Two variable statistics

Transcription

1 Chpter11 Two vrile sttistis Syllus referene: 4.2, 4.3, 4.4 Contents: A B C D E Correltion Mesuring orreltion Line of est fit y eye Liner regression The  2 test of independene

2 316 TWO VARIABLE STATISTICS (Chpter 11) OPENING PROBLEM At junior tournment, group of young thletes throw disus. The ge nd distne thrown re reorded for eh thlete. Athlete A B C D E F G H I J K L Age (yers) Distne thrown (m) Things to think out: Do you think the distne n thlete n throw is relted to the person s ge? Wht hppens to the distne thrown s the ge of the thlete inreses? How ould you grph the dt to more lerly see the reltionship etween the vriles? d How n we mesure the reltionship etween the vriles? Sttistiins re often interested in how two vriles re relted. For emple, in the Opening Prolem, we wnt to know how hnge in the ge of the thlete will ffet the distne the thlete n throw. We n oserve the reltionship etween the vriles y plotting the dt on stter digrm. We ple the independent vrile ge on the horizontl is, nd the dependent vrile distne on the vertil is. We then plot eh dt vlue s point on the stter digrm. For emple, the red point represents thlete H, who is 10 yers old nd threw the disus 15 metres. From the generl shpe formed y the dots, we n see tht s the ge inreses, so does the distne thrown distne (m) ge (yers) A CORRELATION Correltion refers to the reltionship or ssoition etween two vriles. There re severl hrteristis we onsider when desriing the orreltion etween two vriles: diretion, linerity, strength, outliers, nd ustion.

3 TWO VARIABLE STATISTICS (Chpter 11) 317 DIRECTION For generlly upwrd trend, we sy tht the orreltion is positive. An inrese in the independent vrile mens tht the dependent vrile generlly inreses. For generlly downwrd trend, we sy tht the orreltion is negtive. An inrese in the independent vrile mens tht the dependent vrile generlly dereses. For rndomly sttered points, with no upwrd or downwrd trend, we sy there is no orreltion. LINEARITY We determine whether the points follow liner trend, or in other words pproimtely form stright line. These points re roughly liner. These points do not follow liner trend. STRENGTH We wnt to know how losely the dt follows pttern or trend. The strength of orreltion is usully desried s either strong, moderte, or wek. strong moderte wek strong positive moderte positive wek positive strong negtive moderte negtive wek negtive

4 318 TWO VARIABLE STATISTICS (Chpter 11) OUTLIERS We oserve nd investigte ny outliers, or isolted points whih do not follow the trend formed y the min ody of dt. If n outlier is the result of reording or grphing error, it should e disrded. However, if the outlier proves to e genuine piee of dt, it should e kept. outlier not n outlier For the stter digrm for the dt in the Opening Prolem, we n sy tht there is strong positive orreltion etween ge nd distne thrown. The reltionship ppers to e liner, with no outliers. CAUSATION Correltion etween two vriles does not neessrily men tht one vrile uses the other. Consider the following: 1 The rm length nd running speed of smple of young hildren were mesured, nd strong, positive orreltion ws found to eist etween the vriles. Does this men tht short rms use redution in running speed or tht high running speed uses your rms to grow long? This would lerly e nonsense. Rther, the strong, positive orreltion etween the vriles is ttriuted to the ft tht oth rm length nd running speed re losely relted to third vrile, ge. Up to ertin ge, oth rm length nd running speed inrese with ge. 2 The numer of television sets sold in Bllrt nd the numer of stry dogs olleted in Bendigo were reorded over severl yers nd strong positive orreltion ws found etween the vriles. Oviously the numer of television sets sold in Bllrt ws not influening the numer of stry dogs olleted in Bendigo. Both vriles hve simply een inresing over the period of time tht their numers were reorded. If hnge in one vrile uses hnge in the other vrile then we sy tht usl reltionship eists etween them. For emple, in the Opening Prolem there is usl reltionship in whih inresing the ge of n thlete inreses the distne thrown. In ses where this is not pprent, there is no justifition, sed on high orreltion lone, to onlude tht hnges in one vrile use the hnges in the other.

5 EXERCISE 11A TWO VARIABLE STATISTICS (Chpter 11) For eh of the stter digrms elow, desrie the reltionship etween the vriles. Consider the diretion, strength, nd linerity of the reltionship, s well s the presene of outliers. y y y d e f y y y 2 The sores wrded y two judges t n ie skting ompetition re shown in the tle. Competitor P Q R S T U V W X Y Judge A 5 6: : Judge B 6 7 8: : :5 Construt stter digrm for this dt with Judge A s sores on the horizontl is nd Judge B s sores on the vertil is. Copy nd omplete the following omments out the stter digrm: There ppers to e...,...,... orreltion etween Judge A s sores nd Judge B s sores. This mens tht s Judge A s sores inrese, Judge B s sores... Would it e resonle to onlude tht n inrese in Judge A s sores uses n inrese in Judge B s sores? You n use tehnology to drw stter digrms. GRAPHICS CALCULATOR INSTRUCTIONS 3 The results of group of students for Mths test nd n Art essy re ompred: Student A B C D E F G H I J Mths test Art essy This dt is lled ivrite dt euse two vriles re reorded for eh individul. Construt stter digrm for the dt. Mke the sles on oth es from 60 to 90. Desrie the reltionship etween the Mthemtis nd Art mrks.

6 320 TWO VARIABLE STATISTICS (Chpter 11) 4 Choose the stter digrm whih would est illustrte the reltionship etween the vriles nd y. d = the numer of pples ought y ustomers, y = the totl ost of pples = the numer of pushups student n perform in one minute, y = the time tken for the student to run 100 metres = the height of people, y = the weight of people = the distne student trvels to shool, y = the height of the student s unle A B C D y y y y 5 The stter digrm shows the mrks otined y students in test out of 50 mrks, plotted ginst the numer of hours eh student studied for the test. Desrie the orreltion etween the vriles. How should the outlier e treted? Eplin your nswer. 60 mrks numer of hours of study 6 The following pirs of vriles were mesured nd strong positive orreltion etween them ws found. Disuss whether usl reltionship eists etween the vriles. If not, suggest third vrile to whih they my oth e relted. The lengths of one s left nd right feet. The dmge used y fire nd the numer of firemen who ttend it. Compny ependiture on dvertising, nd sles. d The height of prents nd the height of their dult hildren. e The numer of hotels nd the numer of hurhes in rurl towns. B MEASURING CORRELATION In the previous setion, we lssified the strength of the orreltion etween two vriles s either strong, moderte, or wek. We oserved the points on stter digrm, nd mde judgement s to how lerly the points formed liner reltionship. However, this method n e quite inurte, so it is importnt to get more preise mesure of the strength of liner orreltion etween two vriles. We hieve this using Person s produt-moment orreltion oeffiient r.

7 TWO VARIABLE STATISTICS (Chpter 11) 321 For set of n dt given s ordered pirs ( 1, y 1 ), ( 2, y 2 ), ( 3, y 3 ),..., ( n, y n ), P ( )(y y) Person s orreltion oeffiient is r = pp P ( ) 2 (y y) 2 where nd y re the mens of the nd y dt respetively, nd P mens the sum over ll the dt vlues. You re not required to lern this formul. Insted, we use tehnology to find the vlue of r. The vlues of r rnge from 1 to +1. GRAPHICS CALCULATOR INSTRUCTIONS The sign of r indites the diretion of the orreltion. ² A positive vlue for r indites the vriles re positively orrelted. An inrese in one of the vriles will result in n inrese in the other. ² A negtive vlue for r indites the vriles re negtively orrelted. An inrese in one of the vriles will result in derese in the other. The size of r indites the strength of the orreltion. ² A vlue of r lose to +1 or 1 indites strong orreltion etween the vriles. ² A vlue of r lose to zero indites wek orreltion etween the vriles. The following tle is guide for desriing the strength of liner orreltion using r. Positive orreltion Negtive orreltion r =1 perfet positive orreltion r = 1 perfet negtive orreltion 0:95 6 r<1 very strong positive orreltion 1 <r6 0:95 very strong negtive orreltion 0:87 6 r<0:95 strong positive orreltion 0:95 <r6 0:87 strong negtive orreltion 0:5 6 r<0:87 moderte positive orreltion 0:87 <r6 0:5 moderte negtive orreltion 0:1 6 r<0:5 wek positive orreltion 0:5 <r6 0:1 wek negtive orreltion 0 6 r<0:1 no orreltion 0:1 <r6 0 no orreltion

8 322 TWO VARIABLE STATISTICS (Chpter 11) Emple 1 Self Tutor The Deprtment of Rod Sfety wnts to know if there is ny ssoition etween verge speed in the metropolitn re nd the ge of drivers. They ommission devie to e fitted in the rs of drivers of different ges. The results re shown in the stter digrm. The r-vlue for this ssoition is +0:027. Desrie the ssoition verge speed (kmh _ -1) ge (yers) As r is lose to zero, there is no orreltion etween the two vriles. We oserve this in the grph s the points re rndomly sttered. EXERCISE 11B.1 1 In reent survey, the Deprtment of Interntionl Commere ompred the size of ompny with its eport ernings. A stter digrm of their dt is shown longside. The orresponding vlue of r is 0:556. Desrie the ssoition etween the vriles eport ernings ($million) numer of employees Mth eh stter digrm with the orret vlue of r. y y y d y e y A r =1 B r =0:6 C r =0 D r = 0:7 E r = 1

9 TWO VARIABLE STATISTICS (Chpter 11) 323 Emple 2 The Botnil Grdens hve een trying out new hemil to ontrol the numer of eetles infesting their plnts. The results of one of their tests re shown in the tle. Drw stter digrm of the dt. Determine Person s orreltion oeffiient r. Desrie the orreltion etween the quntity of hemil nd the numer of surviving lwn eetles. Smple Quntity of hemil (g) Self Tutor Numer of surviving eetles A 2 11 B 5 6 C 6 4 D 3 6 E 9 3 We first enter the dt into seprte lists: Csio f-cg20 TI-84 Plus TI-nspire Csio f-cg20 TI-84 Plus TI-nspire Csio f-cg20 TI-84 Plus TI-nspire So, r ¼ 0:859. There is moderte, negtive orreltion etween the quntity of hemil used nd the numer of surviving eetles. In generl, the more hemil tht is used, the fewer eetles tht survive.

10 324 TWO VARIABLE STATISTICS (Chpter 11) 3 For the following dt sets: i drw stter digrm of the dt ii lulte Person s orreltion oeffiient r iii desrie the liner orreltion etween X nd Y. X Y X Y X Y A seletion of students were sked how mny phone lls nd tet messges they hd reeived the previous dy. The results re shown elow. Student A B C D E F G H Phone lls reeived Tet messges reeived Drw stter digrm of the dt. Clulte r. Desrie the liner orreltion etween phone lls reeived nd tet messges reeived. 5 Consider the Opening Prolem on pge 316. Clulte r for this dt. Hene desrie the ssoition etween the vriles. 6 A sketller tkes 20 shots from eh of ten different positions mrked on the ourt. The tle elow shows how fr eh position is from the gol, nd how mny shots were suessful: Position A B C D E F G H I J Distne from gol ( m) 2 5 3:5 6:2 4:5 1:5 7 4:1 3 5:6 Suessful shots (y) Drw stter digrm of the dt. Do you think r will e positive or negtive? Clulte the vlue of r. d Desrie the liner orreltion etween these vriles. e Copy nd omplete: As the distne from gol inreses, the numer of suessful shots generlly... f Is there usl reltionship etween these vriles? CALCULATING r BY HAND (EXTENSION) In emintions you re epeted to lulte r using tehnology. However, lulting r using the formul r = this oeffiient works. P ( )(y y) pp ( ) 2 P (y y) 2 my help you understnd how

11 TWO VARIABLE STATISTICS (Chpter 11) 325 Emple 3 Sue investigtes how the volume of wter in pot ffets the time it tkes to oil on the stove. The results re given in the tle. Find nd interpret Person s orreltion oeffiient etween the two vriles. Pot Volume ( L) Self Tutor Time to oil (y min) A 1 3 B 2 5 C 4 7 D 5 9 P ) = n = 12 4 =3 y y y ( )(y y) ( ) 2 (y y) Totls: P y y = n = 24 4 =6 r = = P ( )(y y) pp ( ) 2 P (y y) 2 14 p ¼ 0:990 There is very strong orreltion etween the volume of wter nd the time for the wter to oil. As the volume of wter inreses, so does the time required. EXERCISE 11B.2 1 The tle elow inludes 4 dt points. Totls: y y y ( )(y y) ( ) 2 (y y) Find nd y. Copy nd omplete the tle. Clulte r. 2 For eh of the following grphs, evlute r = P ( )(y y) pp ( ) 2 P (y y) 2 nd omment on its vlue. y y y (3, 4) (2, 3) (1, 2) 2 (1, 2) (2, 1) 1 (3, 0) 1 2

12 326 TWO VARIABLE STATISTICS (Chpter 11) 3 A teher sked 5 students how muh time they spent prepring speeh. The results re shown in the tle, long with the grde wrded to the student. Drw stter digrm to illustrte this dt. Preprtion time Evlute r using the formul Student Grde (y%) ( hours) P ( )(y y) r = pp P ( ) 2 (y y) 2 Hene desrie the strength nd diretion of the liner orreltion etween preprtion time nd the grde wrded. A 5 95 B 4:5 80 C 7 90 D 1 65 E 3 75 THE COEFFICIENT OF DETERMINATION r 2 (EXTENSION) To help desrie the orreltion etween two vriles, we n lso lulte the oeffiient of determintion r 2. This is simply the squre of Person s orreltion oeffiient r, nd s suh the diretion of orreltion is eliminted. Given set of ivrite dt, we n find r 2 using our lultor in the sme wy we find r. Alterntively, if r is lredy known, we n simply squre this vlue. INTERPRETATION OF THE COEFFICIENT OF DETERMINATION If there is usl reltionship then r 2 indites the degree to whih hnge in the independent vrile eplins hnge in the dependent vrile. For emple, n investigtion into mny different rnds of muesli found tht there is strong positive orreltion etween the vriles ft ontent nd kilojoule ontent. It ws found tht r ¼ 0:862 nd r 2 ¼ 0:743. An interprettion of this r 2 vlue is: If 74:3% of the vrition in kilojoule ontent of muesli n e eplined y the ft ontent of muesli, then we n ssume tht the other 100% 74:3% = 25:7% of the vrition in kilojoule ontent of muesli n e eplined y other ftors. Emple 4 dependent vrile independent vrile 74:3% of the vrition in kilojoule ontent of muesli n e eplined y the vrition in ft ontent of muesli. At fther-son mp, the heights of the fthers nd their sons were mesured. Self Tutor Fther s height ( m) Son s height (y m) Drw stter digrm of the dt. Clulte r 2 for the dt nd interpret its vlue.

13 TWO VARIABLE STATISTICS (Chpter 11) son s height (m) fther s height (m) Using tehnology, r 2 ¼ 0:683. Csio f-cg20 TI-84 Plus TI-nspire 68:3% of the vrition in the son s height n e eplined y vrition in the fther s height. EXERCISE 11B.3 1 From n investigtion t n quti entre, the oeffiient of determintion for the vriles numer of visitors nd mimum temperture is found to e 0:578. Complete the following interprettion of the oeffiient of determintion:... % of the vrition in the... n e eplined y the vrition in... 2 An investigtion hs found the ssoition etween the vriles time spent gmling nd money lost hs n r vlue of 0:4732. Find the oeffiient of determintion nd interpret its mening. 3 For group of hildren produt-moment orreltion oeffiient of 0:365 is found etween the vriles hert rte nd ge. Find the oeffiient of determintion nd interpret its mening. 4 A smple of 8 tyres ws tken to emine the ssoition etween the tred depth nd the numer of kilometres trvelled. depth of tred tyre ross-setion Kilometres ( thousnd) Tred depth (y mm) 5:7 6:5 4:0 3:0 1:9 2:7 1:9 2:3 Drw stter digrm of the dt. Clulte r 2 for the dt nd interpret its mening.

14 328 TWO VARIABLE STATISTICS (Chpter 11) C LINE OF BEST FIT BY EYE If there is strong liner orreltion etween two vriles X nd Y, we n drw line of est fit to illustrte their reltionship. The line formed is lled line of est fit y eye. This line will vry from person to person. We drw line of est fit onneting vriles X nd Y s follows: Step 1: Clulte the men of the X vlues, nd the men of the Y vlues y. Step 2: Step 3: Mrk the men point (, y) on the stter digrm. Drw line through the men point whih fits the trend of the dt, nd so tht out the sme numer of dt points re ove the line s elow it. Consider gin the dt from the Opening Prolem: Athlete A B C D E F G H I J K L Age (yers) Distne thrown (m) We hve seen tht there is strong positive liner orreltion etween ge nd distne thrown. We n therefore model the dt using line of est fit. The men point is (15, 29), so we drw our line of est fit through (15, 29). We n use the line of est fit to estimte the vlue of y for ny given vlue of, nd vie vers. distne (m) men point INTERPOLATION AND EXTRAPOLATION Consider the dt in the stter digrm longside. The dt with the highest nd lowest vlues re lled the poles. A line of est fit hs een drwn so we n predit the vlue of one vrile for given vlue of the other ge (yers) y upper pole line of est fit If we predit y vlue for n vlue in etween the poles, we sy we re interpolting in etween the poles. If we predit y vlue for n vlue outside the poles, we sy we re etrpolting outside the poles. lower pole interpoltion etrpoltion etrpoltion The ury of n interpoltion depends on how liner the originl dt ws. This n e guged y the orreltion oeffiient nd y ensuring tht the dt is rndomly sttered round the line of est fit. The ury of n etrpoltion depends not only on how liner the originl dt ws, ut lso on the ssumption tht the liner trend will ontinue pst the poles. The vlidity of this ssumption depends gretly on the sitution we re looking t.

15 For emple, using our line of est fit from the Opening Prolem dt, the ge of 14 is within the rnge of ges lredy supplied. It is resonle to predit tht 14 yer old will e le to throw the disus 26 m. However, it is unresonle to predit tht 30 yer old will throw the disus 78 m. The ge of 30 is outside the rnge of vlues lredy supplied, nd it is unlikely tht the liner trend shown in the dt will ontinue up to the ge of 30. Emple 5 On hot dy, si rs were left in the sun in r prk. The length of time eh r ws left in the sun ws reorded, s well s the temperture inside the r t the end of the period. Cr A B C D E F Time (min) Temperture y ( ± C) Clulte nd y. Drw stter digrm for the dt. TWO VARIABLE STATISTICS (Chpter 11) 329 Plot the men point (, y) on the stter digrm. Drw line of est fit through this point. d Predit the temperture of r whih hs een left in the sun for: i 35 minutes ii 75 minutes. e Comment on the reliility of your preditions in d. = , 60 temperture ( C) ~ =30, y = men point (30, 38) distne (m) ~78 ~ ge (yers) =38 Self Tutor time (min) d i When =35, y ¼ 40. The temperture of r left in the sun for 35 minutes will e pproimtely 40 ± C. ii When =75, y ¼ 55. The temperture of r left in the sun for 75 minutes will e pproimtely 55 ± C.

16 330 TWO VARIABLE STATISTICS (Chpter 11) e The predition in diis relile, s the dt ppers liner, nd this is n interpoltion. The predition in diimy e unrelile, s it is n etrpoltion, nd the liner trend displyed y the dt my not ontinue eyond the 50 minute mrk. EXERCISE 11C 1 Fifteen students were weighed, nd their pulse rtes were mesured: Weight ( kg) Pulse rte (y ets per min) Drw stter digrm for the dt. Clulte r. d e f Desrie the reltionship etween weight nd pulse rte. Clulte the men point (, y). Plot the men point on the stter digrm, nd drw line of est fit through the men point. Estimte the pulse rte of student who weighs 65 kg. Comment on the reliility of your estimte. 2 To investigte whether speed mers hve n impt on rod sfety, dt ws olleted from severl ities. The numer of speed mers in opertion ws reorded for eh ity, s well s the numer of idents over 7 dy period. Numer of speed mers () Numer of r idents (y) d e Construt stter digrm to disply the dt. Clulte r for the dt. Desrie the reltionship etween the numer of speed mers nd the numer of r idents. Plot the men point (, y) on the stter digrm, nd drw line of est fit through the men point. Where does your line ut the y-is? Interpret wht this nswer mens. 3 The trunk widths nd heights of the trees in grden were reorded: Trunk width ( m) Height (y m) d e f Drw stter digrm of the dt. Whih of the points is n outlier? How would you desrie the tree represented y the outlier? Clulte the men point (, y). Plot the men point on the stter digrm, nd drw line of est fit through the men point. Predit the height of tree with trunk width 120 m. Comment on the reliility of your predition.

17 D TWO VARIABLE STATISTICS (Chpter 11) 331 LINEAR REGRESSION The prolem with drwing line of est fit y eye is tht the line drwn will vry from one person to nother. Insted, we use method known s liner regression to find the eqution of the line whih est fits the dt. The most ommon method is the method of lest squres. THE LEAST SQUARES REGRESSION LINE Consider the set of points longside. For ny line we drw to model the points, we n find the y vertil distnes d 1, d 2, d 3,... etween eh point nd the line. We n then squre eh of these distnes, nd find their d2 sum d1 2 + d d :::: d1 If the line is good fit for the dt, most of the distnes will e smll, nd so will the sum of their squres. The lest squres regression line is the line whih mkes this sum s smll s possile. The demonstrtion longside llows you to eperiment with vrious dt sets. Use tril nd error to find the lest squres regression line for eh set. d3 d4 DEMO In prtie, rther thn finding the regression line y eperimenttion, we use lultor or sttistis pkge. GRAPHICS CALCULATOR INSTRUCTIONS STATISTICS PACKAGE Emple 6 Self Tutor The nnul inome nd verge weekly groery ill for seletion of fmilies is shown elow: Inome ( thousnd pounds) Groery ill (y pounds) Construt stter digrm to illustrte the dt. Use tehnology to find the lest squres regression line. Estimte the weekly groery ill for fmily with n nnul inome of $ Comment on whether this estimte is likely to e relile groery ill ( $ ) inome ( $ thousnds)

18 332 TWO VARIABLE STATISTICS (Chpter 11) Csio f-cg20 TI-84 Plus TI-nspire Using tehnology, the line of est fit is y ¼ 4:18 56:7 When =95, y ¼ 4:18(95) 56:7 ¼ 340 So, we epet fmily with n inome of $ to hve weekly groery ill of pproimtely $340. This is n etrpoltion, however, so the estimte my not e relile. EXERCISE 11D 1 A newspper reports strting slries for reently grduted university students whih depend on whether they hold Bhelor degree or PhD. Drw stter digrm for the dt. Determine r. Desrie the ssoition etween strting slries for Bhelor degrees nd strting slries for PhDs. d Find the eqution of the line of est fit. Field Bhelor degree ($) PhD ($y) Chemil engineer Computer oder Eletril engineer Soiologist Applied mthemtiin e The strting slry for n eonomist with Bhelor degree is $ i ii Predit the strting slry for n eonomist with PhD. Comment on the reliility of your predition. Aountnt Steve wnted to see whether there ws ny reltionship etween the temperture when he leves for work in the morning, nd the time it tkes to get to work. He olleted dt over 14 dy period: Temperture ( ± C) Time (y min) Drw stter digrm of the dt. Clulte r. Desrie the reltionship etween the vriles. d Is it resonle to try to find line of est fit for this dt? Eplin your nswer.

19 TWO VARIABLE STATISTICS (Chpter 11) The tle elow shows the prie of petrol nd the numer of ustomers per hour for siteen petrol sttions. d Petrol prie ( ents per litre) 105:9 106:9 109:9 104:5 104:9 111:9 110:5 112:9 Numer of ustomers (y) Petrol prie ( ents per litre) 107:5 108:0 104:9 102:9 110:9 106:9 105:5 109:5 Numer of ustomers (y) Clulte r for the dt. Desrie the reltionship etween the petrol prie nd the numer of ustomers. Use tehnology to find the line of est fit. Interpret the grdient of this line. e Estimte the numer of ustomers per hour for petrol sttion whih sells petrol t 115:9 ents per litre. f Comment on the vlidity of your estimte in e. 4 The tle elow ontins informtion out the mimum speed nd mimum ltitude otinle or eiling for nineteen World Wr II fighter plnes. The mimum speed is given in thousnds of km/h, nd the eiling is given in km. m. speed eiling 0:46 8:84 0:42 10:06 0:53 10:97 0:53 9:906 0:49 9:448 0:53 10:36 0:68 11:73 m. speed eiling 0:68 10:66 0:72 11:27 0:71 12:64 0:66 11:12 0:78 12:80 0:73 11:88 m. speed eiling 0:67 12:49 0:57 10:66 0:44 10:51 0:67 11:58 0:70 11:73 0:52 10:36 Drw stter digrm for this dt. Clulte r. d e Desrie the ssoition etween mimum speed () nd eiling (y). Use tehnology to find the line of est fit. Estimte the eiling for fighter plne with mimum speed of 600 km/h. 5 A group of hildren were sked the numer of hours they spent eerising nd wthing television eh week. Eerise ( hours per week) Television (y hours per week) Drw stter digrm for the dt. Clulte r. d Desrie the orreltion etween time eerising nd time wthing television. Find the eqution of the lest squres line of est fit. e Give n interprettion of the grdient nd the y-interept of this line. f Another hild eerises for 5 hours eh week. Estimte how long he spends wthing television eh week.

20 334 TWO VARIABLE STATISTICS (Chpter 11) 6 The yield of pumpkins on frm depends on the quntity of fertiliser used. Fertiliser ( gm 2 ) Yield (y kg) 1:8 2:9 3:8 4:2 4:7 5:7 4:4 d e Drw stter digrm of the dt nd identify the outlier. Clulte the orreltion oeffiient: i with the outlier inluded ii without the outlier. Clulte the eqution of the lest squres regression line: i with the outlier inluded ii without the outlier. If you wish to estimte the yield when 15 gm 2 of fertiliser is used, whih regression line from should e used? Cn you eplin wht my hve used the outlier? E THE  2 TEST OF INDEPENDENCE This tle shows the results of smple of 400 rndomly seleted dults lssified ording to gender nd regulr eerise. We ll this 2 2 ontingeny tle. Regulr eerise No regulr eerise sum Mle Femle sum We my e interested in how the vriles gender nd regulr eerise re relted. The vriles my e dependent, for emple femles my e more likely to eerise regulrly thn mles. Alterntively, the vriles my e independent, whih mens the gender of person hs no effet on whether they eerise regulrly. The hi-squred or  2 test is used to determine whether two vriles from the sme smple re independent. CALCULATING  2 To test whether gender nd regulr eerise re independent, we first onsider only the sum vlues of the ontingeny tle. We then lulte the vlues we would epet to otin if the vriles were independent. Regulr eerise No regulr eerise sum Mle 216 Femle 184 sum For emple, if gender nd regulr eerise were independent, then P(mle \ regulr eerise) =P(mle) P(regulr eerise) = So, in smple of 400 dults, we would epet ³ = = 112:32 to e mle nd eerise regulrly

21 TWO VARIABLE STATISTICS (Chpter 11) 335 We n perform similr lultions for eh ell to omplete n epeted frequeny tle. This displys the vlues we would epet to otin if the vriles were independent. Mle Femle Regulr eerise = 112: =95:68 No regulr eerise sum = 103: =88: sum For eh ell, we multiply the row sum y the olumn sum, then divide y the totl. The  2 test emines the differene etween the oserved vlues we otined from our smple, nd the epeted vlues we hve lulted.  2 l = X (f o f e ) 2 f e where nd f o f e is n oserved frequeny is n epeted frequeny. If the vriles re independent, the oserved nd epeted vlues will e very similr. This mens tht the vlues of (f o f e ) will e smll, nd hene  2 l will e smll. If the vriles re not independent, the oserved vlues will differ signifintly from the epeted vlues. The vlues of (f o f e ) will e lrge, nd hene  2 l will e lrge. For our emple on gender nd regulr eerise, our  2 lultion is f o f e f o f e (f o f e ) 2 (f o f e ) :32 2:32 5:3824 0: :68 2:32 5:3824 0: :68 2:32 5:3824 0: :32 2:32 5:3824 0:0609 In this se,  2 l ¼ 0:217, whih is very smll. This indites tht gender nd regulr eerise re independent. USING TECHNOLOGY f e Totl 0:2170 You n lso use your lultor to find  2 l. You must first enter the ontingeny tle s mtri. Using Csio f-cg20:

22 336 TWO VARIABLE STATISTICS (Chpter 11) Using TI-84 Plus: Using TI-nspire: Consult the grphis lultor instrutions for more detiled help. EXERCISE 11E.1 1 Construt n epeted frequeny tle for the following ontingeny tles: Likes hiken Dislikes hiken sum Likes fish 60 Dislikes fish 40 sum Drove to work Cyled to work Puli trnsport sum Mle 44 Femle 36 sum Junior shool Middle shool High shool sum Plys sport Does not ply sport sum GRAPHICS CALCULATOR INSTRUCTIONS d Wore ht nd sunsreen Wore ht or sunsreen Wore neither sum Sunurnt Not sunurnt sum

23 TWO VARIABLE STATISTICS (Chpter 11) Consider the ontingeny tle: Pss Mths test Fil Mths test sum Mle Femle sum Construt n epeted frequeny tle. Interpret the vlue in the top left orner of the epeted frequeny tle. Clulte  2 l y opying nd ompleting this tle: 3 For the following ontingeny tles: f o f e f o f e (f o f e ) 2 (f o f e ) Totl i onstrut the epeted frequeny tle ii find  2 l without using tehnology. Likes footll Dislikes footll sum Mle Femle sum Full-time jo Prt-time jo Unemployed sum Left hnded Right hnded sum Age Mrried Single d Visits Museum f e Visits Art Gllery Often Rrely Never Often Rrely Never Chek your nswers using your lultor. They my differ slightly due to rounding. FORMAL TEST FOR INDEPENDENCE We hve seen tht smll vlue of  2 indites tht two vriles re independent, while lrge vlue of  2 indites tht the vriles re not independent. We will now onsider more forml test whih determines how lrge  2 must e for us to onlude the vriles re not independent. This is known s the ritil vlue of  2. The ritil vlue of  2 depends on: ² the size of the ontingeny tle, mesured y degrees of freedom ² the signifine level used.

24 338 TWO VARIABLE STATISTICS (Chpter 11) DEGREES OF FREEDOM In ontingeny tle, the numer of degrees of freedom (df) is the numer of vlues whih re free to vry. Consider the 2 2 ontingeny tle longside, with the sum vlues given. The vlue in the top left orner is free to vry, s it n tke mny possile vlues, one of whih is 9. However, one we set this vlue, the remining vlues re not free to vry, s they re determined y the row nd olumn sums. So, the numer of degrees of freedom is 1, whih is (2 1) (2 1). A 1 A 2 sum B 1 12 B 2 8 sum A 1 A 2 sum B B sum In 3 3 ontingeny tle, we n hoose (3 1) (3 1) = 4 vlues efore the remining vlues re not free to vry. C 1 C 2 C 3 sum D 1 12 D 2 8 D 3 13 sum C 1 C 2 C 3 sum D D D sum The row nd olumn numers do not inlude sums. For ontingeny tle whih hs r rows nd olumns, df =(r 1)( 1). SIGNIFICANCE LEVEL As the  2 vlue gets lrger, it eomes inresingly unlikely tht the vriles involved re independent. The signifine level indites the minimum eptle proility tht the vriles re independent. We usully use either 10%, 5%, or 1% for the signifine level. For given signifine level nd degrees of freedom, the tle longside gives the ritil vlue of  2, ove whih we onlude the vriles re not independent. For emple, t 5% signifine level with df =1, the ritil vlue is 3:84. This mens tht t 5% signifine level, the deprture etween the oserved nd epeted vlues is too gret if  2 l > 3:84. Likewise, t 1% signifine level with df =7, the deprture etween the oserved nd epeted vlues is too gret if  2 l > 18:48. Degrees of Signifine level freedom (df) 10% 5% 1% 1 2:71 3:84 6:63 2 4:61 5:99 9:21 3 6:25 7:81 11:34 4 7:78 9:49 13:28 5 9:24 11:07 15: :64 12:59 16: :02 14:07 18: :36 15:51 20: :68 16:92 21: :99 18:31 23:21

25 TWO VARIABLE STATISTICS (Chpter 11) 339 Clik on the ion for more detiled tle of ritil vlues. CRITICAL VALUES In emintions the ritil vlue of  2 will e provided. Importnt: In order for  2 to e distriuted ppropritely, the smple size n must e suffiiently lrge. Generlly, n is suffiiently lrge if no vlues in the epeted vlue tle re less thn 5. THE p-value When finding  2 on your lultor, p-vlue is lso provided. This n e used, together with the  2 vlue nd the ritil vlue, to determine whether or not to ept tht the vriles re independent. For given ontingeny tle, the p-vlue is the proility of otining oserved vlues s fr or further from the epeted vlues, ssuming the vriles re independent. If the p-vlue is smller thn the signifine level, then it is suffiiently unlikely tht we would hve otined the oserved results if the vriles hd een independent. We therefore onlude tht the vriles re not independent. It is not lwys essentil to use the p-vlue when testing for independene, s we n perform the test y simply ompring  2 l with the ritil vlue. However, the p-vlue does give more meningful mesure of how likely it is tht the vriles re independent. THE FORMAL TEST FOR INDEPENDENCE Step 1: Stte H 0 lled the null hypothesis. This is sttement tht the two vriles eing onsidered re independent. Stte H 1 lled the lterntive hypothesis. This is sttement tht the two vriles eing onsidered re not independent. Step 2: Stte the rejetion inequlity  2 l >k where k is the ritil vlue of Â2. Step 3: Construt the epeted frequeny tle. Step 4: Use tehnology to find  2 l. Step 5: We either rejet H 0 or do not rejet H 0, depending on the result of the rejetion inequlity. Step 6: We ould lso use p-vlue to help us with our deision mking. For emple, t 5% signifine level: If p<0:05, we rejet H 0. If p>0:05, we do not rejet H 0. We write we do not rejet H0 rther thn we ept H0 euse if we perform the test gin with different level of signifine, we my then hve reson to rejet H. 0

26 340 TWO VARIABLE STATISTICS (Chpter 11) Emple 7 A survey ws given to rndomly hosen high shool students from yers 9 to 12 on possile hnges to the shool s nteen. The ontingeny tle shows the results. At 5% signifine level, test whether the student s nteen preferene depends on the yer group. Self Tutor Yer group hnge no hnge H 0 is tht yer group nd nteen preferene re independent. H 1 is tht yer group nd nteen preferene re not independent. df =(2 1)(4 1) = 3 nd the signifine level is 5% or 0:05. ) the ritil vlue is 7:81 ffrom the tle of ritil vluesg We rejet H 0 if  2 l > 7:81. The 2 4 ontingeny tle is: Yer group sum C C sum The epeted frequeny tle is: Yer group C 10:6 10:6 11:1 10:6 C 0 10:4 10:4 10:9 10:4 Csio f-cg20 TI-84 Plus TI-nspire Using tehnology,  2 l ¼ 5:81, whih is < 7:81: Therefore, we do not rejet H 0. p ¼ 0:121 whih is > 0:05, providing further evidene to not rejet H 0. We onlude tht t 5% level of signifine, the vriles yer group nd nteen preferene re independent.

27 TWO VARIABLE STATISTICS (Chpter 11) 341 EXERCISE 11E.2 1 This ontingeny tle shows the responses of rndomly hosen smple of dults regrding the person s weight nd whether they hve dietes. At 5% signifine level, the ritil vlue of  2 is 5:99. Test t 5% level whether there is link etween weight nd suffering dietes. 2 The tle opposite shows the wy in whih rndom smple of people intend to vote in the net eletion. For 10% signifine level, wht is the ritil vlue of  2? Weight light medium hevy Dieti Non-dieti Age of voter 18 to to Prty A Prty B Test t 10% level whether there is ny ssoition etween the ge of voter nd the prty they wish to vote for. 3 Noh wnted to find out whether there is Fvourite seson reltionship etween person s gender Summer Autumn Winter Spring nd their fvourite seson. He smpled 100 people, nd otined the results longside. Mle At 1% signifine level, the ritil vlue for this test is 11:34. Femle Test, t 1% level, whether the vriles gender nd fvourite seson re independent. 4 The guests stying t hotel re sked to provide their reson for trvelling, nd to rte the hotel on sle from Poor to Eellent. The results re shown elow. Reson for trvelling Rting Poor Fir Good Eellent Business Holidy Show tht, t 5% signifine level, the vriles reson for trvelling nd rting re dependent. By emining the ontingeny tle, desrie how guest s rting is ffeted y their reson for trvelling. 5 The hir nd eye olours of 150 rndomly seleted individuls re shown in the tle elow. Eye olour Hir olour Blond Blk Brunette Red Blue Brown Green At 5% signifine level, the ritil vlue for  2 is 12:59. Test, t 5% level, whether there is n ssoition etween hir olour nd eye olour.

28 342 TWO VARIABLE STATISTICS (Chpter 11) 6 Hokey plyer Julie wondered whether the position you plyed ffeted your likelihood of eing injured. She sked rndom smple of hokey plyers wht position they plyed, nd wht injuries they hd sustined in the lst yer. Injury type Position Forwrd Midfielder Defender Golkeeper No injury Mild injury Serious injury Test, t 10% signifine level, whether the vriles position nd injury type re independent. LIMITATIONS OF THE  2 TEST (EXTENSION) There re two situtions in whih the  2 test my e unrelile: 1 Any of the epeted frequenies re less thn 5. This n e resolved y omining dt. 2 The degrees of freedom is 1. This n e resolved using Ytes ontinuity orretion. These situtions my rise in internl ssessment tsks, ut you will not e required to del with them in emintions. COMBINING DATA The  2 test my e unrelile if ny of the epeted frequeny vlues re less thn 5. Consider the ontingeny tle longside. Wth TV Rrely Sometimes Often Very often For this ontingeny tle,  2 ¼ 8:52. Mle For 5% signifine level nd df =3, the Femle ritil vlue is 7:81. Sine  2 l > 7:81, we would rejet H 0, nd onlude tht gender nd television wthing re dependent. However, on inspeting the epeted Wth TV frequeny tle, there re two epeted Rrely Sometimes Often Very often frequenies whih re less thn 5. This Mle 15:3 16:2 9:45 4:05 indites tht our onlusion my not e relile. Femle 18:7 19:8 11:55 4:95 We n improve the reliility of this test y Wth TV omining rows or olumns so tht there re no Rrely Sometimes Often/Very often ells with epeted frequeny less thn 5. In this se we omine the often nd very often olumns to produe: Mle Femle

29 TWO VARIABLE STATISTICS (Chpter 11) 343 The epeted frequeny tle is now: Wth TV Rrely Sometimes Often/Very often Mle 15:3 16:2 13:5 Femle 18:7 19:8 16:5 Now  2 l ¼ 4:18, nd for 5% level with df =2, the ritil vlue is 5:99. Sine  2 l < 5:99, we now onlude tht the vriles re independent. This is different from our originl onlusion. EXERCISE 11E.3 1 Consider the ontingeny tle longside: Construt the epeted frequeny tle. Are ny of the epeted frequenies less thn 5? Comine the dt so tht none of the ells hve n epeted frequeny less thn 5. Age Own pet? Yes No The following tle is result of mjor investigtion onsidering the two ftors of intelligene level nd igrette smoking. Intelligene level low verge high very high Non smoker Medium level smoker Hevy smoker Test t 1% level whether there is link etween intelligene level nd igrette smoking. Construt the epeted frequeny tle. Comine pproprite olumns so tht none of the epeted frequenies is less thn 5. d Perform this test gin t 1% level. Is your onlusion the sme s in? YATES CONTINUITY CORRECTION The  2 test my lso e unrelile if the numer of degrees of freedom is 1. This ours when we hve 2 2 ontingeny tle. To improve the reliility of the  2 test for 2 2 ontingeny tles, we n pply Ytes ontinuity orretion. We use modified formul to find  2 l. If df =1, we use  2 l = P (jf o f e j 0:5) 2 where jf o f e j is the solute vlue or modulus of f o f e. f e

30 344 TWO VARIABLE STATISTICS (Chpter 11) Emple 8 80 people were surveyed to find whether they enjoyed surfing nd skiing. The results re shown longside. Test, t 1% level, whether there is n ssoition etween enjoying surfing nd enjoying skiing. Enjoy skiing? Self Tutor Enjoy surfing? Yes No Yes No 8 40 H 0 : The vriles enjoying surfing nd enjoying skiing re independent. H 1 : The vriles enjoying surfing nd enjoying skiing re not independent. At 1% level with df =1, the ritil vlue is 6:63. So, we rejet H 0 if  2 l > 6:63. The 2 4 ontingeny tle is: Enjoy skiing? Enjoy surfing? Yes No sum Yes No sum The epeted frequeny tle is: Enjoy skiing? Enjoy surfing? Yes No Yes No We will now find  2 l using Ytes ontinuity orretion: f o f e f o f e jf o f e j jf o f e j 0:5 (jf o f e j 0:5) 2 (jf o f e j 0:5) :5 42:25 4: :5 42:25 1: :5 42:25 2: :5 42:25 1:280 Totl 10:242 So,  2 l ¼ 10:2 Sine  2 l > 6:63, we rejet H 0 nd onlude tht, t 1% signifine level, enjoying surfing nd enjoying skiing re dependent. f e EXERCISE 11E.4 1 Hore lims tht he n predit the outome of oin toss. To test this, he tosses oin 200 times, nd tries to guess the outome of eh toss. The results re shown longside. Construt the epeted frequeny tle. Use Ytes ontinuity orretion to find  2 l. The ritil vlue t 5% level with df =1 is 3:84. Test whether Hore s guess nd the result re independent. d Comment on the vlidity of Hore s lim. Guess Result Heds Tils Heds Tils In emintions, the numer of degrees of freedom will lwys e greter thn 1, so Ytes ontinuity orretion will not e required.

31 TWO VARIABLE STATISTICS (Chpter 11) The prtil test for motorike liene differs in Frne Result nd Germny. An inquiry into the two systems yielded Pss Fil the following results for rndomly seleted ndidtes. Frne A hi-squred test t 10% signifine level is used Country Germny to investigte whether the result of motorike test is independent of the ountry where it took ple. Construt the epeted frequeny tle. Write down the ritil vlue of the hi-squred test sttisti. Using Ytes ontinuity orretion, find the hi-squred vlue for this dt. d Wht onlusion n e drwn from this hi-squred test? THEORY OF KNOWLEDGE In the previous eerise we sw emples of dt whih ws non-liner, ut for whih we ould trnsform the vriles so liner model ould e used. In other situtions we n use qudrti or trigonometri funtions to model dt. 1 Cn ll dt e modelled y known mthemtil funtion? 2 How relile is mthemtis in prediting rel-world phenomen? The Lotk-Volterr predtor-prey model ws developed independently y Alfred Lotk ( ) nd Vito Volterr ( ). The model is used to predit the popultions of two speies of nimls over time, where one speies is predtor of the other. Alfred Lotk 3 Is the Lotk-Volterr model defined y nture or y mn? 4 Is nture governed y mthemtis, or re we imposing our own retion upon it? REVIEW SET 11A 1 Thoms rode for n hour eh dy for eleven dys. He reorded the numer of kilometres he rode long with the temperture tht dy. Temperture (T ± C) 32:9 33:9 35:2 37:1 38:9 30:3 32:5 31:7 35:7 36:3 34:7 Distne (d km) 26:5 26:7 24:4 19:8 18:5 32:6 28:7 29:4 23:8 21:2 29:7 d Using tehnology, onstrut stter digrm of the dt. Find nd interpret Person s orreltion oeffiient for the two vriles. Find the eqution of the lest squres regression line. How hot must it get efore Thoms does not ride t ll?

32 346 TWO VARIABLE STATISTICS (Chpter 11) 2 The ontingeny tle elow shows the results of motor vehile idents in reltion to whether the trveller ws wering set elt. Serious injury Permnent dislement Deth Wering elt Not wering elt At 5% level with df =2, the ritil vlue is 5:99. Test t 5% level whether wering of set elt nd severity of injury re independent ftors. 3 A rft shop sells nvsses in Are ( m 2 ) vriety of sizes. The tle elow shows the re nd prie of eh Prie ($y) nvs type. Construt stter digrm for the dt. Clulte r. Desrie the orreltion etween re nd prie. d Find the eqution of the lest squres regression line. e Drw the line of est fit on your stter digrm. f Estimte the prie of nvs with re 1200 m 2. Is your estimte likely to e relile? 4 A lothing store reorded the length of time ustomers were in the store nd the mount of money they spent. Time (min) Money (E) Drw stter digrm of the dt. Clulte the men point. Plot the men point on your digrm nd drw line of est fit through the men point. d Desrie the reltionship etween time in the store nd money spent. e Estimte the mount of money spent y person who is in the store for 15 minutes. Comment on the reliility of your estimtion. 5 A drinks vendor vries the prie of Sup-fizz on dily sis. He reords the numer of sles of the drink s shown: Prie (p) $2:50 $1:90 $1:60 $2:10 $2:20 $1:40 $1:70 $1:85 Sles (s) d Produe stter digrm for the dt. Are there ny outliers? If so, should they e inluded in the nlysis? Clulte the lest squres regression line. Do you think the lest squres regression line would give n urte predition of sles if Sup-fizz ws pried t 50 ents? Eplin your nswer. 6 Eight identil flower eds ontin petunis. The different eds were wtered different numers of times eh week, nd the numer of flowers eh ed produed ws reorded in the tle elow: Numer of wterings (n) Flowers produed (f)

33 TWO VARIABLE STATISTICS (Chpter 11) 347 d e Whih is the independent vrile? Clulte the eqution of the lest squres regression line. Is it likely tht usl reltionship eists etween these two vriles? Eplin your nswer. Plot the lest squres regression line on stter digrm of the dt. Violet hs two eds of petunis. One she wters five times fortnight (2 1 2 times week), nd the other ten times week. i How mny flowers n she epet from eh ed? ii Whih is the more relile estimte? 7 Emine the following ontingeny tle for the independene of ftors P nd Q. Use  2 test: t 5% level of signifine t 1% level of signifine. Q 1 Q 2 Q 3 Q 4 P P P REVIEW SET 11B 1 The following tle gives the verge numer of hildren for different fmily inomes. Inome (I thousnd $) Numer of hildren, n 4:2 3:4 3:2 2:9 2:7 2:5 2:3 2:1 1:9 Construt n pproprite grph to disply the dt. Find r. Find the eqution of the line of est fit. d Estimte the verge numer of hildren for fmily inome of: i $ ii $ e Comment on the reliility of your estimtes. 2 For the following pirs of vriles, disuss: i whether the orreltion etween the vriles is likely to e positive or negtive ii whether usl reltionship eists etween the vriles. prie of tikets nd numer of tikets sold ie rem sles nd numer of drownings. 3 The tle shows the responses to survey out whether the ity speed limit should e inresed. Test t 10% level whether there is ny ssoition etween the ge of driver nd inresing the speed limit. Age of driver 18 to to Inrese No inrese

34 348 TWO VARIABLE STATISTICS (Chpter 11) 4 The following tle shows the results from mjor investigtion onsidering the two ftors intelligene level nd usiness suess. Business suess Intelligene level Low Averge High Very high No suess Low suess Suess High suess At 1% level with df =9, the ritil vlue is 21:67. Test t 1% level whether there is link etween intelligene level nd usiness suess. 5 Sfety uthorities dvise drivers to trvel three seonds ehind the r in front of them. This provides the driver with greter hne of voiding ollision if the r in front hs to rke quikly or is itself involved in n ident. A test ws rried out to find out how long it would tke driver to ring r to rest from the time red light ws flshed. It involved one driver in the sme r under the sme test onditions. Speed (v km h 1 ) Stopping time (t s) 1:23 1:54 1:88 2:20 2:52 2:83 3:15 3:45 3:83 Produe stter digrm of the dt. Find the liner model whih est fits the dt. Hene estimte the stopping time for speed of: i 55 km h 1 ii 110 km h 1 d Interpret the vertil interept of the model. 6 Two supervillins, Silent Predtor nd the Furry Reper, terrorise Metropolis y duting fir midens (most of whom hppen to e journlists). The superhero Supermn elieves tht they re ollorting, lterntively duting fir midens so s not to ompete with eh other for rnsom money. He plots their dution rte elow, in dozens of midens. Silent Predtor (p) Furry Reper (r) d e f g Plot the dt on stter digrm with Silent Predtor on the horizontl is. Find the lest squres regression line. Clulte r, nd hene desrie the strength of Silent Predtor nd Furry Reper s reltionship. Is there ny evidene to support Supermn s suspiions? Estimte the numer of the Furry Reper s dutions when the Silent Predtor s were 6 dozen. Why is the model inpproprite when the Furry Reper duts more thn 20 dozen midens? Clulte the p- nd r-interepts of the regression line. Wht do these vlues represent? If Supermn is fed with hoie of pturing one supervillin ut not the other, whih should he hoose?