ 6-3

24 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Secto Stem-ad-leaf dsplay for Force: ut = 1 1 represets (5) Tme Seres Plot of Pull-off Force Pull-off Force Ide I the tme seres plot there appears to be a dowward tred begg after tme 30. The stem-ad-leaf plot does ot reveal ths Stem-ad-leaf of Wolfer suspot N = 0 Leaf Ut = The data appears to decrease betwee 1790 ad 1835, the stem ad leaf plot dcates skewed data. 6-4

25 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Tme Seres Plot of Wolfer suspot 1 Wolfer suspot Ide Stem-ad-leaf of Number of Earthquakes N = 1 Leaf Ut = (13) Tme Seres Plot 45 Tme Seres Plot of Number of Earthquakes Number of Earthquakes Year

26 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, a) 0.75 Tme Seres Plot of Aomaly 0.50 Aomaly Year There s a creasg tred the most recet data. b) Tme Seres Plot of CO CO Year c) The plots crease appromately together. However, ths relatoshp aloe does ot prove a cause ad effect. 6-6

27 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Secto a) Scatterplot of Prce vs Taes 45 Prce Taes As the taes crease, the prce teds to crease. b) From computer software the correlato coeffcet s b) Values for y ted to crease as values for 1 crease. However, values for y ted to decrease as values for or 3 decrease The patter of the data dcates that the sample may ot come from a ormally dstrbuted populato or that the largest observato s a outler. Note the slght bedg dowward of the sample data at both eds of the graph. Normal Probablty Plot for 6-1 Psto Rg Dameter ML Estmates - 95% CI ML Estmates Mea StDev Percet Data 6-7

28 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, A ormal dstrbuto s reasoable for these data. Normal Probablty Plot for 6-5 Vsual Accomodato Data ML Estmates - 95% CI ML Estmates Mea StDev Percet Data The data appear to be appromately ormally dstrbuted. However, there are some departures from the le at the eds of the dstrbuto. Normal Probablty Plot for temperature Data from eercse ML Estmates Mea StDev Percet Data

29 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Normal Probablty Plot for cycles to falure Data from eercse 6-15 Percet ML Estmates Mea StDev Data The data appear to be appromately ormally dstrbuted. However, there are some departures from the le at the eds of the dstrbuto. Normal Probablty Plot for cocetrato Data from eercse ML Estmates Mea StDev Percet Data

30 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Normal Probablty Plot for ML Estmates - 95% CI Female Studets Male Studets Percet Data Both populatos seem to be ormally dstrbuted. Moreover, the les seem to be roughly parallel dcatg that the populatos may have the same varace ad dffer oly the value of ther mea. Supplemetal Eercses 6-3. Based o the dgdot plot ad tme seres plots of these data, each year the temperature has a smlar dstrbuto. I each year, the temperature creases utl the md year ad the t starts to decrease. Dotplot of 000, 001, 00, 003, 004, 005, 006, 007, 008, Global Temperature

31 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Global Temperature Tme Seres Plot of 000, 001, 00, 003, 004, 005, 006, Varable Moth Stem-ad-leaf of Global Temperature N = 117 Leaf Ut = (11) Tme-seres plot by moth over years 17 Tme Seres Plot of Global Temperature 16 Global Temperature Moth

32 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, The uemploymet rate s steady from 00-00, the t creases utl 004, decreases steadly from 004 to 008, ad the creases aga dramatcally 009, where t peaks. Tme Seres Plot of Uemploymet 9 8 Uemploymet Moth a) Sample 1 Rage = 4, Sample Rage = 4 Yes, the two appear to ehbt the same varablty b) Sample 1 s = 1.604, Sample s = 1.85 No, sample has a larger stadard devato. c) The sample rage s a relatvely crude measure of the sample varablty as compared to the sample stadard devato because the stadard devato uses the formato from every data pot the sample whereas the rage uses the formato cotaed oly two data pots - the mmum ad mamum From the stem-ad-leaf dagram, the dstrbuto looks lke the uform dstrbuto. From the tme seres plot, there s a creasg tred eergy cosumpto. Stem-ad-leaf of Eergy N = 8 Leaf Ut = (3)

33 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Tme Seres Plot of Egergy 3500 Egergy Year sales 5 0 Ide There appears to be a cyclc varato the data wth the hgh value of the cycle geerally creasg. The hgh values are durg the wter holday moths. b) We mght draw aother cycle, wth the peak smlar to the last year s data (1969) at about 1.7 thousad bottles a) Stem-ad-leaf dsplay for Problem -35: ut = 1 1 represets 1 1 0T 3 8 0F S (7) 0o * T F S

34 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, o 899 b) Sample Average = 9.35, Sample Stadard Devato = c) 0 15 sprgs 5 Ide 0 30 The tme seres plot dcates there was a crease the average umber of ocoformg sprgs durg the days. I partcular, the crease occurred durg the last days The golf course yardage data appear to be skewed. Also, there s a outlyg data pot above 7500 yards yardage

35 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Normal Probablty Plot for Temperature ML Estmates Mea StDev Percet Data A ormal dstrbuto s reasoable for these data. There are some repeated values the data that cause some pots to fall off the le Boplot for problem Dstace yards The plot dcates that most balls wll fall somewhere the rage. Ths same type of formato could have bee obtaed from the stem ad leaf graph

36 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Tral 1 Tral Tral 3 Tral 4 Tral 5 - There s a dfferece the varablty of the measuremets the trals. Tral 1 has the most varablty the measuremets. Tral 3 has a small amout of varablty the ma group of measuremets, but there are four outlers. Tral 5 appears to have the least varablty wthout ay outlers. - All of the trals ecept Tral 1 appear to be cetered aroud 850. Tral 1 has a hgher mea value - All fve trals appear to have measuremets that are greater tha the true value of The dfferece the measuremets Tral 1 may dcate a start-up effect the data. There could be some bas the measuremets that s ceterg the data above the true value a) Stem-ad-leaf of Drowg Rate N = 35 Leaf Ut = (1) Tme Seres Plots 6-36

37 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, Tme Seres Plot of Drowg Rate Drowg Rate Year b) Descrptve Statstcs: Drowg Rate Varable N N* Mea SE Mea StDev Mmum Q1 Meda Q3 Drowg Rate Varable Mamum Drowg Rate c) Greater awareess of the dagers ad drowg preveto programs mght have bee effectve. d) The summary statstcs assume a stable dstrbuto ad may ot adequately summarze the data because of the tred preset a) From computer software the sample mea = 34,3.05 ad the sample stadard devato = 0,414.5 b) 6-37

38 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 There s substatal varablty mleage. There are umber of vehcles wth mleage ear the mea, but aother group wth mleage ear or eve greater tha 70,000. c) Stem-ad-leaf of Mleage N = 0 Leaf Ut = (13) d) For the gve mleage of 4534, there are 9 observatos greater ad 71 observatos less tha ths value. Therefore, the percetle s appromately 71% a) Stem-ad-leaf of Force N = 68 Leaf Ut = (9) b) The sample mea s 1.65 ad the sample stadard devato s 6.4. c) 6-38

39 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Probablty Plot of Force Normal Percet Mea 1.6 StDev 6.4 N 68 AD.035 P-Value < Force 30 There a umber of repeated values for force that are see as pots stacked vertcally o the plot. Ths s probably due to roud off of the force to two dgts. There are fewer lower force values tha are epected from a ormal dstrbuto. d) From the stem-ad-leaf dsplay, 9 caps eceed the force lmt of 30. Ths s 9/68 = e) The mea plus two stadard devatos equals (6.4) = From the stem-ad-leaf dsplay, caps eceed ths force lmt of 30. Ths s /68 = f) I the followg bo plots Force1 deotes the subset of the frst 35 observatos ad Force deotes the remag observato. The mea ad varablty of force s greater for the secod set of data Force. Boplot of Force1, Force Data Force1 Force g) A separate ormal dstrbuto for each group of caps fts the data better. Ths s to be epected whe the mea ad stadard devatos of the groups dffer as they do here. 6-39

40 Appled Statstcs ad Probablty for Egeers, 6 th edto December 31, 013 Probablty Plot of Force1, Force Normal Percet Varable Force1 Force Mea StDev N AD P Data