AS MATHEMATICS HOMEWORK S1

Transcription

1 Name Teacher AS MATHEMATICS HOMEWORK S1 Mathematcs Department September 015 Verson 1.0

2 Contents Contents... AS Maths Homework S HW1 Data1 dscrete data, bo plots, stem and leaf dagrams... 4 HW Data grouped data, hstograms... 7 HW3 Probablt HW4 Correlaton and Lnear Regresson... 1 HW5 Dscrete Random Varables HW6 Normal Dstrbutons HWX S1 June Statstcs 1 Formula Sheet... 5 Normal Dstrbuton Tables... 6

3 AS Maths Homework S1 014 Am to complete all the questons. If ou fnd the work dffcult then get help [lunchtme workshops n room 16, onlne, frends, teacher etc]. To learn effectvel ou should check our work carefull and mark answers? If ou have questons or comments, please wrte these on our homework. Your teacher wll then revew and mark our mathematcs. If ou spot an error n ths pack please let our teacher know so we can make changes for the net edton! Homework Tasks These cover the man topcs n S1. Your teacher ma set homework from ths or other tasks. has vdeo solutons to eam queston and clear eplanatons of man topcs. Topc Date completed Comment HW1 Data 1 HW Data HW3 Probablt HW4 Correlaton and lnear regresson HW5 Dscrete random varables HW6 Normal dstrbutons HWX S1 June 010 3

4 HW1 Data1 dscrete data, bo plots, stem and leaf dagrams Wrte our answers on a separate pece of paper. Show clear workng and mark answers Ke words: Stem and leaf dagram, hstogram, boplot, outler, skewness, IQR Formulae: Quartles Q 1, Q, Q 3 see page 9 for detals mean = Σ n, varance σ = Σ n Σ n stdevaton = varance Shape: postve skew, negatve skew, smmetrcal 1. [Jan 006 Q1] Over a 7 da perod, the number of people stang n a hotel was recorded. Ths data s summarsed n the stem and leaf dagram below. Number leavng 3 means 3 Totals a wrte down the mode, b fnd the values of the three quartles. 1 Gven that = 1335 and = , fnd c the mean and the standard devaton of these data. 4 One measure of skewness s found usng [ see pages for notes on skewness] mean mode. standarddevaton d Evaluate ths measure to show that these data are negatvel skewed. e Gve two other reasons wh these data are negatvel skewed. 4. [Jan 008 Q] Cotnne s a chemcal that s made b the bod from ncotne whch s found n cgarette smoke. A doctor tested the blood of 1 patents, who clamed to smoke one packet of cgarettes a da, for cotnne. The results, n approprate unts, are shown below. Patent A B C D E F G H I J K L Cotnne level, [You ma use = ] a Fnd the mean and standard devaton of the level of cotnne n a patent s blood. 4 4

5 b Fnd the medan, upper and lower quartles of these data. A doctor suspects that some of hs patents have been smokng more than a packet of cgarettes per da. He decdes to use Q Q 3 Q 1 to determne f an of the cotnne results are far enough awa from the upper quartle to be outlers. c Identf whch patents ma have been smokng more than a packet of cgarettes a da. Show our workng clearl. 4 Research suggests that cotnne levels n the blood form a skewed dstrbuton. Q1 Q Q3 One measure of skewness s found usng. Q Q 3 1 d Evaluate ths measure and descrbe the skewness of these data. 3. [Ma 006 Q1] a Descrbe the man features and uses of a bo plot. Chldren from schools A and B took part n a fun run for chart. The tmes, to the nearest mnute, taken b the chldren from school A are summarsed n Fgure 1. Fgure 1 School A Tme mnutes b Wrte down the tme b whch 75% of the chldren n school A had completed the run. State the name gven to ths value. c Eplan what ou understand b the two crosses on Fgure 1. For school B the least tme taken b an of the chldren was 5 mnutes and the longest tme was 55 mnutes. The three quartles were 30, 37 and 50 respectvel. d On graph paper, draw a bo plot to represent the data from school B. e Compare and contrast these two bo plots. 4 4 HW1 Answers 1a mode 56, b Q 1 = 35, Q = 5, Q 3 = 60 c = 49.4, σ = 14.6 accept 14.9 d skew = e mean < medan < mode or Q Q 1 > Q 3 Q. a 30, 87.3 b 198; 170, 54 c F and Bd 0. 3, postve skew 3a Indcates ma, mn, quartles; ndcates range, IQR; llustrates skewness; allows comparsons; ndcates outlers. b 37 mns Q 3 or upper quartle c outlers 5

6 6 d mn 5, ma 55, Q 1 = 30, Q = 37, Q 3 = 50 e see offcal mark scheme

7 HW Data grouped data, hstograms Wrte our answers on a separate pece of paper. Show clear workng and mark answers Ke words: Range, Quartles, Percentles, Interquartle range, Varance, Standard devaton Formulae: When calculatng quartles and percentles of grouped data, use lnear nterpolaton. Consder as contnuous data - quartles n 4, n, 3n 4 Interquartle range IQR= Q 3 Q 1 Mean for grouped data: = f f Varance = n = n n Varance grouped data/frequenc tables= f f = n = mean Standard devaton = Varance f or f f f f f = frequenc, 1. [Jan 011 Q5] On a randoml chosen da, each of the 3 students n a class recorded the tme, t mnutes to the nearest mnute, the spent on ther homework. The data for the class s summarsed n the followng table. Tme, t Number of students cf a Cop and complete the table for cumulatve frequenc, then use nterpolaton to estmate the value of the medan. Gven that t = 1414 and t = , b fnd the mean and the standard devaton of the tmes spent b the students on ther homework. c Comment on the skewness of the dstrbuton of the tmes spent b the students on ther homework. Gve a reason for our answer based on mode, medan, mode 7

8 . [Ma 007 Q5] Frequenc Denst Hstogram of tmes t Fgure Fgure shows a hstogram for the varable t whch represents the tme taken, n mnutes, b a group of people to swm 500 m. a Cop and complete the frequenc table for t. t Frequenc b Estmate the number of people who took longer than 0 mnutes to swm 500 m. c Fnd an estmate of the mean tme taken. d Fnd an estmate for the standard devaton of t. e Fnd the medan and quartles for t usng lnear nterpolaton. 4 4 One measure of skewness s found usng 3mean medan standarddevaton. f Evaluate ths measure and descrbe the skewness of these data. 8

9 3. [Jan 008 Q3]The hstogram n Fgure 1 shows the tme taken, to the nearest mnute, for 140 runners to complete a fun run. Fgure 1 Use the hstogram to calculate the number of runners who took between 78.5 and 90.5 mnutes to complete the fun run. 5 HW Answers 1. a 41.3 b 14.7 c postve skew: mode < medan < mean. a 35, 15 b 40 c d 7.6 e Medan = 18, Lower Quartle = 13.75, Upper Quartle = 3 f 0.376, postve skew 3. 1 runners 9

10 HW3 Probablt Wrte our answers on a separate pece of paper. Show clear workng and mark answers Ke words: Venn dagram, Tree dagram, mutuall eclusve, ndependent event, condtonal probablt, sample space, event Formulae: Probablt of A gven B PA B = PA + PB PA B PA B = PA B, Independent f PA B = PA PB PB Read pages Take note of the summar page 107 Venn Dagrams 1. [Ma 007 Q4] A surve of the readng habts of some students revealed that, on a regular bass, 5% read qualt newspapers, 45% read tablod newspapers and 40% do not read newspapers at all. a Draw a Venn dagram to represent ths nformaton. b Fnd the proporton of students who read both qualt and tablod newspapers. 6 A student s selected at random. Gven that ths student reads newspapers on a regular bass, c fnd the probablt that ths student onl reads qualt newspapers.. [Jan 006 Q 6] For the events A and B, PA B = 0.3, PA B = 0.11 and PA B = a Draw a Venn dagram to llustrate the complete sample space for the events A and B. b Wrte down the value of PA and the value of PB c Fnd PAB. d Determne whether or not A and B are ndependent. 3. [Jan 01 Q6]The followng shows the results of a surve on the tpes of eercse taken b a group of 100 people. 65 run 48 swm 60 ccle 40 run and swm 30 swm and ccle 35 run and ccle 5 do all three a Draw a Venn Dagram to represent these data. 4 Fnd the probablt that a randoml selected person from the surve 10

11 b takes none of these tpes of eercse, c swms but does not run, d takes at least two of these tpes of eercse. Jason s one of the above group. Gven that Jason runs, e fnd the probablt that he swms but does not ccle. Tree dagrams 4. [Jan 006 Q4]A bag contans 9 blue balls and 3 red balls. A ball s selected at random from the bag and ts colour s recorded. The ball s not replaced. A second ball s selected at random and ts colour s recorded. a Draw a tree dagram to represent the nformaton. Fnd the probablt that b the second ball selected s red, c both balls selected are red, gven that the second ball selected s red. 5. [June 009 Q] On a randoml chosen da the probablt that Bll travels to school b car, b bccle or on foot s 1, 6 1 and 3 1 respectvel. The probablt of beng late when usng these methods of travel s 5 1, 5 and 10 1 respectvel. a Draw a tree dagram to represent ths nformaton. b Fnd the probablt that on a randoml chosen da Bll travels b foot and s late Bll s not late. c Gven that Bll s late, fnd the probablt that he dd not travel on foot [June 008 Q1] A dsease s known to be present n % of a populaton. A test s developed to help determne whether or not someone has the dsease. Gven that a person has the dsease, the test s postve wth probablt Gven that a person does not have the dsease, the test s postve wth probablt a Draw a tree dagram to represent ths nformaton. A person s selected at random from the populaton and tested for ths dsease. 11

12 b Fnd the probablt that the test s postve. A doctor randoml selects a person from the populaton and tests hm for the dsease. Gven that the test s postve, c fnd the probablt that he does not have the dsease. d Comment on the usefulness of ths test. 1 HW3 Answers 1. c 5% Venn dagram see eamsolutons 3 b PA = 0.54; PB = 0.33 c 67 d Not Independent 3 b 0.07 c 0.08 d 0.55 e b b 30 c c b c

13 HW4 Correlaton and Lnear Regresson Wrte our answers on a separate pece of paper. Show clear workng and mark answers Topcs covered n ths homework Pearsons product moment correlaton coeffcent measure of lnear correlaton between two varables - calculaton and nterpretaton Lnear regresson fndng a lnear relatonshp between varables lne of best ft Dependant/Independent or Eplanator/Response varable The dfference between correlaton and causaton Calculatons n, Σ, Σ, Σ, Σ, Σ S n S n n S Correlaton r S S S Lnear regresson a b b = S S, a = b Eercse A 1. Match up these correlaton coeffcents r= - 0.7, 0, 0.9, -0.99, 0.5, -0.3 A B C D E F 1b Calculate the Product moment correlaton coeffcent usng the nformaton below S = 36,, S = 9, S = 4 b Calculate the lne of best ft = a + b, usng the nformaton below S =, S = 5, = 6, = 7 13

14 . Before ou start ths queston thnk. Do ou epect there to be a lnk between rse n unemploment and wage ncrease? % Unemploment % ncrease n wages n a Calculate and Interpret Pearsons Product Moment Correlaton Coeffcent b Calculate the Lnear Regresson equaton = a + b c Usng our equaton estmate the % ncrease n wages when Unemploment s %. Ths s nterpolaton d Usng our equaton estmate the % ncrease n wages when Unemploment s 8%. Ths s etrapolaton. Comment on our result. Eercse B eam questons 1. [Jan 010 Q6] The blood pressures, p mmhg, and the ages, t ears, of 7 hosptal patents are shown n the table below. Patent A B C D E F G t P [ n=7, t = 341, p = 833, t = , p = , tp = ] a Fnd, S tt, S pp and S tp for these data. 4 b Calculate the product moment correlaton coeffcent for these data. c Interpret the correlaton coeffcent. 1 d Draw the scatter dagram of blood pressure aganst age for these 7 patents. e Fnd the equaton of the regresson lne of p on t. [P = a + bt] 4 f Plot our regresson lne on our scatter dagram. [calculate ponts on ths lne and plot them eg when t = 0, t = 80] g Use our regresson equaton to estmate the blood pressure of a 40 ear old patent.. [June 008 Q4] Crckets make a nose. The ptch, v khz, of the nose made b a crcket was recorded at 15 dfferent temperatures, t C. These data are summarsed below. t , v , tv , t 401.3, v 5.08 a Fnd S tt, S vv and S tv for these data. 14

15 4 b Fnd the product moment correlaton coeffcent between t and v. c State, wth a reason, whch varable s the eplanator varable. d Gve a reason to support fttng a regresson model of the form v a bt to these data. e Fnd the value of a and the value of b. Gve our answers to 3 sgnfcant fgures. f Usng ths model, predct the ptch of the nose at 19 C Etenson 1. Correlaton coeffcents can be ver deceptve. It s mportant to look at the graphs of the data too. Tr a google search on Anscombe s quartet wk.. Fnd out about other tpes of regresson. For eample: polnomal regresson, eponental regresson. 3. Practse our ICT sklls b dong queston or 3 usng a spreadsheet. HW4 Answers Eercse A 1a r = 4 3 b = a r= b = c 3.36% d % Eercse B 1. ab S tt = 1569, S PP = 770, S tp = 369, r = c Older patent have hgher blood pressure. e P = t g t = 40, P = 106 See eamsolutons for more detal. a S tt = S vv = S tv = b c temperature t as ths wll affect the crckets not vce versa d r s close to 1 whch supports lnear regresson e , f

16 HW5 Dscrete Random Varables Wrte our answers on a separate pece of paper. Show clear workng and mark answers Topcs covered n ths homework Epectaton EX = PX = Varance VarX = EX [EX] EX = PX = Standard Devaton StdevX = VarX Cumulatve Dstrbuton Functon F = PX Epectaton Algebra EaX + b = aex + b VaraX + b = a VarX StdevaX + b = astdevx 1. [June 008 Q3]The random varable X has probablt dstrbuton gven n the table below PX = p q Gven that EX = 0.55, fnd a the value of p and the value of q, 5 b Var X, 4 c EX 4.. [June 008 Q6]The dscrete random varable X can take onl the values, 3 or 4. For these values the cumulatve dstrbuton functon s defned b F = k 5 for =, 3, 4, where k s a postve nteger. a Fnd k. b Fnd the probablt dstrbuton of X. 3. [Jan 007 Q3]The random varable X has probablt functon Fnd 1 P X 1,, 3, 4, 5, a Construct a table gvng the probablt dstrbuton of X. b P X 5 c the eact value of EX. d Show that VarX = 1.97 to 3 sgnfcant fgures. 4 e Fnd Var 3X. 16

17 4. [June 011 Q3]The dscrete random varable Y has the probablt dstrbuton PY = a b 0.3 c where a, b and c are constants. The cumulatve dstrbuton functon F of Y s gven n the followng table F d 1.0 where d s a constant. a Fnd the value of a, the value of b, the value of c and the value of d. b Fnd P3Y [Jan 013 Q] The dscrete random varable X can take onl the values 1, and 3. For these values the cumulatve dstrbuton functon s defned b F = 3 k, = 1,, a Show that k = 13. b Fnd the probablt dstrbuton of X Gven that VarX =, 30 c fnd the eact value of Var4X 5. HW5 Answers 1 c a 1 q, p 0. 4 b.5 c =.9 b 3a 3 4 PX = PX = 36 b c 4.47 d 1.97 e a a = 0.1, b = 0.4, c = 0., d = 0.8 b b PX = 1 = 14 40, PX = = , PX = 3 = c Var4X 5 =

18 HW6 Normal Dstrbutons Wrte our answers on a separate pece of paper. Show clear workng and mark answers Ke words: Dstrbuton, mean, standard devaton, standard normal dstrbuton, contnuous, Formulae: Z = X μ σ Eample -The heghts of a group of students are Normall dstrbuted mean 174 cm Stdev 5 cm. What s the probablt that a randoml selected student has a heght less than 180 cm X~N174, PX < 180 = % X Z Stdev = 1. 5 Eercse A 1. The weghts of new born babes are normall dstrbuted wth mean 7.8lbs and standard devaton 0.6lbs. A bab s regarded as underweght f the are less than 5.5lbs. a Shade the area representng the probablt that a bab s underweght. b Wrte down 3 propertes of a Normal Dstrbuton.. Z ~ N0,1, draw a dagram for each queston and use the tables to fnd: a PZ < 0.3 d PZ < -0.4 b PZ < 0.36 e P0.78 < Z < 1.04 c PZ > 0.81 f P-0.3 < Z < Z ~ N0,1, draw a dagram for each queston and fnd the value of a such that: a PZ < a = d PZ < a = b PZ < a = e PZ > a = 0. [Use the lttle table] c PZ > a = f PZ < a = 0.95 [Use the lttle table] Eercse B Eam questons 1. [Jan 006 Q7] The heghts of a group of athletes are modelled b a normal dstrbuton wth mean 180 cm and a standard devaton 5. cm. The weghts of ths group of athletes are modelled b a normal dstrbuton wth mean 85 kg and standard devaton 7.1 kg. Fnd the probablt that a randoml chosen athlete [draw clear, labelled dagrams] a s taller than 188 cm, b weghs less than 97 kg. 18

19 c Assumng that for these athletes heght and weght are ndependent, fnd the probablt that a randoml chosen athlete s taller than 188 cm and weghs more than 97 kg. d Comment on the assumpton that heght and weght are ndependent. 1. [Jan 007 Q7]The measure of ntellgence, IQ, of a group of students s assumed to be Normall dstrbuted wth mean 100 and standard devaton 15. a Fnd the probablt that a student selected at random has an IQ less than The probablt that a randoml selected student as an IQ of at least k s b Fnd, to the nearest nteger, the value of k [Jan 008 Q6] The weghts of bags of popcorn are normall dstrbuted wth mean of 00 g and 60% of all bags weghng between 190 g and 10 g. a Wrte down the medan weght of the bags of popcorn. b Fnd the standard devaton of the weghts of the bags of popcorn. A shopkeeper fnds that customers wll complan f ther bag of popcorn weghs less than 180 g. 1 5 c Fnd the probablt that a customer wll complan. 4. [Jan 011 Q8] The weght, X grams, of soup put n a tn b machne A s normall dstrbuted wth a mean of 160 g and a standard devaton of 5 g. A tn s selected at random. a Fnd the probablt that ths tn contans more than 168 g. The weght stated on the tn s w grams. b Fnd w such that PX < w = The weght, Y grams, of soup put nto a carton b machne B s normall dstrbuted wth mean grams and standard devaton grams. c Gven that PY < 160 = 0.99 and PY > 15 = 0.90, fnd the value of and the value of. 6 19

20 HW6 Answers Eercse A 1a bell shaped; mean = mode = medan; smmetrcal; area under curve = probablt, tends to 0 for etreme values a b c 0.09 d e f a.1 b 0.14 c 1.19 d e f Eercse B 1a b c d Evdence suggests heght and weght are postvel correlated Assumpton of ndependence s not sensble a 0.74 b k = 1 3 a 00 g b 11.9 c a b w = c = 155, =. 0

21 HWX S1 June Gar compared the total attendance,, at home matches and the total number of goals,, scored at home durng a season for each of 1 football teams plang n a league. He correctl calculated: S = 10500, S = 130.9, S = 885. a Calculate the product moment correlaton coeffcent for these data. b Interpret the value of the correlaton coeffcent. 1 Helen was gven the same data to analse. In vew of the large numbers nvolved she decded to dvde the attendance fgures b 100. She then calculated the product moment correlaton coeffcent between 100 and. c Wrte down the value Helen should have obtaned. 1. An eperment conssts of selectng a ball from a bag and spnnng a con. The bag contans 5 red balls and 7 blue balls. A ball s selected at random from the bag, ts colour s noted and then the ball s returned to the bag. When a red ball s selected, a based con wth probablt 3 of landng heads s spun. When a blue ball s selected a far con s spun. a Cop and complete the tree dagram below to show the possble outcomes and assocated probabltes. Shvan selects a ball and spns the approprate con. b Fnd the probablt that she obtans a head. 1

22 Gven that Tom selected a ball at random and obtaned a head when he spun the approprate con, c fnd the probablt that Tom selected a red ball. Shvan and Tom each repeat ths eperment. d Fnd the probablt that the colour of the ball Shvan selects s the same as the colour of the ball Tom selects. 3. The dscrete random varable X has probablt dstrbuton gven b PX = 5 a 10 a 5 where a s a constant. a Fnd the value of a. b Wrte down EX. c Fnd VarX. 1 The random varable Y = 6 X. d Fnd VarY. e Calculate PX Y. 4. The Venn dagram n Fgure 1 shows the number of students n a class who read an of 3 popular magaznes A, B and C. One of these students s selected at random. Fgure 1

23 a Show that the probablt that the student reads more than one magazne s 6 1. b Fnd the probablt that the student reads A or B or both. c Wrte down the probablt that the student reads both A and C. 1 Gven that the student reads at least one of the magaznes, d fnd the probablt that the student reads C. e Determne whether or not readng magazne B and readng magazne C are statstcall ndependent. 5. A teacher selects a random sample of 56 students and records, to the nearest hour, the tme spent watchng televson n a partcular week. Hours Frequenc Md-pont a Fnd the md-ponts of the 1 5 hour and hour groups. A hstogram was drawn to represent these data. The 11 0 group was represented b a bar of wdth 4 cm and heght 6 cm. b Fnd the wdth and heght of the 6 30 group. c Estmate the mean and standard devaton of the tme spent watchng televson b these students. 5 d Use lnear nterpolaton to estmate the medan length of tme spent watchng televson b these students. The teacher estmated the lower quartle and the upper quartle of the tme spent watchng televson to be 15.8 and 9.3 respectvel. e State, gvng a reason, the skewness of these data. 6. A travel agent sells flghts to dfferent destnatons from Beerow arport. The dstance d, measured n 100 km, of the destnaton from the arport and the fare f are recorded for a random sample of 6 destnatons. Destnaton A B C D E F d f

24 [You ma use d = f = 3686 fd = 73.1] a On graph paper, draw a scatter dagram to llustrate ths nformaton. b Eplan wh a lnear regresson model ma be approprate to descrbe the relatonshp between f and d. 1 c Calculate S dd and S fd. d Calculate the equaton of the regresson lne of f on d gvng our answer n the form f = a + bd. 4 e Gve an nterpretaton of the value of b. Jane s plannng her holda and wshes to fl from Beerow arport to a destnaton t km awa. A rval travel agent charges 5p per km. f Fnd the range of values of t for whch the frst travel agent s cheaper than the rval The dstances travelled to work, D km, b the emploees at a large compan are normall dstrbuted wth D N 30, 8. a Fnd the probablt that a randoml selected emploee has a journe to work of more than 0 km. b Fnd the upper quartle, Q 3, of D. c Wrte down the lower quartle, Q 1, of D. 1 An outler s defned as an value of D such that D < h or D > k where h = Q Q 3 Q 1 and k = Q Q 3 Q 1. d Fnd the value of h and the value of k. An emploee s selected at random. e Fnd the probablt that the dstance travelled to work b ths emploee s an outler. TOTAL FOR PAPER: 75 MARKS END 4

25 5 Statstcs 1 Formula Sheet Probablt P P P P B A B A B A P P P A B A B A P P P P P P P A A B A A B A A B B A Dscrete dstrbutons For a dscrete random varable X takng values wth probabltes PX = Epectaton mean: EX = = PX = Varance: VarX = = PX = = PX = For a functon g X : EgX = g PX = Standard contnuous dstrbuton: Dstrbuton of X P.D.F. Mean Varance Normal, N 1 e 1 Correlaton and regresson For a set of n pars of values, n S n S n S The product moment correlaton coeffcent s n n n S S S r } }{ { The regresson coeffcent of on s S S b Least squares regresson lne of on s b a where b a

26 Normal Dstrbuton Tables z z z z z z z z z z PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION p z p z