Outside the high school program / Statistics 67 exercises corrected

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1. Table reading E.154 The table below shows greenhouse gas emissions in the European Union in millions of tonnes of CO 2 . equivalent In the last column, the Kyoto Protocol targets for greenhouse gas emission reductions or maximum allowable in-creases have been indicated for each country. For example: Germany must reduce its emissions by at least 21 % be-tween the years 1990 and 2010; Spain can increase them by a maximum of 15 % between the years 1990 and 2010. Some data have been deleted, and we propose to recover some of them in the following Q . C . M . . For information : to express the emissions of each greenhouse gas in tonnes of CO 2 equivalent, the emissions of each gas are weighted by a coefficient taking into account its warming power compared with that of CO 2 . This coefficient is 1 for CO 2 , 21 for CH 4 , 310 for N 2 O , 23 900 for SF 6 , 140 to 11 700 for HFCs and 2100 to 9 200 for PFCs. Émissions en 1990 Emissions en 2001 Variation entre 1990 et 2001 (en % ) Variation prévue entre 1990 et 2010 (en % ) Allemagne 1218 -18.3 -21 Autriche 78.4 9.6 -13 Belgique 141.3 150.2 6.3 -7.5 Danemark 69.5 69.4 -0.1 -21 Espagne 289.8 382.8 32.1 15 Finlande 77.3 4.7 France 558.6 0.4 Grèce 132.2 23.5 25 Irlande 53.4 70 13 Italie 545.4 7.1 -6.5 Luxembourg 6.1 -44.2 -28 Pays-Bas 219.7 4.1 -6 Portugal 61.4 83.8 36.5 27 Royaume Uni 657.2 -12 -12.5 Suède 70.5 -3.3 4 Ensemble De l’Union européenne 4108.3 -2.3 -8 Source : European Environment Agency, 2003. Part A Q.C.M. each question has three statements identified by the letters a , b , c of which only one is correct. In Table 1, provided in the appendix, the candidate must cir-cle the correct answer for each question. No justification is required. A correct answer earns 1 point ; an incorrect answer deducts 0.5 points ; no answer earns or deducts no points. If the total is negative, the mark is reduced to zero. 1 For the European Union as a whole, the quantity of green- house gases emitted between 1990 and 2001 was multi-plied by: a 0.977 b 1.023 c 0.023 2 Greenhouse gas emissions in Austria for the year 2001 accounted for around 0.1 million tonnes of CO 2 equiva-lent : a 85.9 million tonnes of CO 2 ; b 153.7 million tonnes of CO 2 equivalent ; c 88 million tonnes of CO 2 equivalent. 3 The percentage change in greenhouse gas emissions in Ire-land between 1990 and 2001 is equal to 0.1 % to within : a 23.7 % b 31.1 % c 16.6 % 4 Greenhouse gas emissions in Luxembourg for the year 1990 amounted to around 0.1 million tonnes of CO 2 equivalent : a 8.8 million tonnes of CO 2 ; b 13.8 million tonnes of CO 2 equivalent ; c 10.9 million tonnes of CO 2 equivalent. Part B We wish to know, for certain countries that have not yet reached their Kyoto Protocol targets in 2001, the rate of re-duction to be applied to 2001 greenhouse gas emissions in order to reach the quantities forecast for 2010. Greenhouse gas emissions are expressed in millions of tonnes of CO 2 . equivalent In the columns D , F and G , the results are rounded to the tenth. The contents of some cells are hidden. 1 a Which formula was entered in cell F2 and then copied down to cell F6 ? b What formula does cell F6 contain? c Complete the column F of table 2 given in the appendix. A result rounded to 0.1 million tonnes of CO 2 equiva-lent will be given. 2 a Belgium wants to meet the targets set under the Kyoto Protocol. Justify that it will have to reduce its greenhouse gas emissions between 2001 and 2010 by approximately 13 % b What formula has been entered in cell G2 and then copied down to cell G6 ? c Which country, listed in Table 2, will have to achieve the highest rate of emissions reduction between 2001 https://chingmath.fr sacados/154 123456ABCDEFGPaysEmissionsen1990Emissionsen2001Variationentre2001et1990(en%)Variationprévueentre1990et2010en%)Emissionsprévuesen2010Variationprévueentre2001et2010(en%)Belgique141,3150,26,3-7,5Danemark69,569,4-0,1-2154,9-20,9Espagne289,8382,832,115333,3-12,9Italie509,2545,47,1-6,5-12,7Portugal61,483,836,52778-6,9
and 2010 to meet its Kyoto Protocol targets? E.146 The F . M . I (international monetary fund) , created in 1945, has ˇ as its main mission to help coun-tries with balance-of-payments problems that are unable to borrow on the financial markets on affordable terms ı. 1 Its outstanding loans in 2003 were 85.3 billion dollars in 2003 , whereas for the financial year 2006 it was only 27 billion dollars. Express this change using a percentage, rounded to the nearest hundredth. 2 Of the year’s 2006 outstanding loans, African countries account for 7 % of the F . M . I ’s outstanding loans. The Democratic Republic of Congo is the country most in-debted to the F . M . I with a slate representing 29.1 % of African outstandings. a Faîtes a diagram representing this situation b Determine the amount of outstanding loans dedicated to African countries. c What is the amount of outstanding R . D . C . ? taken from Jeune Afrique 2006 E.156 new Caledonia - March 2004 - 9 points A private individual is fitting out the house he has just bought : he is having a new gas heater installed. He makes a model of the future bill on the basis of the informa-tion provided by his installer; the latter gives him the prices HT (excluding tax) . To obtain the prices TTC (inclusive of all taxes) , he must add to the price HT the amount of the TV A (value-added tax) ; this TV A is expressed as a per-centage of the price HT : it is 5.5 % for supplies (radiators, thermostat, boiler) and 19.6 % for labor. The unit price for labor is counted by the hour. The table, provided in Appendix 1, to be returned with the copy, represents elements of the spreadsheet on which the in-dividual has created his invoice model. Throughout the exercise will be rounded to the hundredth 1 From the information provided on the attached table : a Calculate the amount of TV A for a radiator of 1.20 m b Calculate the price HT of the thermostat. c Calculate the price HT of the boiler d Complete the cells C 2 , B 4 and B 5 of the table with the missing numerical values 2 What formula can be entered in cell C2 , before automati-cally copying it down to line 5, to obtain the amounts of TV A ? Complete cells C3 and C4 of the table with the missing numerical values. 3 What formula can be entered in cell E2 , before automat-ically copying it down to line 5, to obtain unit prices TTC ? 4 Calculate the price TTC per hour of labor and fill in cell E6 of the table with the missing numerical value. 5 What formula can be entered in cell G2 , before automat-ically copying it down to line 6, to obtain prices TTC ? Then complete column G of the table with the missing numerical values. 6 a The installer gives a discount of 4 % on the price HT of the boiler: which table cells will the numerical content change? b By how much will the final bill fall? c Will this amount be the same if the discount of 4 % is made on the price TTC of the boiler? Justify answer E.149 In 2004, in a medium-sized town, a census was taken of young people aged 10 to 15 who regularly played a team sport (soccer, handball) or an individual sport (tennis, judo) . It is assumed that each young person counted in this way plays just one sport. The city has been divided into four sec-tors : north, south, east, west. The results are grouped in the table given in appendix I (to be completed and returned with the copy) 1 a We want to calculate the totals per row. What for-mula should we write in cell F2 to obtain by copying it down to F6 the total number of young people per line? b We want to calculate, by sector, the frequencies of young people practicing an individual or team sport, relative to the census population. What formula should we write in cell B7 to obtain, by copying it to the right to F7 , these frequencies? In the following questions, percentages will be rounded to the tenth. 2 Complete the table given in appendix1 (to be returned with the copy) 3 Can we say that less than a third of the teenagers who responded to this survey seem to be more attracted to an individual sport than to a team sport? Justify the answer with a calculation 4 Assuming that every year the number of teenagers play-ing team sports increases by 5 % and that the number of teenagers practicing an individual sport decreases by 10 % , calculate: a the number of teenagers playing a team sport in 2005 in this city; b the number of teenagers practicing an individual sport in 2005 in this city; c the percentage change between 2004 and 2005 in the number of teenagers who will practice a sport in this city. https://chingmath.fr chapExoCorrec/146 sacados/146 sacados/156 12345678ABCDEFGDésignationdel’articlePrixunitaire(HT)TVAà5;5%TVAà19;6%PrixunitaireTTCQuantitésPrixTotalTTCradiateur1;20m49,004radiateur0;80m37,002thermostat17,343chaudière66,751maind’oeuvre25,5065Total chapExoCorrec/149 sacados/149
E.186 A company has two production lines : blue line, yellow line. An alarm system detects incidents that may occur on each of these two lines. In order to monitor the effectiveness of this alarm system, we observed, over a period of one month, the number of triggers of this system, as well as these failures (i.e. incidents occurring without the alarm being triggered) . The results were taken from a spreadsheet (table 1) . Some cells in this table have been masked. There were 52 incidents on the blue chain, 46 of which were detected by the alarm system. For the yellow chain, the alarm system was triggered 72 times, i.e. in 96 % of incidents on this chain. 1 a Calculate the total number of incidents on the yellow chain. b Complete Table 1. c What formula has been entered in cell D2 knowing that it has been copied down to D4 ? 2 In Table 2, whose cells are in percentage format, we seek to obtain the percentages in relation to the number of incidents observed on each channel. a Calculate the percentage of incidents on the blue chain for which the alarm did not work. The result will be rounded to 0.1 % . b Complete Table 2. c What formula has been written into cell B7 , then copied down into cells B8 and B9 ? 3 We’re interested in Table 3. a In cell C13 , we find the number 33 % . Give an inter-pretation of this number. b What formula has been entered in cell B12 , then copied throughout Table 3? 4 An alarm system’s failure rate is the percentage of in-cidents in which the alarm failed as a proportion of all incidents on the line as a function of the alarm system’s failure rate. The company considers that the alarm system is not ef-fective when the average cost of an intervention becomes greater than 1 500 e . a At what failure rate is the alarm system no longer con-sidered effective? The result will be given to the near-est 0.2 % . b Is the alarm system effective for the blue chain? for the yellow chain? Justify https://chingmath.fr 1234567ABCDEFNordSudEstOuestTOTALFootball15012575250600Handball50753085240Tennis35301550130Judo705020100240TOTAL3052801404851210Fréquenceen% sacados/186 12345678910111213ABCDtableau1chaînebleuechaînejauneTotalnombred’incidentspourlesquelsl’alarmeafonctionné4672nombred’incidentspourlesquelsl’alarmen’apasfonctionnéTotal52tableau2pourcentaged’incidentspourlesquelsl’alarmeafonctionné96%pourcentaged’incidentspourlesquelsl’alarmen’apasfonctionnéTotal100%100%tableau3chaînebleuechaînejauneTotalpourcentaged’incidentspourlesquelsl’alarmeafonctionné39%61%100%pourcentaged’incidentspourlesquelsl’alarmen’apasfonctionné67%33%100% Tauxdedéfaillancedusystèmed’alarmeen%0123456789101112Coûtmoyend’uneinterventioneneuros5001000150020002500
E.203 The three parts of the exercise are independent. Before children start elementary school, doctors and nurses from the Ministry of Education carry out a health check and measure the height (in meters) and weight (in kilograms) of each child. These two parameters are used to calculate the body mass index (BMI) , which is an indicator of possible overweight. Part I The graph in Appendix 2 shows different curves of body sur-face area S giving the BMI as a function of weight (between 16 and 32 kilograms) and height (between 1 meter and 1.3 meters) of the child. This appendix must be returned with the copy with all the construction lines necessary to solve part I. 1 Give a rounded value for the BMI of a child weighing 19 kilograms and measuring 1.09 meters. 2 Give a rounded value for the weight of a child with a BMI of 20 and a height of 1.22 meters. 3 What is the weight range for a child measuring 1.25 me-ters with a BMI between 12 and 16 ? Part II BMI is calculated using the following formula : BMI = P T 2 P denotes the child’s weight (in kilograms) and T their height (in meters) . 1 The weight and height of a sample of 15 six-year-old boys were recorded in order to calculate their BMI. These data are provided in the appendix. Without using a calculator, determine the median height, the first and third quar-tiles associated with height, and specify the interquartile range. Explain your approach. 2 Calculate the BMI of a child weighing 19 kilograms and measuring 1.09 meters. Which question in Part 2 does this result correspond to? 3 Based on the document provided in the appendix, is a 6-year-old boy measuring 1.20 meters tall and weighing 26 kilograms overweight? Justify your answer. 4 When a child is overweight but not obese, they are said to be moderately overweight. What BMI condition would apply to a 6-year-old boy to be considered moderately overweight? 5 Using the sample in the appendix, calculate the percent-age of boys who are overweight. What percentage of these boys are moderately overweight? Part III During a survey conducted during the 1999-2000 school year, the percentages of 6-year-old children who were overweight or not overweight were recorded according to their place of residence. These results are presented in the appendix. 1 Calculate the percentage of overweight children in rural areas. 2 Calculate the percentage of obese children in rural areas. 3 Clearly state whether the following statements are true or false, or whether the data does not allow for a clear conclusion : In the Paris metropolitan area, there are more than 3 obese children for every 10 overweight children. The number of overweight children in towns with fewer than 50 000 inhabitants is very slightly lower than the number of overweight children in towns with between 50 000 and 200 000 inhabitants. Weight (P) 15 18 19 20 22 23 23 23 Height (T) 1.05 1.05 1.09 1.10 1.02 1.17 1.15 1.16 IMC 13.61 16.33 16.53 21 ; 15 16.80 17.39 17.09 Weight (P) 24 24 25 25 26 27 27 Size (T) 1.20 1.20 1.25 1.30 1.20 1.24 1.30 IMC 16.67 16.67 16.00 14.79 18 ; 06 17.56 15.98 International thresholds for body mass index (BMI) to define overweight and obesity in children Age BMI for overweight BMI for obesity (in years) Boys Girls Boys Girls 5 ans 17.42 17.15 19.30 19.17 5 years and a half 17.45 17. 20 19.47 19.34 6 ans 17.55 17.35 19.78 19.65 6 years and a half 17.71 17.53 20. 23 20.08 Lecture : for a 6-year-old boy measuring 120 cm , the over-weight BMI is 17.55, which corresponds to a weight of 25.272 kg ; above this, he’s considered overweight. The threshold for obesity is 19.78 , which corresponds to a weight of 28.483 kg . For a girl of the same age and height, the threshold for overweight is 24.977 kg et that for obe-sity 29.296 kg . Source : COLE et al. British medical journal 2000. 320 https://chingmath.fr sacados/203 France - Septembre 2006 - 12 points IMC10IMC12IMC14IMC16IMC18IMC20IMC22IMC24IMC26IMC28IMC30Taille11.051.11.151.21.251.3Poids1617181920212223242526272829303132
Type of agglomeration meration % of children without being overweight % children who are overweight % children with moderate overweight % children who are obese rurales 87.2 9.2 moins de 50 000 habitants 86.9 13, 1 9.9 3.2 entre 50 000 et 200 000 habitants 86, 8 13.2 9.7 3.5 entre 200 000 et 2 000 000 habitants 85.7 14, 3 10.2 4.1 agglo-mération parisienne 83.4 16.6 11, 6 5 E.1772 Les two parts are independent Part A In 2003, on average, every day in France, almost 49.1 people aged 12 to 18 were victims (injured or killed) of road acci-dents. Mopeds have the highest average number of victims (26.0 victims per day) , followed by passenger cars (12.8 vic-tims per day) , pedestrians (5.0 victims per day) and cyclists (2.5 victims per day) . The table below gives the breakdown of road accident victims aged 12 to 18 by age and user category for 2003. Age Pié-tons Cyclis te Cyclo moto ristes Moto cycliste Usagers de voiture de tourisme Autres usagers* Total 12 ans 330 147 49 22 247 24 819 13 ans 263 165 122 23 279 26 878 14 ans 234 135 1 010 35 292 37 1 743 15 ans 260 139 1 701 57 396 27 2 580 16 ans 269 111 2 549 136 610 25 3 700 17 ans 223 111 2 457 259 925 39 4 013 18 ans 248 115 1 605 214 1 940 59 4 181 Total 1 827 923 9 493 745 4 689 237 17 914 * Users of vans, heavy goods vehicles, transport in com-mum. . . Data from : http://eduscol.education.fr/D0187/default.htm 1 Moped riders a Check that in 2003, every day in France there was an average of 26.0 moped riders aged 12 to 18 killed in road accidents. b Calculate the percentage, among moped riders, of ac-cidents involving under-13s, rounded to 0.1. c Calculate the percentage of injured moped riders un-der 18 years of age out of all injured riders, rounded to 0.1. 2 Motorcyclists a In 2003 there were, all ages combined, 16 629 motor-cyclists involved in road accidents. Is the following statement true or false? Justify the answer: ˇAmong motorcyclists involved in accidents in 2003, less than one in twenty was a young person aged be-tween 12 and 18ı. b All ages combined, there were 18 518 motorcyclists killed in road accidents in 2002 and 16 629 in 2003. Calculate the percentage change in the number of mo-torcyclists involved in road accidents between 2002 and 2003. Round the result to 0.1. Part B The graph below gives, for the 12-24 age group, the number of road accident casualties in 2003 by age for three categories of user : moped riders, motorcyclists and passenger car users. Victimes of accidents involving mopeds, motorcycles or passenger cars 1 a Indicate the ages for which, in 2003 the number of passenger car casualties exceeded 2 000. the number of moped casualties is approximately equal to 1 000. b With the precision allowed by the graph, determine the number of moped riders aged 20 who were victims of a road accident. 2 a At what age is the maximum number of passenger car accident victims? b What phenomenon(s) can explain the fact that as age increases, the number of passenger car accident victims increases rapidly and then gradually decreases? https://chingmath.fr chapExoCorrec/1772 sacados/1772 France - Septembre 2006 - 9 points
E.4745 In a city, a concert hall has scheduled 40 concerts during the 2004/2005 season. The expected audience numbers are shown in the histogram in Appendix 3. For example, the manager believes that six concerts will attract between 500 and 700 spectators during the 2004/2005 season. 1 Complete the following staffing table : Classe [100;500[ [500;700[ [700;900[ [900;1100[ [1100;1300[ [1300;1500[ Number 2 Use the calculator to determine the mean and standard deviation of this series. 2. Medians and quartiles E.1958 Give the range, median, first and third quartile of the following series : 34 38 39 41 42 43 44 45 45 47 47 48 49 50 51 51 52 52 53 54 55 55 55 55 55 55 55 56 56 57 58 58 58 59 59 59 60 62 62 62 62 63 64 65 66 66 66 66 67 68 68 73 74 74 75 75 79 81 81 85 E.4740 Give the range, median, first and third quartile of the following series : 34 38 39 41 42 43 44 45 45 47 47 48 49 50 51 51 52 52 53 54 55 55 55 55 55 55 55 56 56 57 58 58 58 59 59 59 60 62 62 62 62 63 64 65 66 66 66 66 67 68 68 73 74 74 75 75 79 81 81 85 E.196 Consider four statistical series whose char-acter values have been represented on a graduated line by points (two individuals do not have the same character value) Series 1: Series 2: Series 3: Series 4: 1 Determine the total number in each series. 2 Plot the first quartile, median and third quartile on each of the graduated lines. E.1920 Determine the range, median, first and third quartile of the following statistical series : 0,250 0,260 0,290 0,290 0,300 0,300 0,310 0,310 0,320 0,320 0,320 0,320 0,330 0,330 0,330 0,340 0,340 0,340 0,340 0,350 0,350 0,350 0,350 0,305 0,360 0,360 0,370 0,370 0,370 0,370 0,380 0,380 0,380 0,390 0,390 0,390 0,390 0,400 0,400 0,410 0,410 0,420 0,420 0,420 0,430 0,430 0,450 0,460 0,470 0,470 E.4737 On a graduated line, a teacher has or-dered the grades of these four classes of 10th graders. Here are their representations : Series 1: Series 2: Series 3: Series 4: The aim of the exercise is to divide each of the classes into four parts ˇ of equal size ı representing : The weakest quarter The medium-low quarter The medium-strong quarter The strong quarter 1 Show the median value of the series on each of the grad-uated lines. 2 Complete the splitting of the series by re-splitting each part in half. 3 Can you give a qualitative judgment of these classes? https://chingmath.fr chapExoCorrec/4745 sacados/4745 Un concert100500700900110013001500G sacados/1958 chapExoCorrec/4740 sacados/4740 sacados/196 N N N N sacados/1920 Extrait Liban -- juin 2004 chapExoCorrec/4737 sacados/4737 N N N N
E.4739 1 Here are the scores of four groups of students on the mock exam. Fill in the boxes for the different indicators below : Group 1 Group 2 Group 3 Group 4 Notes 5 - 6 - 10 10 - 11 12 - 12 14 6 - 8 - 8 8 - 10 - 11 14 - 15 8 - 8.5 8.5 - 9 11 - 11 12 - 12 6 - 6 - 7 8 - 10 - 11 11 - 15 Average Range Median 2 Compare qualitatively in light of the indicators calcu-lated previously: a Group 1 and group 2 b Group 2 and group 4 c Group 1 and group 3 3. Mustache box E.1970 Consider four statistical series whose char-acter values have been represented on a graduated line by points (two individuals do not have the same character value) Series 1: Series 2: Series 3: Series 4: Draw the box plots corresponding to each of these series di-rectly on the graduated lines. E.4751 On a graduated line, a teacher has or-dered the grades of these four classes of 10th graders. Here are their representations : 1 Draw the box diagram for each of these classes. 2 Compare these statistical series qualitatively. E.167 On 1 er January 2005, the table below gives the prices, in euros, per liter of diesel in fifteen European countries. The table shows : price HT (price excluding tax) the amount of domestic tax each country adds to the pre-tax price ; the rate of TV A applied after adding the domestic tax; the price TTC (price inclusive of all taxes) Prix diesel in Europe at 1 er January 2005 Example : In the UK, the price of diesel excluding tax is 0.302 euro plus domestic tax of 0.693 e . Then to the price ex-cluding TV A of 0.995 e is applied a TV A of 17.5 % , leading to a price TTC 1.169 e . Part A - Table work The table is a spreadsheet. The columns D and F required the https://chingmath.fr chapExoCorrec/4739 sacados/4739 sacados/1970 chapExoCorrec/4751 sacados/4751 02468101214161820Classe 2 - AClasse 2 - BClasse 2 - CClasse 2 - D sacados/167 1234567891011121314151617ABCDEFPaysPrixHTTaxeintérieurePrixhorsTVATVAen%PrixTTCLuxembourg0,3250,2530,57815,000,665Grèce0,3240,2510,57518,000,679Portugal0,3060,2920,59817,000,700Espagne0,3300,2970,62716,000,727Belgique0,3070,3050,61221,000,740Autriche0,3400,2890,62920,000,755Irlande0,3420,3270,66921,000,809France0,3010,3920,69319,600,829Pays-Bas0,3400,3580,69819,000,831Danemark0,3400,3580,69819,000,831Suède0,3300,3470,70125,000,846Finlande0,3540,3470,70125,000,876Allemagne0,3090,4700,77916,000,904Italie0,3510,4030,75420,000,905RoyaumeUni0,3020,6930,99517,501,169Moyenne
use of formulas to be filled in. 1 What formula was entered in D2 , then recopied to D16 to obtain the price excluding TV A ? 2 What formula was entered at F2 , then recopied to F16 to obtain the price TTC ? 3 We want to obtain in cell B17 the average price before tax. What formula can be entered in this cell? Part B - Study of prices before tax The statistical series made up of prices before tax has been studied and the characteristics given below have been ob-tained : mean : X ≈ 0.324 : standard deviation: ff ≈ 0.019 first and third quartiles: Q 1 =0.306 and Q 3 =0.34 ; median : M e =0.325 1 Using the graded axis below, construct the box plot for this series. The first and third quartiles, median, maxi-mum and minimum of the series will be shown. 2 Give the number of countries whose price HT of diesel belongs to the interval [ x − ff ; x + ff ] . Part C - Price study TTC We are now interested in the TTC price of diesel fuel in these fifteen countries. 1 Calculate the mean x of this series. The result will be rounded to the thousandth. 2 Read from the diagram below, the median, first and third quartile of this series of TTC diesel prices. Values will be given with the precision permitted by the diagram. 3 Q.C.M. Answer the two questions below by choosing the correct answer from the three proposals. No justification is re-quired. To answer, copy the chosen answer on your copy. A cor-rect answer earns 0.5 point, a wrong answer deducts 0.25 point and no answer adds or deducts no points. A nega-tive point total is reduced to zero. a The median of the series of fifteen prices TTC is : The 8 e value ; the half-sum of the 7 e and the 8 e value ; the half-sum of the 8 e and the 9 e value. b The first quartile of the series of fifteen TTC prices is : the 3 e value ; la 4 e value ; the half-sum of the 3 e and the 4 e value. 4 A journalist writes :ıIn Europe, taxes on diesel harmonize prices within the European Union.ı Confirm or deny this statement by arguing. Part D - Study of prices in France Q.C.M. Answer the two questions below by choosing the correct an-swer from the three proposals (where the results have been rounded to 1 % ) . No justification is required. To answer, copy the chosen answer on your copy. A correct answer earns 1 point, a wrong answer deducts 0.5 point and no answer adds or deducts no points. A negative point total is reduced to zero. 1 In France, the domestic tax on diesel represents : 230 % of the price HT ; 57 % price HT ; 130 % of price HT . 2 In France, the overall percentage of taxes applied to price HT to obtain price TTC is : 175 % 150 % 64 % E.169 lebanon ffl June 2006 ffl 11 points In a region of eastern France, air pollution is monitored daily, hour by hour, by a network of 21 measuring stations. Among these, let’s consider the station noted U , which is located in an urban area, the station I , in an industrialized area and the station R , in a rural mid-mountain area. Part A In this part, we compare the measurements obtained at sta-tions U and I for sulfur dioxide ( SO 2 ) , one of the pollutants measured. Concentrations of this pollutant are expressed in millionths of a gram per cubic meter of air (in this exercise, this unit is noted µ = g ) . The box plots drawn below relate to hourly measurements of the pollutant at stations U and I , for the day of November 16, 2004. The ends of the diagrams correspond to the minimum and maximum values. For example, at station U , the maximum value recorded was 8 µg = m 3 1 For each of the two stations, indicate the median and calculate the interquartile range as well as the range of the measurement series 2 Indicate, by graphical reading and specifying the statis-tical parameters used, on which station(s), on that day: a measurement dispersion was greatest b at least half the measurements were less than or equal to 5? c 75 % of measurements at least were less than or equal to 6? Part B In this part, station R . is considered The table below gives the hourly readings, for the same day of November 16, 2004, what concerns the pollu-tant ozone ( O 3 ) . Concentrations are expressed in millionths of a gram per cubic meter of air. https://chingmath.fr 0,3100,3200,3300,3400,3500,360 0,60,70,80,90,100,110,12 chapExoCorrec/169 sacados/169 Liban -- Juin 2006 -- 11 points 012345678910111213StationIStationU
Heure 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10 h 11 h 12 h Concen-tration 78 79 77 59 57 65 65 67 68 67 59 54 Heure 13 h 14 h 15 h 16 h 17 h 18 h 19 h 20 h 21 h 22 h 23 h 24 h Concen-tration 64 68 78 74 72 72 76 77 76 74 77 76 1 What are the minimum and maximum values of this se-ries? 2 Determine, with justification, the median and quartiles Q 1 and Q 3 of this series. 3 Construct the box diagram, taking half a centimetre as the graphical unit. Part C The graph below gives, for the pollutant, the average daily results for November 2004 at stations U , I and R . 1 a On which days in November was the ozone concen-tration at least 30 µg = m 3 for station U ? b Which days of the same month was it at most 60 µg = m 3 for station R ? 2 Station I sensor read 58 µg = m 3 on November 11 and 45 µg = m 3 on November 13. On November 12, the sensor was out of order. The technician decided to replace the missing value with one obtained using linear interpola- tion. What value did he obtain? Justify the answer, specifying the procedure followed. E.159 Part A The table below gives the breakdown of 225 cross-country skiers from two sports clubs : the ˇ Fondus de la glisse ı and the ˇ Glisse plaisir ı, according to their average time on a typ-ical race. Times are grouped into half-hour amplitude bands. Club Time in heures TOTAL [0.5 ;1[ [1 ;1.5[ [1.5 ;2[ [2 ;2.5[ [2.5 ;3[ [3 ;3.5[ [3.5 ;4[ Fondu de glisse 6 21 37 45 22 7 0 138 Glisse plaisir 0 0 1 10 44 29 3 87 TOTAL 6 21 38 55 66 36 3 225 1 Of the ˇ Fondus club runners at glisse ı, what percentage have an average time in the [1.5 ; 2[ ? 2 Of all the riders, what percentage have average times in the range [1.5 ; 2[ ? 3 Lucas claims that more than half the riders have an av-erage time strictly under 2.5 h . Is he right? Justify your answer with a calculation. Part B We’re interested in the club ˇ Glisse plaisir ı. 1 Below is an extract from the ranking of the 87 runners in this club according to their average time over this race. Using this extract, determine the median, first and third quartile of the series of average times of these 87 runners. Coureur n o 1 2 · · · 19 20 21 22 23 24 25 · · · 38 39 40 Temps h 1.98 2.01 · · · 2.69 2.7 2.7 2.74 2.75 2.76 2.77 · · · 2.87 2.87 2.88 Coureur n o 41 42 43 44 45 46 · · · 63 64 65 66 67 · · · 86 87 Temps h 2.89 2.89 2.89 2.89 2.89 2.9 · · · 3.08 3.1 3.1 3.11 3.11 · · · 3.6 3.67 2 The box plot of the average time series of the ˇ Fondus de la Glisse ı club runners is given. The ends of the whiskers correspond to the minimum and maximum times. Construct on the same drawing, with the precision al-lowed by the scale, the box diagram of the series of aver-age times of the riders of the ˇ Glisse plaisir ı club. 3 Using the two box plots, compare the results of skiers from the two clubs. Argue. Part C To study performances, two friends Théo and Clément recorded in a table their times achieved during 8 training sessions on this typical race. The table was created using a spreadsheet program. The table cells are in the format : num-ber, 2 decimal places. https://chingmath.fr stationRstationUstationI024681012141618202224262830concentrationenmillionièmedegrammesparmètrescubed’air0102030405060708090100 chapExoCorrec/159 sacados/159 Asie - Juin 2006 - 8 points 0123453equartile:2,42Médiane:2,061erquartile:1,63Fondus de la glisseGlisse plaisir
We read that Clément put in for his 1 er training : 2 hours 25 minutes 57 seconds, or 2.43 hours. 1 a Which formula has been written into cell E4 , then copied down to E11 ? b Which formula was written into cell F4 , then copied down into formula F11 ? c What formula was entered in cell E12 to calculate Clé-ment’s average time? 2 The two friends want to join one of the two clubs next year. They compare their average times with those of skiers in both clubs. They would like to be in the same club and be in the top quarter of skiers. Is their wish achievable? Explain your answer. E.172 new Caledonia ffl March 2006 ffl 10 points We want to tile a rectangular room of 6 m in length and 4 m in width, therefore of area 24 m 2 , using tiles of 2 col-ors : red and gray. What’s more, there are two types of tile in each color: patterned tiles and plain tiles. So there are a total of 4 tile patterns. Parts A , B and C of this exercise are independent. Part A 1 Each tile is a square of 0.2 m side. What is the minimum number of tiles required to tile the room? 2 For installation purposes, it is necessary to buy at least 672 tiles, of which 25 % in red, the rest in grey. It is also planned to lay 1 = 3 patterned tiles for each color, the rest being plain. Calculate the minimum number of tiles of each pattern to be purchased and complete the table below. Rouges Gris Total Motifs 56 Unis Total 504 3 Tiles are sold in packs of 15 of the same model. a Calculate the minimum number of packs of each kind required and complete the table below : Rouges Gris Total Motifs 4 Unis Total 47 b Justify that the minimum total number of packets re-quired is indeed equal to 47. Part B In packs of 15 tiles, we studied, out of a batch of 1 000 packs, the number of packs containing n defective tiles ( n being a number between 0 and 15) and we obtained the following ta-ble : n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Nombre de paquets compor-tant n carreaux défec-tueux 30 80 170 440 190 70 20 0 0 0 0 0 0 0 0 0 1 Determine the median of this statistical series. 2 a Determine first quartile Q 1 b Recall the definition of the third quartile Q 3 . Calcu-late it. c Draw a box plot of this series in the reference frame below. https://chingmath.fr 12345678910111213141516171819202122232425ABCDEFTempsdeClémentTempsenheure/minutes/secondesTempsdeThéoTempsenheure/minutes/secondestempsensecondestempsenheuresheuresminutessecondes1erentraînement2255787572,432≥èmeentraînement2232686062,393≥èmeentraînement2203984392,344≥èmeentraînement227788272,455≥èmeentraînement2241986592,416≥èmeentraînement2213784972,367≥èmeentraînement2252187212,428≥èmeentraînement2195683962,33Tempsmoyen:2233386132,39tempsensecondestempsenheuresheuresminutessecondes1erentraînement2503102032,832≥èmeentraînement24918101582,833≥èmeentraînement24827101072,814≥èmeentraînement24759100792,805≥èmeentraînement24939101792,836≥èmeentraînement24826101062,817≥èmeentraînement2503102032,838≥èmeentraînement24847101272,81Tempsmoyen:24912101452,82 chapExoCorrec/172 sacados/172 012345678910
Part C We note (in euros) a the price of a packet of tiles and b the fixed delivery charge whatever the number of packets deliv-ered. The graph calculates the price paid by the customer as a func-tion of the number of tile packs purchased. 1 Graphically determine the price paid by the customer for the purchase of 50 packs of tiles. 2 Calculate the price a of a packet of tiles and the amount b of the delivery charge. Then calculate the price paid by the customer for 47 packages delivered. E.170 A competition features two rounds. 30 participants take part. Here is the table showing the scores obtained by the participants : Notes on 20 Effectifs Proof n o 1 Proof n o 2 5 0 3 6 6 0 7 5 5 8 8 0 9 1 8 10 3 0 11 0 3 12 2 4 13 0 0 14 1 1 15 2 4 16 2 2 1 Determine the median, first and third quartile of each of the two series. 2 Construct on the graduated lines below the box plots corresponding to the two series. Statistical series E 1 - box plot Statistical series E 2 - box plot https://chingmath.fr Nombre de paquets0102030405060Prix à payer1002003004005006007008009001000 sacados/170 Extrait Juin 2004 - Centres Etrangers 01020 01020
E.164 The table (incomplete) belowbelow gives the distribution of the 800 farm managers in a region ac-cording to their age and the Useful Agricultural Area (UAA) of their farm. Area is expressed in hectares ( ha ) and age in years. 0;10 10;30 30;50 50;100 Total 15;25 2 1 5 3 25;35 21 16 28 84 35;45 40 33 59 148 45;55 17 53 123 55;65 110 60 70 57 297 Total 190 180 270 800 Part A 1 Complete the table (the completed columns correspond-ing to a Useful Agricultural Area of 10 shouldbecopiedontothecopy ; 30 and 30 ; 50 ) . 2 Percentages asked in this question will be rounded to 0.1 % . a Among farm managers, what percentage are in the age range 25 ; 35 . b Among farm managers, what percentage are strictly under 45 years of age and own at least 30 ha of Useful Agricultural Area? c Among farm managers aged 55 or over, what percent-age have a Useful Agricultural Area of less than 10 ha ? d Among farm managers with a Useful Agricultural Area of less than 10 ha , what percentage are aged 55 or over? Part B 1 a How many farm managers are strictly under 45? Strictly under 55? b Explain why the median age of farm managers is nec-essarily between 45 and 55. To determine the median age, the distribution of ages in the class 45 ; 55 is given by the following table : Age 45 46 47 48 49 50 51 52 53 54 Number 18 21 24 31 30 31 3 27 28 20 How many farm managers are 45 years old or younger? Justify that the median age is 51. c The first and third quartiles of the age series are 42 and 58. Construct a box plot of this series, taking as extreme values 18 and 65. We’ll choose 2 mm as the scale for one year. 2 The box plot of the ages of farm managers from 50 ha to 100 ha and that of farm managers under 10 ha are shown below. A journalist wrote : ˇAs a whole, heads of 50 to 100 ha are younger than heads of less than 10 ha .ı Comment on this statement using these box plots. E.1824 In India, a population census is taken every ten years. The last census was carried out in 2001. It revealed the dis-tribution of India’s population according to various criteria including age, gender, place of residence, and took stock of India’s literacy. (source: Census of india 2001) A literate person being one who can read and write, children aged 0-6 have been excluded from the statistics. Here is an extract from the data collected on the population aged 7 and over (in millions of inhabitants) . Population aged 7 and plus (in millions of inhabitants) Hommes Femmes Total Rural 318 301 619 Urban environment 131 119 250 Total 449 420 869 Population aged 7 and over non alphabetic (in million inhabitants) Hommes Femmes Total Rural 91 161 252 Urban environment 18 32 50 Total 109 193 302 1 The following statements relate to India’s population aged 7 and over in 2001. For each of these statements, say whether it is true or false, justifying the answer. a Less than one in four men is non-literate. b At least two-thirds of non-literates are women. c In urban areas, one in five people is non-literate. d More than 80 % of non-literate women live in rural ar-eas. 2 In an article published by UNESCO, we read : ˇRecent statistics show a steady decline in the number of non-literates in the world: literacy is progressing slowlyı. a Calculate the percentage (rounded to the unit) of lit-erates among India’s inhabitants aged 7 and over, in 2001. b India is made up of 35 states. A state’s literacy rate is said to equal x when the per-centage of literate inhabitants in that state is x % . The following table gives the literacy rate, rounded to the unit, recorded in 2001 in each of India’s 35 states : https://chingmath.fr sacados/164 tranched’âgeSAU Exploitation de50haà100ha2442475465 Exploitation de moins de10ha1843576063 chapExoCorrec/1824 sacados/1824
48 54 54 55 57 60 61 61 63 64 64 64 65 67 67 69 69 69 70 70 70 72 73 74 77 77 81 81 81 82 82 82 88 88 91 Thus, in 2001, in one of India’s states, the literacy rate is equal to 48, which means that the percentage of literate inhabitants in this state is equal to 48 % . Determine the median, first quartile and third quartile of the series of literacy rates found in the 2001 census in each of India’s 35 states. 3 The literacy series recorded in the 1991 census in each of the 35 states of India is represented by the box plot given in appendix 2 where values 37 and 90 are the minimum and maximum values of the series. a Show on the same graph, the box plot of the series of literacy rates recorded in the 2001 census in each of India’s 35 states. b By comparing these two diagrams, give two precise arguments for asserting that literacy has clearly pro-gressed in India between 1991 and 2001. E.2028 A car rental company owns a fleet of 800 vehicles of three different makes A , B and C . Within each brand, the company has two vehicle models : ˇEssenceı or ˇDieselı. During the year, each vehicle may be immobilized to undergo maintenance, adjustments, oil changes, repairs, etc. For all 500 vehicles ˇDieselı of the company, we studied, dur-ing the year 2005, the number of days of immobilization. The following statistical series S was obtained : Nombre de journées d’immobilisation 1 2 3 4 5 6 7 8 Nombre de véhicules concernés 11 34 86 121 120 88 28 12 1 Calculate the mean x of the series S (the result will be rounded to the nearest 0.1) . 2 Determine the median m of the series S . 3 Determine the first quartile Q 1 of the series S . The third quartile Q 3 is assumed to equal 6. 4 Using the axis shown in appendix , to be returned with the copy, draw the box plot of the S series. 5 Calculate the standard deviation of the series S (the re-sult should be rounded to the nearest hundredth) . 4. Mustache box: reading E.4748 Below are the box plots for the 6 second-year classes in a high school during the second-quarter com-mon assessment : 1 Give the marks corresponding to the median, first, third quartile of the Seconde 1 mark series. 2 In Seconde 2, can we say that at least one in two students has a mark of 10 or less? Justify. 3 In which Seconde classes can we say that at least 75 % of students have a mark of 13 or below? Justify. 4 For Seconde class 3, give the interquartile range. 5 In Seconde 5, what percentage of students scored 10 or above? https://chingmath.fr sacados/2028 chapExoCorrec/4748 sacados/4748 01234567891011121314151617181920Seconde 1Seconde 2Seconde 3Seconde 4Seconde 5Seconde 6
E.4744 The Paris Montsouris meteorological ob-servatory has been continuously recording outdoor tempera-tures since 1872, and provides annual averages based on these readings. The aim of this exercise is to compare these aver-ages by twenty-year periods between 1880 and 2000. To clar-ify vocabulary, we’ll call ˇannual temperatureı the average of temperatures recorded in a given year (days and nights) , expressed in degrees Celsius and rounded to 0.05 o C . Sources Météo France The document below shows box plots constructed from annual temperatures over each twenty-year period between 1881 and 1980. On each of these diagrams, the median, first and third quartiles have been plotted. The ends of the ˇmoustachesı mark the minimum and maximum of this series. For each of the following propositions, indicate whether it is true, false or undecidable (in the case where the document would not allow us to know whether the proposition is true or false) . Justify the answer. 1 The maximum annual temperature was 12.65 o C pendant a century, from 1881 to 1980. 2 The annual temperature range was 2.25 o C during one century, from 1881 to 1980. 3 During one century, from 1881 to 1980, at least thirty years had their annual temperature below 11.5 o C . 4 1961 was the coldest year over the period 1901-1980. E.1957 The results obtained by 199 students from six classes in the second year of secondary school were used to construct the six box plots shown below 1 Give the scores corresponding to the median, first to third quartile of the Seconde 1 score series. 2 In Seconde 2, can we say that at least one in two students has a mark of 10 or less? Justify. 3 In which Seconde classes can we say that at least 75 % of students have a mark of 13 or below? Justify. 4 How can we explain that in Seconde 5, the average mark is lower than the median mark? 5 For Seconde 3, give the interquartile range. 6 In Seconde 5, how many students scored 10 or above? E.4747 A pétanque boules factory designs com-petition boules of different masses and diameters. The three masses offered are 700 g , 720 g and 745 g and, for each of these masses, three diameters are proposed : 71 mm , 75 mm , 79 mm . A regional champion decides to buy 720 g balls, but hesitates over the diameter. To make his choice. He places a jack at 9 meters, points 200 times with each of the balls of different diameters and measures the distance to the jack. Here are the decile-pruned box plots representing this test. The ends of the diagram are the first and ninth decile respectively. Here are some of the player’s feelings after the test : 1 " With the 79 mm ball, I managed some very good throws, but also some very bad ones. " 2 " With the 71 mm ball, I had very good sensations, half my throws were within 16 cm of the jack and I managed some very nice ones. " 3 " But my preference is for the 75 mm ball, with which I’m more consistent. " Associate the corresponding box diagram with each type of ball. Justify your answer. https://chingmath.fr chapExoCorrec/4744 sacados/4744 1010.51111.51212.5131881à19001901à19201921à19401941à19601961à1980 sacados/1957 chapExoCorrec/4747 sacados/4747 46810121416182022242628303234cmCBA
E.4783 The diagram below shows box plots of the averages obtained by students in three classes of second year at a school: 1 a Give the range of averages for the class 2 o B . b Give the interquartile range of the class 2 o A . 2 Say whether the following statements are true, false or undecidable : a At least 50 % of second graders at this school had an average above 9 . b At most 25 % of second graders at this school had an average below 7 . E.4827 A school has three sophomore classes. The box plots below represent their scores on a common test : Based on their scores, we also constructed the following three tables of student numbers : a Note 0;4 4;8 8;12 12;16 16;20 Staff 4 4 10 3 1 b Note 0;4 4;8 8;12 12;16 16;20 Staff 3 2 6 8 4 c Note 0;4 4;8 8;12 12;16 16;20 Number of employees 6 4 5 5 2 Associate each box plot with the corresponding table of em-ployee numbers. 5. Cumulative increasing frequency curve E.4738 Here is the table showing the students’ grades in the middle school diploma exam: Note 0 ; 4 4 ; 8 8 ; 12 12 ; 16 16 ; 20 Number 5 32 61 80 15 Freq. Cumulative frequency Ascending 1 What is the modal class of this statistical series? 2 Calculate the school’s average for this exam, rounded to the nearest tenth. 3 a Complete the table by rounding the frequencies to the nearest thousandth. b Construct an orthonormal coordinate system where the scores will be represented on the x-axis ( 1 cm =2 points) and on the y-axis ( 1 cm =8 students) . Plot the cumulative frequency curve on this graph. c Deduce the value of the median. (Leave the construc-tion lines visible) E.4750 66 Météo-France weather stations spread across France have provided annual temperature data for each year between 1901 and 2006. The cumulative frequency polygon shown below was ob-tained : 1 Using the graph above, determine the values for this sta-tistical series of the first quartile, the median, and the third quartile. 2 a Complete the following frequency table : https://chingmath.fr chapExoCorrec/4783 sacados/4783 024681012141618202oA2oB2oC chapExoCorrec/4827 sacados/4827 012345678910111213141516171819202o12o22o3 chapExoCorrec/4738 sacados/4738 chapExoCorrec/4750 sacados/4750 010.51111.51212.51313.514102030405060708090100110
Température moyenne [10;10.5[ [10.5;11[ [11;11.5[ [11.5;12[ Number of years Température moyenne [12;12.5[ [12.5;13[ [13;13.5[ Number of years b Using the calculator, determine the mean and stan-dard deviation of this statistical series (results should be given to two decimal places) . E.4784 In an amusement park, we studied vis-itor waiting times at the entrance to a household. The data was compiled in the table below : Waiting time (in min) 0 ; 5 5 ; 10 10 ; 20 20 ; 30 30 ; 60 Number 26 76 82 24 8 Frequency F.C.C. 1 Complete the two lines for frequencies and cumulative fre-quencies, rounding the results to the nearest thousandth. 2 Construct the cumulative frequency curve for this statis-tical series on the graph below : 3 a Graphically determine the median and quartiles of this statistical series (leave the construction lines) . b Construct the box plot for this statistical series using the scale : 1 cm represents 5 min expected E.4825 The 2011 census of the French popula-tion yielded the following headcount table : Classe d’âges 0;10 10;20 20;30 30;50 50;70 70;100 Effectif (in thousands) 8 054 7 981 8 077 17 372 15 630 8 234 Fréquence F. C.C. 1 Determine the frequency of the French population repre-sented by the population aged under 20 ans, rounded to the nearest thousandth. 2 Complete the rows, in the table below, of the frequencies and increasing cumulative frequencies (frequencies will be rounded to the nearest thousandth) . 3 a Plot the curve of increasing cumulative frequencies in the frame below. b Determine graphically the value of the first quartile, median and third quartile of this statistical series. (construction lines should be left visible) . 4 Plot the box plot using the position indicators for this se-ries obtained in question 3 b . The scale will be 1 cm for 10 ans. https://chingmath.fr chapExoCorrec/4784 sacados/4784 01020304050600.10.20.30.40.50.60.70.80.91 chapExoCorrec/4825 sacados/4825 01020304050607080901000.10.20.30.40.50.60.70.80.91AgeFréquence
E.4742 We consider the students in a junior high school class and study their height. Here is the statistical series associated with height in centime-ters : 167 − 181 − 173 − 179 − 165 − 169 − 170 − 174 − 160 172 − 173 − 164 − 170 − 156 − 161 − 171 − 174 − 162 183 − 176 − 170 − 163 − 175 − 155 − 173 − 167 − 168 175 − 170 − 162 − 169 − 170 − 159 1 Complete the table of numbers below and the associated lines : frequencies will be given to 10 − 3 near. Classe [155;160[ [160;165[ [165;170[ [170;175[ [175;180[ [180;185[ Staff Fréq Fréq cum crois sante 2 Consider the graph below : a Draw the cumulative frequency polygon. b Determine the position of the first quartile, the median, and the third quartile. 3 Plot the box plot corresponding to this statistical series using the following scale : 1 cm on the graduated line ←→ 2 cm for size. E.221 Here is the table showing the distribution of students’ scores on the middle school graduation exam: Score 0 ; 4 4 ; 8 8 ; 12 12 ; 16 16 ; 20 Number of Students 5 32 61 80 15 Fréquence en % Fréquence cumulée croissante 1 Find the modal class for this statistical series. 2 Calculate the school’s average score on this exam, rounded to the nearest hundredth. 3 a Complete the rows for frequencies and cumulative frequencies in ascending order in the table above. b Construct an orthonormal coordinate system where the x-axis represents ( 1 cm =2 points) and the y-axis represents ( 1 cm =10 % students) . Plot the cumulative frequency curve on this coordinate system. c Determine the median from this curve. (Leave your construction lines visible) https://chingmath.fr chapExoCorrec/4742 sacados/4742 15501601651701751801850.10.20.30.40.50.60.70.80.91 chapExoCorrec/221 sacados/221
E.224 In a group of 60 athletes, resting heart rates (FCR) were studied. That is, the number of heartbeats per minute after a long period of calm and rest. Here are the ages and FCRs of the population Age FCR Age FCR Age FCR Age FCR 42 42 50 50 37 52 41 54 41 43 35 50 42 52 31 55 61 45 24 50 21 52 50 55 51 45 23 50 40 53 32 55 41 46 52 50 34 53 22 55 27 46 36 51 35 53 42 55 33 46 31 51 28 53 52 55 40 48 35 51 55 53 18 57 55 48 60 51 49 53 51 59 31 48 29 52 31 53 22 59 32 48 30 52 35 53 23 59 35 48 49 52 38 54 53 59 44 49 32 52 53 54 50 59 40 50 40 52 42 54 28 59 36 50 47 52 54 54 47 60 The CRF of the study population will be taken as the study character : 1 Give the median of this statistical series. 2 Complete the following table : Classe [40 ; 45[ [45 ; 50[ [50 ; 55[ [55 ; 60] Effectif Fréquence Fréq cum croissante Frequencies will be expressed as percentages to the near-est 0.1 % . 3 a Draw in the table below, the curve of increasing cu-mulative frequencies in the frame below. b Leaving your constructions on the figure, give the ab-scissa of the point on the curve with ordinate 50 % c Give an interpretation of this result. https://chingmath.fr chapExoCorrec/224 sacados/224 40455055600102030405060708090100
E.229 Jean, a video game enthusiast, recorded the duration in seconds of the 40 games he played : 49 52 57 57 57 58 58 59 60 60 60 62 63 63 63 63 64 64 64 64 65 65 66 67 69 69 70 70 72 74 74 75 75 76 77 78 79 80 80 80 1 Give the median value of this series. That is, the duration such that : The number of games with a duration less than this represents 50 % of the games considered. Thus, that the games representing a shorter duration also represent 50 % 2 a Complete the table below : Classe 45;50 50;55 55;60 60;65 65;70 70;75 75;80 80;85 Eff. Fréq. Fréq cum croiss. b Construct the cumulative increasing frequency curve : c Find the value of the median from this graph E.2016 Here are the average monthly tempera-tures recorded between 1994 and 2003 in the town of Sète in the south of France : Mois Janvier Février Mars Avril Mai Juin Température 8.3 9.4 12.1 13.5 17.5 21.2 Mois Juillet Août Sept. Octob. Novem. Decem. Température 23.4 22.8 18.6 15.7 10.8 8.5 1 Determine the average annual temperature for the city of Sète. 2 Scientists estimate a global temperature rise of 1.5 o C in 2020. a Reproduce the previous table, accepting the scientists’ predictions. b Then calculate the average annual temperature of this city in 2020. E.2065 We study the journey made by students in the second grade each day to get from their homes to the lycée. The time (expressed in minutes) taken for this journey is recorded : Temps de transport par jour 0 ; 10 10 ; 30 30 ; 40 40 ; 50 Nombre d’élèves 32 54 17 3 1 a Recopy the above table, adding the frequencies and increasing cumulative frequencies expressed as percent-ages to the nearest 0.1 % . b Determine the median class of this statistical series. 2 Determine the average travel time, rounded to the minute, per day of a second-grade student at this high school. 3 a In an orthogonal frame of reference O ; I ; J such that : OI = 0.2 cm OJ = 0.1 cm Plot the polygon of increasing cumulative frequencies. b Using this curve, determine the median value of this statistical series. E.3075 i’ll have to redo all the exercises ; for the cumulative frequencies curve, I’ll have to take not the middle of a class, but the bounds of each bound, which will better represent linear interpolation. E.230 By studying, on a class basis, the travel time to school. Here’s how long it takes : 5 ; 15 ; 15 ; 25 ; 5 ; 38 ; 37 ; 20 ; 3 ; 15 ; 7 ; 2 ; 30 ; 10 16 ; 2 ; 5 ; 5 ; 20 ; 25 ; 25 ; 30 ; 3 ; 11 ; 25 ; 8 ; 13 1 Complete the table below, rounding frequencies to the nearest thousandth. Classe (en min) 0 ; 10 10 ; 20 20 ; 30 30 ; 40 Effectif Fréquence Fréquence Cumulé Croissant 2 Construct the frequency histogram below : 3 Construct the cumulative increasing frequency curve : https://chingmath.fr chapExoCorrec/229 sacados/229 45556575850102030405060708090100 chapExoCorrec/2016 sacados/2016 chapExoCorrec/2065 sacados/2065 sacados/3075 chapExoCorrec/230 sacados/230 0102030400.10.20.30.40.5Duréedutrajet(enmin)Fréquence(en%)
E.2363 The graph opposite shows the polygon of increasing cumulative frequencies of a statistical series rep-resenting the size of a sample of high school students. This diagram has been drawn from a table of student num-bers, in which the students have been sorted into classes : [155 ; 160[ ; [160 ; 165[ ; [165 ; 170[ [170 ; 175[ ; [175 ; 180[ ; [180 ; 185[ 1 a Determine, approximately, the frequency associated with the class [155 ; 160[ b Determine, approximately, the frequency associated with the class [175 ; 180[ 2 Leaving the construction lines on the graph : a Determine the median of this statistical series. b Determine the first and third quartile of the series. 6. Sampling E.283 The table below shows the number of couples with children under 25 still living at home. Number of children Number of couples (in thousands) 0 5420 1 3048 2 2880 3 1233 4 and more 651 1 What is the modal class of this statistical series? 2 If each of the 651 000 couples in the "4and over"row had exactly 4 children, what would be the average number of children per couple? 3 Another study states that the average number of children under 25 still living with their parent is 1.18. Calculate the average number of children, among the 651 000 cou-ples with 4 or more children, living still living in their household. E.284 During a competition, 40 judokas. are weighed The calculated average weight is 72 kg . A check of the scale shows that it was incorrectly set and indicates 500 g less than the actual weight on each of the weighings. What is the actual average weight of the 40 judokas? What property is used for this statement? E.285 In a high school, there are four second-year classes containing 30, 32 ,28 and 27 pupils re-spectively. The average physical education grades for these classes are 12, 11, 13 and 14 respectively. What is the average, rounded to the nearest hundredth, of the physical education grades for all four of the lycée’s second-year classes? https://chingmath.fr 0102030400.10.20.30.40.50.60.70.80.91Duréedutrajet(enmin)Fréquence(en%) chapExoCorrec/2363 sacados/2363 chapExoCorrec/283 sacados/283 chapExoCorrec/284 sacados/284 chapExoCorrec/285 sacados/285
E.286 The latest French population census by INSEE (Institut national de la statistique et des études économiques - http ://www.insee.fr) in 1999 gives us the fol-lowing table : Age de la population Less de 20 ans de 20 ans à 39 ans de 40 à 59 ans de 60 à 74 ans de 75 à 99 ans Fréquence (%) 24.6 28.1 26 13.6 7.7 Calculate the average age of the French population from this table (It is assumed that people over 100 won’t influence the calculations too much) . E.287 Étude du tirage du January prints: 02/01/2002 5 39 7 28 18 43 24 3 19 27 16 24 14 26 05/01/2002 3 19 27 16 24 14 26 9 34 13 25 7 28 6 09/01/2002 8 12 48 19 2 45 46 21 10 44 7 5 36 22 12/01/2002 42 12 3 45 26 47 2 20 29 31 48 17 34 16 16/01/2002 13 26 48 41 30 29 37 35 26 18 14 17 42 5 19/01/2002 46 3 17 44 25 37 2 29 1 27 45 17 40 21 23/01/2002 5 32 19 45 34 13 36 34 44 4 43 45 3 9 26/01/2002 23 40 33 10 14 3 4 18 24 7 41 27 47 44 30/01/2002 25 5 46 22 21 1 33 14 10 26 21 19 40 37 1 Dividing the results into the following five classes : [1 ; 10[ , [10 ; 20[ , [20 ; 30[ , [30 ; 40[ and [40 ; 50[ , draw up a table showing the numbers and frequencies frequencies of each class for the month of January . 2 You are given the numbers in each class for the draws for the month of February : 1 ; 10 10 ; 20 2 ; 30 30 ; 40 40 ; 50 Effectifs 27 18 14 24 29 Give the frequency distributions for the whole of January and February. 3 You are given the numbers in each class for the draws in the month of March : 1 ; 10 10 ; 20 2 ; 30 30 ; 40 40 ; 50 Effectifs 22 27 24 30 23 Give the frequency distributions for all of January, Febru-ary and March. 4 Do you agree with the assertion : ˇPlus, we increase the size of the study sam-ple, the more the frequency of each class stabilizesı 5 On the website of Française des Jeux (http ://www.fdjeux.com) , a statistician has studied since the game first appeared the frequency distribution between all the balls. Here are his results : 1 ; 10 10 ; 20 2 ; 30 30 ; 40 40 ; 50 Effectifs (en % ) 18.5 20.21 20.1 20.71 20.4 Can you give an explanation as to why the frequency of the class [1 ; 10[ is significantly lower than the frequencies of the other classes? E.289 Here are the draws for the month of Jan-uary: 02/01/2002 5 39 7 28 18 43 24 3 19 27 16 24 14 26 05/01/2002 3 19 27 16 24 14 26 9 34 13 25 7 28 6 09/01/2002 8 12 48 19 2 45 46 21 10 44 7 5 36 22 12/01/2002 42 12 3 45 26 47 2 20 29 31 48 17 34 16 16/01/2002 13 26 48 41 30 29 37 35 26 18 14 17 42 5 19/01/2002 46 3 17 44 25 37 2 29 1 27 45 17 40 21 23/01/2002 5 32 19 45 34 13 36 34 44 4 43 45 3 9 26/01/2002 23 40 33 10 14 3 4 18 24 7 41 27 47 44 30/01/2002 25 5 46 22 21 1 33 14 10 26 21 19 40 37 Here are the February draws : 02/02/2002 48 14 5 45 6 29 42 47 19 4 10 30 44 36 06/02/2002 33 18 14 10 7 3 36 2 45 12 43 38 41 30 09/02/2002 41 4 46 31 32 43 29 9 7 3 31 4 49 6 13/02/2002 44 12 37 31 24 2 45 2 39 23 38 17 13 37 16/02/2002 47 21 23 7 37 46 35 35 47 31 42 4 34 17 20/02/2002 6 46 4 19 8 42 40 16 2 24 4 45 44 37 23/02/2002 3 49 17 43 11 1 2 39 20 11 23 3 27 32 27/02/2002 43 29 45 6 32 24 25 18 4 37 42 46 23 14 Here are the draws for March : 02/03/2002 10 39 1 37 40 48 29 32 29 49 23 45 22 30 06/03/2002 16 3 1 33 10 32 19 34 49 9 17 22 30 18 09/03/2002 22 30 38 15 25 20 48 8 21 25 48 34 31 39 13/03/2002 26 21 45 25 6 38 33 39 18 42 11 8 25 22 16/03/2002 12 34 39 17 40 15 44 31 33 26 40 3 37 18 20/03/2002 14 17 27 16 37 39 8 43 48 38 14 49 21 9 23/03/2002 2 47 39 13 36 9 12 10 8 42 3 46 5 1 27/03/2002 47 35 21 30 49 14 12 23 32 19 16 26 7 47 30/03/2002 48 25 40 13 24 16 4 10 2 1 4 9 36 22 https://chingmath.fr chapExoCorrec/286 sacados/286 chapExoCorrec/287 sacados/287 sacados/289
7. Gaussian, normal series E.7860 One company claims that 80 % of its customers are satisfied with its products. 1 A consumer association wishes to verify this claim and commissions a study of 50 customers of this company. a Determine the fluctuation interval at the threshold of 95 % . b The study obtains a satisfaction rate of 71 % . Accord-ing to this study, what can we say about the company’s affirmation at the threshold of 5 % ? 2 The association renews its study, this time focusing on 100 customers. This new study still obtains a satisfaction rate of 71 % . What can we say about the company’s assertion at the threshold of 5 % . E.161 The questions 2 et 3 are inde-pendent of the question 1 . A forestry service carried out reforestation on a hill. One year after the initial planting, it seeks to evaluate the quality of this reforestation and, to do so, selects two observation points on the hill, around which it measures the heights of a sample of young trees. 1 Locating the two observation points In Graph 1 in Appendix 2 (to be turned in with the exam) , there is a map of the hill on which only the contour lines (spaced 20 meters apart) . are plotted. Each contour line represents points at the same elevation. This hill reaches an elevation of 410 meters, a point marked by a cross on Figure 1. Two axes placed along the edges of the drawing allow us to locate each point : both axes are graduated in incre-ments of fifty meters starting from the lower left edge ; the horizontal axis of the drawing will be called the x-axis, and the vertical axis, the y-axis. The graph shows that point A , with an x-coordinate of 150 and a y-coordinate of 100, is located at an elevation between 300 and 320 meters a Plot point B with an x-coordinate of 250, knowing that its elevation is 360 meters and that it is located on the steepest side of the hill. b Draw a path on the diagram connecting point A to point B without ever going back down. c Graph 2 shows the profile of the hill as a south-north cross-section. (Points S and N , indicated on the dia-gram, are both at an elevation of 285 meters.) . This profile contains two errors. Identify them on Graph 2: circle the incorrectly placed points and explain your answer (You can draw the line ( NS ) on Graph 1. 2 The data near point A The heights (were recorded at cm ) for forty-four trees around point A ; the series below lists them in ascending order : 210 215 215 215 218 219 220 225 225 227 230 230 230 232 232 233 234 234 236 236 236 236 236 240 240 245 245 245 245 245 248 248 250 250 250 250 252 252 253 254 254 256 259 260 a Use a calculator to find the mean of this series. b Calculate the median and quartiles of this series, then plot the corresponding box plot. 3 Comparison of observations at A and B with ex-pected results The heights of fifty-six trees near point B were recorded at cm : the observed average height is 220 cm ; ten trees are shorter than 200 cm , and the maximum height is 250 cm . a What is the average height of the sample of 100 trees observed around points A and B ? b A study of the same tree species suggests that the heights can be considered Gaussian data with a mean of — =240 cm and a standard deviation of ff =10 cm . Determine the normality range containing 95 % of the population according to this study. Explain why we can conclude that the heights recorded for the 100 trees observed do not conform to the results of the study. Figure 2: Hill profile (south-north cross-section) https://chingmath.fr chapExoCorrec/7860 sacados/7860 sacados/161 SN280300320340360380400420
Figure 1: Hill map (contour lines) E.188 An Internet auction site wishes to carry out a statistical study of its clientele. The study uses a sample of 3.000 of the site’s most regular customers. Part A The first question concerns the age of the customers consid-ered. The results are given in the histogram below. 1 Complete, without justification, table 3 in the appendix. 2 Using the calculator, determine without justification (re-sults should be rounded to the tenth) : a the average age of the auction site’s 3.000 customers. b the standard deviation ff of the customer age series. 3 Can we estimate that the percentage of individuals who have an age belonging to the range [ m − ff ; m + ff ] is greater than or equal to 75 % . Part B The second question posed to the 3.000 customers concerns average connection time in minutes over a one-week period. 1 The study showed that the series of average connection times follows a Gaussian distribution with mean — ∼ 83.5 and standard deviation s ∼ 26.6 a Determine the normal range at 95 % of this series. b How many customers can we estimate whose average connection time per week is outside this range? 2 For this series, the first quartile Q 1 is 65, the median Me is 85 and the third quartile Q 3 is 100. a What is the minimum number of customers whose av-erage connection time per week on the site is less than or equal to 65 minutes? b The site managers hoped that at least 1.000 people would log on for an average of 1 hour and 40 minutes or more per week. Has this goal been achieved? Annexe Class Centre de la classe Effectif fréquence (en % ) [13 ; 18[ [18 ; 20[ [20 ; 25[ 22.5 [25 ; 30[ 27.5 32 [30 ; 35[ 32.5 18.6 [35 ; 45[ 40 7.4 [45 ; 55[ 1.9 [55 ; 70[ 1 Total × 3000 100 https://chingmath.fr 400300A05050 sacados/188 1481318815209592555930221355745315521070Age (ans)40 individus
E.195 French Polynesia - 8 points - June 2005 The following series shows the number of days of snow-fall per year in Paris from 1900 to 1948. The 49 values in this series are not listed in chronological order, but in ascending order. 1 - 5 - 6 - 6 - 6 - 7 - 7 - 7 - 8 - 8 9 - 10 - 10 - 11 - 11 - 11 - 12 - 12 - 12 - 12 13 - 13 - 13 - 14 - 14 - 14 - 14 - 15 - 16 - 17 17 - 17 - 18 - 18 - 18 - 18 - 19 - 19 - 20 - 20 20 - 23 - 26 - 29 - 29 - 31 - 32 - 32 - 34 1 a Calculate the average number x of snow days per year in Paris between 199 and 1948. (the result will be rounded to the nearest tenth.) b Determine the median, Me , as well as the first and third quartiles, Q 1 and Q 3 , of this series. Justify each answer. 2 The number of days of snowfall per year in Paris was also recorded from 1949 to 1997. The characteristics of this new statistical series are provided below. Minimum First quartile Q 1 Médiane med’ Troisième quartile Q 3 Maximum Moyenne x 1 7 12 18 36 13.3 a Give the interquartile range for each of the two statis-tical series showing the number of days of snowfall per year in Paris for the period 1900-1948 and then for the period 1949-1997. b Build on the appendix the box plot for each of the two series studied. c Compare these two box plots. 3 The series of numbers of days of snowfall per year in Paris over the period 1900-1948 has a standard deviation of s , and the series of numbers of days of snowfall per year in Paris over the period 1949-1997 has a standard deviation of s . We assume that : s ≈ 7.82 ; s ≈ 8.01 . What does it mean that the standard deviation is higher for the second period than for the first? 4 Various comments have been made in the press about these weather records, including the following : ˇWinters with less and less snow over the course of the centuryı. What can we make of this comment? Period 1900-1948 Period 1949-1997 E.2024 The Paris Montsouris meteorologi-cal observatory has been continuously recording outdoor tem-peratures since 1872, and provides annual averages based on these readings. The aim of this exercise is to compare these averages over twenty-year periods between 1880 and 2000. To clarify the vocabulary, we’ll call ˇannual temperatureı the av-erage temperature recorded in a given year (days and nights) , expressed in degrees Celsius and rounded to 0.05 o C . Sources Météo France Part A. Temperatures at the end of the XX e siècle Document 2 in Appendix 1 shows the annual temperature se-ries for the years 1981 to 2000, arranged chronologically in ascending order. 1 Calculate the median, first and third quartiles of this series. Justify each answer. 2 Draw the box plot corresponding to this last period on document I in appendix 1, which will be returned with the copy. Show the median, first and third quartiles, minimum and maximum of this temperature series. 3 Determine the mean of the annual temperature series from 1981 to 2000 using the calculator (the result will be rounded to 0.05 o C ) Part B. A century of temperatures A closer analysis of annual temperatures between 1881 and 1980 shows that they are Gaussian data with mean m = 11.49 o C and standard deviation ff =0.54 o C . Recall that for Gaussian data, the interval [ m − ff ; m + ff ] is the normality range at 68 % . 1 Determine the normality range at 68 % of the annual temperature series between 1881 and 1980. To how many years can we estimate the number of years between 1881 and 1980 whose annual temperature is greater than m + ff ? 2 Document I in Appendix 1 shows box plots constructed from annual temperatures over each twenty-year period between 1881 and 1980. On each of these diagrams, the median, first and third quartiles have been plotted. The ends of the ˇmoustachesı mark the minimum and maxi-mum of this series. For each of the following propositions, indicate whether it is true, false or undecidable (in case the document would not allow to know whether the proposition is true or false) . Justify the answer. a The maximum annual temperature was 12.65 o C for a century, from 1881 to 1980. b The annual temperature range was 2.25 o C for one cen-tury, from 1881 to 1980. c For a century, from 1881 to 1980, at least thirty years had their annual temperature below 11.5 o C . d 1961 was the coldest year over the period 1901-1980. Part C. Comparative study Making reasoned use of Parts A and B , compare the temper-atures observed in Paris in the last twenty years of the XX e century with those observed over the previous hundred years. Document 2: Annual temperatures in Paris between 1981 and 2000: sorted in chronological order https://chingmath.fr sacados/195 010203040 010203040 sacados/2024
Année 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 Tempé rature en o C 11.50 12.40 12.30 11.85 11.10 11.25 11.15 12.40 12.95 13.10 Année 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Tempé rature en o C 11.75 12.30 11.85 13.10 12.85 11.40 12.90 12.40 13.05 12.90 sorted in ascending order Tempé rature en o C 11.10 11.15 11.25 11.40 11.50 11.75 11.85 11.85 12.30 12.30 Tempé rature en o C 12.40 12.40 12.40 12.85 12.90 12.90 12.95 13.05 13.10 13.10 Document 1: Annual temperatures in Paris by twenty-year periods E.165 The results obtained by the 199 stu-dents from six classes of seconde in a common assignment were used to construct the six box plots given in the appendix. The results obtained by all second-year students are given in the following Table 1: Notes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Effectifs 1 2 8 11 14 16 17 20 30 26 15 11 11 7 2 2 2 2 2 The numbers and averages for the second-year classes are given in the following Table 2: Seconde 1 2 3 4 5 6 Effectifs 34 33 34 33 32 33 Moyennes 8.76 9.52 6.82 9.97 8.41 9.52 Part A Study of Seconde classes using box diagrams 1 Give the scores corresponding to the median, first, third quartile of the Seconde 1 score series. 2 In Seconde 2, can we say that at least one in two students has a mark of 10 or less? Justify. 3 In which Seconde classes can we say that at least 75 % of students have a mark of 13 or below? Justify. 4 How can we explain that in Seconde 5, the average mark is lower than the median mark? 5 For Seconde 3, give the interquartile range. 6 In Seconde 5, how many students scored 10 or above? Part B Study of all Seconde students 1 Determine the median, first and third quartile of the grade series for all Seconde students. On the axis given in the appendix, construct the box plot for this series 2 a Using Table 1 above, calculate the average score x of Seconde students (a value rounded to the hundredth will be given) . b Explain the method for calculating this average using Table 2 above exclusively. Perform this calculation. 3 It is assumed that the grades of all Seconde students are Gaussian data and that the standard deviation ff is equal to 3.45. Comment s’appelle l’intervalle x − 2 ff ; x +2 ff ] ? Give the percentage of Secondes students who scored outside this range? https://chingmath.fr sacados/165 02468101214161820Seconde 1Seconde 2Seconde 3Seconde 4Seconde 5Seconde 6
E.189 The three parts are independent Part A A pétanque ball factory manufactures competition balls of various masses and diameters. The three masses offered are 700 g , 720 g , and 745 g , and for each of these masses, three diameters are available: 71 mm , 75 mm , 79 mm . 1 How many types of balls are manufactured by this com-pany? 2 A regional champion decides to buy 720 g balls but is un-sure about the diameter. To make his choice, he places a jack 9 meters away, aims 200 times with each ball of a different diameter, and measures the distance to the jack. Here are the box plots, trimmed to the deciles, rep-resenting this test. The ends of the plots correspond to the first and ninth deciles, respectively. Here are some of the player’s impressions after the test : With the 79 mm ball, I had some very good throws, but also some very bad ones. With the 71 mm ball, I felt great≠half of my throws were within 16 cm of the jack, and I made some really nice ones But I prefer the 75 mm ball, which allows me to be more consistent. Match each type of ball with the corresponding box plot. Explain your answer. Part B: Annual Sales in 2004 Tables 1 and 2, taken from a spreadsheet, show the break-down of pétanque ball sales for the year 2004 by diameter and mass. Note: Round the percentages to the nearest tenth of a percent. 1 Calculate the percentage, rounded to 0 ; 1 % , of 720-gram balls with a diameter of 75 mm sold in 2004. 2 Calculate the percentage, rounded to 0 ; 1 % , of balls with a diameter of 75 mm sold in 2004. 3 Among the balls of 700 g sold in 2004, what is the per-centage, rounded to 0 ; 1 % , of balls with a diameter of 79 mm ? 4 The table of percentages is in percentage format. The formulas below should be entered into cell B10 and copied to the rest of the table. Select (or the correct (s) formula (s)) . a =B3/E6 b =B3/$E$6 c =B$3/E6 d =B3/23547 You are not required to complete the table. Part C : Rejected Balls The table below shows the distribution of the actual masses of 2 500 720-gram competition balls that were manufactured : Masse (in g ) 719,5 719,6 719,7 719,8 719,9 720 720,1 720,2 720,3 720,4 720,5 Sample size 10 53 178 385 441 524 478 201 165 51 14 1 Calculate the mean — of the masses of these 2 500 balls. Round the result to 3 decimal places. 2 Assume that the data follow a Gaussian distribution. The standard deviation s of this set is equal to 0 ; 185 . Determine the normal range at 95 % for this set. 3 The company rejects balls that fall outside the normal-ity range as unsuitable for competition. Calculate the percentage, rounded to 0 ; 1 % , of rejected balls. https://chingmath.fr chapExoCorrec/189 sacados/189 46810121416182022242628303234cmCBA 12345678910111213ABCDETableau2:enpourcentageparrapportàl’e∑ectiftotalTableau1:e∑ectifMasse(eng)\Diamètre(mm)717579Total700215731231803708372030034122231094357452124298219237029Total728410227603623547Masse(eng)\Diamètre(mm)717579Total700720745Total100%
E.192 There are two concert halls in a city, each of which programmed 40 concerts during the 2004 = 2005 season. Salle G specializes in classical music and Salle J in jazz. 1 For the venue G , the results in number of expected spec-tators are shown by the histogram given in Appendix 3. For example, the manager thinks that 6 concerts will attract between 500 and 700 spectators during the 2004 = 2005 season. a Calculate, using class middles, the mean m G of this statistical series. b The data in this series are considered to be Gaussian (i.e. they approximately follow a normal distribution) . The normality range at 95 % is [302 ; 1438] . Using this interval find the mean m G and calculate the standard deviation ff G of the series. 2 Statistics for room J are given on a spreadsheet made using a table (appendix 4) ? Recall that C3 , for example, denotes the address of the cell located at the intersection of column C and row 3 . The cells A5 to A11 contain the spectator number classes, all with amplitude 200. The cells B5 to B11 contain the class middles. The cells C5 to C11 contain the numbers of concerts corresponding to the classes in the column A . a The manager wants to obtain, using the table, the av-erage number of spectators per concert for the season 2004 = 2005 . In the cell D5 figure 400 which represents the number of spectators likely to have attended the four concerts relating to the first class. What formula has the manager entered in D5 , knowing that it must be recopied to D11 , to obtain the num-bers relating to the other classes? Enter the results from cells D6 to D11 on Appendix 4 (to be returned with copy) . b What formula has the manager entered in D13 ? What formula should he enter in D15 to get the average num-ber of spectators per concert in the venue J ? Enter this number in cell D15 in Appendix 4. 3 Find, for the series concerning room J , the respective classes containing the median and quartiles of the num-ber of shows. 4 To boost attendance during the 2005 = 2006 season, the manager decides to offer subscriptions for several con-certs during the year. He hopes to increase by 10 % the number of spectators at each concert from under 800. a Which formula must be entered in cell E5 (recopied to E8 ) to find the expected number of spectators in 2005 = 2006 for these first 4 classes? Enter the 4 results in the table in Appendix 4. b What formulas need to be entered in cellsenter in the cells E13 and E15 to obtain the expected number of spectators for 2005 = 2006 and the average per concert? c Calculate, under this assumption, the relative percent-age change between the expected average in 2004 = 2005 and that expected in 2005 = 2006 . The result will be rounded to the nearest 0.1 % . https://chingmath.fr sacados/192 Un concert100500700900110013001500G 12345678910111213141516ABCDEClassemilieuxdenombredespectateursspectateursclassesconcerts2004/20052005/2006[0;200[1004400[200;400[3008[400;600[5004[600;800[7002[800;1000[90065400[1000;1200[11001011000[1200;1400[130067800somme:403040031020moyenne:775,5
E.197 We studied the heart rates of a group of 60 amateur sportsmen and women (called group I.) , prac-ticing their sport 2 to 4 times a week. Heart rate is the number of heartbeats per minute. For each of these athletes in the I group, we measure resting heart rate (RHR) , i.e. the lowest heart rate encountered in that person, measured after several trials following a long pe-riod of calm and rest. The results of this study are summarized in the table below, where the resting heart rates (RCF) of the 60 athletes in the I group are ranked in ascending order. Age FCR Age FCR Age FCR Age FCR 42 42 50 50 37 52 41 54 41 43 35 50 42 52 31 55 61 45 24 50 21 52 50 55 51 45 23 50 40 53 32 55 41 46 52 50 34 53 22 55 27 46 36 51 35 53 42 55 33 46 31 51 28 53 52 55 40 48 35 51 55 53 18 57 55 48 60 51 49 53 51 59 31 48 29 52 31 53 22 59 32 48 30 52 35 53 23 59 35 48 49 52 38 54 53 59 44 49 32 52 53 54 50 59 40 50 40 52 42 54 28 59 36 50 47 52 54 54 47 61 1 a Determine the median and the first and third quar-tiles of the FCR series. b Construct on the D 1 axis given in Appendix (to be returned with copy) , a box plot for this series. 2 a Complete the table in appendix and plot a graphi-cal representation of the FCR series of the 60 athletes in the I group. b Calculate the mean x of this series. 3 a It is assumed that the FCR of the athletes in the I group are Gaussian data with a standard deviation ff equal to 4.06 . Déterminer l’intervalle 52 − 2 ff ; 52+2 ff . What do we call this interval? b Calculate the percentage of athletes whose FCR lies within this interval. Was it possible to predict this result? Explain. 4 We wish to compare the FCR of athletes in the I group with the FCR of a group of 60 people doing little phys-ical activity (called the II group) . The FCR study of people in the II group yielded the following results : Mean : 59.8 Median : 60 Standard deviation: 6.23 Third quartile: 63 First quartile: 57 Maximum value : 70 a On the D 2 axis given in the appendix, draw a box diagram for the FCR of the people in the II group. b What impact does the regular practice of sports activ-ities seem to have on an individual’s FCR ? Axe D 1 Axe D 1 FCR 42 43 45 46 48 49 50 51 52 53 54 55 57 59 61 Nombre d’individus Table E.190 A company wishes to purchase a large number of motors. Before choosing between M 1 or M 2 motors, it wishes to compare their respective service lives. Part A Comparison of the lifetimes of a sample of 60 motors of type M 1 and a sample of 60 motors of type M 2 . 1 Study of a sample of 60 motors of type M 1 . In the table given in document 1 of appendix 2 is the life-time in months of each motor of a sample of 60 motors of type M 1 . The values were sorted in ascending order using a spreadsheet program. What percentage of M 1 motors have a service life of 4 years or more? (result will be rounded to 0.1% ) . 2 a Determine the median and then the first and third quartiles of the series of lifetimes for the 60 M 1 motors. b Represent this series by a box plot on document 2 in appendix 2, to be handed in with the copy. The me-dian, first and third quartiles and extremes of the series studied will be shown. 3 Study of a sample of 60 motors of type M 2 . On document 2 in appendix 2 is the box plot represent-ing the statistical series of service lives in months of a sample of 60 motors of type M 2 . The median of this series is 59 months. a What is the minimum number of motors of type M 2 whose lifetime is greater than or equal to 59 months? b What is the minimum percentage of M 2 type motors with a service life less than or equal to 66 months? c Compare the series of lifetimes of motors of type M 1 and type M 2 using their box diagrams. Part B Study of a sample of 1000 motors of type M 2 A statistical series on a sample of 1000 motors of type M 2 showed that the series of mean lifetimes, expressed in months, is Gaussian with mean m ≈ 59.6 and standard devia-tion s ≈ 0.8 . 1 Determine the range of normality at 95 % . 2 How many M 2 motors with a life of less than 58 months can be estimated? 3 The company wishes to purchase motors of type M 2 such that at least 99% have a service life greater than or equal to 58 months. What comments can you make? Annexe Document 1 : Lifetimes in months of 60 motors of type M 1 sorted in ascending order in rows : https://chingmath.fr sacados/197 42455055606570 42455055606570 chapExoCorrec/190 sacados/190
34 38 39 41 42 43 44 45 45 47 47 48 49 50 51 51 52 52 53 54 55 55 55 55 55 55 55 56 56 57 58 58 58 59 59 59 60 62 62 62 62 63 64 65 66 66 66 66 67 68 68 73 74 74 75 75 79 81 81 85 Document 2: box plots of the service life series of 60 motors of type M 1 and M 2 : E.191 A mouse runs down a pipe (shown in the figure below) leading to exits 0, 1, 2, 3. It is assumed that she progresses towards the finish by ran-domly heading each level to the right or left to access the level below. A possible route can therefore be coded GGD , where G means ˇ going towards gaucheı and D ˇaller towards droiteı, on each of the three levels. We are then interested in the number of the mouse output. Part A Theoretical study Find all possible paths (possibly using a tree) and then com-plete the theoretical frequency table (appendix 1, return with copy) Part B Spreadsheet simulation Using a spreadsheet, we run a simulation of 100 mouse pro-gressions through the pipe : we obtain the frequencies corre-sponding to each of the possible mouse outputs. We then note the frequency corresponding to the output n o 1 obtained. Running 50 simulations, we obtain 50 frequencies correspond-ing to the output n o 1 . (These frequencies are recorded in Appendix 2) 1 The series of 50 frequencies is assumed to have mean m =0.364 and standard deviation x =0.051 , results given with 3 decimal places. Calculate the percentage of values in the series lying within the interval m − 2 s ; m +2 s . Does this result correspond to what we would expect from a Gaussian or normal series? Justify. 2 We then run two series of 50 simulations, one correspond-ing to 500 mouse progressions, the other to 1000 progres-sions, and obtain 50 frequencies of the output n o 1 for each series. Draw, on the same graph, the box plot that corresponds to the series of 50 simulations performed in question 1 , calculating all the elements needed to build this type of box. 3 a Using the three diagrams, determine which series ap-pears to give the frequencies closest to the theoretical frequency. b What would it take to get even closer? Output n o 0 1 2 3 Number of possible paths Theoretical frequency (in % ) Fréquences theoretical 0,250 0,260 0,290 0,290 0,300 0,300 0,310 0,310 0,320 0,320 0,320 0,320 0,330 0,330 0,330 0,340 0,340 0,340 0,340 0,350 0,350 0,350 0,350 0,305 0,360 0,360 0,370 0,370 0,370 0,370 0,380 0,380 0,380 0,390 0,390 0,390 0,390 0,400 0,400 0,410 0,410 0,420 0,420 0,420 0,430 0,430 0,450 0,460 0,470 0,470 Fréquences (arranged in ascending order) obtenues for output n o 1 (for 50 simulations) Boîtes with whiskers https://chingmath.fr 3236404448525660646872768084M2 sacados/191 Liban - Juin 2004 - 8 points 1234o 0,260,280,300,320,340,360,380,400,420,44
E.168 Jean, an enthusiast of a computer game, recorded the duration in seconds of the 40 games he played. These durations were ranked in ascending order in the table below : 49 55 57 57 57 58 58 59 60 60 60 62 63 65 65 66 67 69 69 70 70 72 74 74 75 75 63 63 63 64 64 64 64 76 77 78 79 80 80 82 So, for example, there were three games played in 60 seconds each. 1 a Determine the median, then the first and third quar-tiles of this series of values. b Represent this series of values by a box plot on which at least the three values obtained in the previous ques-tion will appear. c Calculate, in seconds, the average duration of the 40 games (round to the tenth) 2 The manufacturer of this game, after surveying a large number of players, estimated that the game durations constituted Gaussian data with a mean — of 62 seconds and a standard deviation ff of 6 seconds. The manufacturer advertises : ˇ You have a 95 % chance of playing every game lasting between 50 seconds and 1 minute 14 seconds. ı a On what is this manufacturer’s claim based? b Can we say that 95 % of the 40 games played by John last between 50 seconds and 1 minute 14 seconds? 8. Linear interpolation E.2023 The graph below shows changes in the number of inhabitants in two neighboring communes, named A and B , from 1980 to 2004 (from four years to four years) . Part A Reading graphique Answer the following questions using only the graph above. 1 In what year was the population of the commune A max-imum. 2 a Specify the years in which the communes A and B had the same number of inhabitants. b What are the periods during which the commune B had more inhabitants than the commune A ? c In which year was the difference between the number of inhabitants of commune A and commune B the great-est? 3 Specify, justifying the answer, during which four-year pe-riod the commune A had the highest population increase. Part B We’re interested in population trends in these communes from 2000 to 2004. The following table shows the number of inhab-itants in these two communes in 2000 and 2004. Years 2000 2004 Commune A 863 795 Commune B 711 947 The two questions are independent. 1 a Justify that, from 2000 to 2004, the commune’s pop-ulation A fell by approximately 7.9 % . b Determine the percentage increase in the commune’s population B in this same period (we’ll give the result rounded to 0.1 % ) . c If we consider the population of the two communes combined, determine the percentage change in this population during this period (we’ll give the result rounded to 0.1 % ) . 2 Estimate by linear interpolation on the interval [2 000 ; 2 003] the number of inhabitants in the munici-pality B in 2003. 9. Old statistics yearbook E.144 A mountain race The three parts are independent. The appendix is to be re- turned with the copy. Part A : Topographical study The route of a mountain foot race is given in the appendix. https://chingmath.fr sacados/168 Antilles-Guyanes - Juin 2006 - 8 points sacados/2023 chapExoCorrec/144 sacados/144
Competitors cross a first hill via its summit S 1 . The finish is at the top of the second hill S 2 . Point P designates the location of an aid station. 1 What is the altitude of the starting point? From the aid station? 2 A runner twists his ankle. He gives his position using a cell phone as follows : ˇ I’m on the descent of the first hill and my altimeter in-dicates an altitude of 1 274 m ı. Indicate in color on the map the minimum search area for this runner by rescue workers. 3 The map is to scale 1 = 50 000 . Calculate the approximate length of the route between the starting point and the summit S 1 (we’ll neglect the difference in altitude between the starting point and the summit neglect the difference in altitude between the starting point and the summit S 1 ) Part B: stroke profile The graph above shows the running profiles of three runners : Julien, Marc and Audrey. 1 Which of these three riders arrives first? 2 What happens at the twentieth minute of the race? 3 At the score of 5.5 km , what is Marc’s time lead over Julien? 4 At the fifteenth minute of the race, what distance sepa-rates Julien and Marc? Part C : Evolution of the number of participants The table below, extracted from an automated spreadsheet, gives the number of participants as a function of the year and the evolution of this number in relation to the year 2000. The column C is in percentage format. Percentages will be rounded to 1 % . 1 What type of growth in the number of participants is this over the period 2000-2004? Justify your answer. 2 a What is the percentage change, rounded to 1 % , in the number of participants from 2000 to 2003? b What formula, to be copied down using only cell refer-ences, has been written in cell C3 ? The race organizer feels that the increase in the number of participants is insufficient. It is therefore launching an advertising campaign and hopes to increase participation by 15 % per year. The effects of this campaign should be felt as early as 2005. a Calculate the expected number of participants in 2005. b What type of expected growth is this from 2005? Jus-tify your answer. c Calculate the expected number of participants in 2010. Appendix: to be returned with copy This topography of the place where the pedestrian race takes place. The course is marked in bold. https://chingmath.fr 01234567510152025303540Distances enkmMarcAudreyJulien 123456ABCAnnéeNombredeparticipantsPourcentaged’évolutiondunombredecandidatsparrapportà2000.20001420%200116214%200218228%20032022004222
E.152 The European Union, noted UE , grew from 15 to 25 member countries on 1 er May 2004. Table 2 gives indications of the European Union at each change in the number of member countries. It was obtained using a table. 1 In this question, we’re interested in the EU’s population increase. a What formula can be written in cell D3 to obtain, by automatic copy down, the percentage increase in the EU population at each date of change in the number of member countries, compared with the previous date of change? b Complete the column D (results will be rounded to the hundredth) c Calculate the percentage increase in EU population from 1957 to 2004. 2 In this question, we’re interested in the population den-sity of EU countries, i.e. the number of inhabitants per km 2 (results will be rounded to the unit) . The table below gives the population densities of EU countries in 2004. Pays Finlande Suède Estonie Lettonie Densité 15 22 30 37 Pays Irlande Lituanie Grèce Espagne Density 55 57 78 81 Pays Chypre Autriche Slovénie Hongrie Densité 91 97 98 107 Pays Slovaquie France Portugal Danemark Densité 110 118 123 Pays Pologne Rép. Tchèque Luxembourg Italie Densité 123 129 155 187 Pays Allemagne Royaume Uni Belgique Pays-Bas Malte Densité 231 248 338 388 1 266 a Knowing that France has 61.2 million inhabitants in 2004 for an area of 543 965 km 2 , calculate the popula-tion density of France in 2004 b Determine the median and quartiles of this density se-ries, then make a box plot (the maximum will not be shown) c Calculate the average population density of the EU countries. Note that the mean is higher than the me-dian. Explain why. 3 In this question, we are interested in France’s place in the EU in 2004 (results will be rounded to the unit) . a What percentage of the EU population does the French population represent in 2004? b What percentage of the surface area of the EU repre-sents the surface area of France in 2004? 4 Answer true or false to the following three statements : a The EU population grew by 108 % (to the nearest unit) between 1957 and 2004. b The surface area of the EU increased by a factor of 2 between 1957 and 2004 c At least 75 % of EU countries had a population density of 150 or more in 2004. E.1899 Part1: In the French rail network, trains ˇGrandes lignesı are of two types : Corail or TGV (Train à Grande Vitesse) and cus-tomers are offered second- or first-class travel. A survey is carried out in a provincial station during the first week of July 2006. Of the 2 450 tickets sold, 82 % are second-class tickets. Of the 850 TGV tickets sold, 14 % are first-class tickets. 1 Copy and complete the following table : Corail tickets TGV tickets Total Billets Seconde classe Billets Première classe Total 850 2 450 2 Verify that the percentage of first-class tickets among Corail tickets sold during the first week of July month of July 2006 is 20 % (rounded to the nearest whole number) . 3 Can the station manager deduce from this survey that approximately 34 % of the tickets sold in his station dur-ing the first week of July 2006 were first-class tickets? Justify. Part2: 1 In 2005, the station generated annual sales in thousands of euros (K e ) , rounded to the 4 108 Ke unit. The sta-tistical series of monthly sales achieved in that year is represented by a box plot given in the appendix. Assertion 1: 50 % at least of the station’s monthly sales in 2005 are less than or equal to 340 K e . Assertion 2: out of the twelve months of 2005, only two generated monthly sales for the station of less than or equal to 310 K e . 2 The monthly and annual sales figures for the station in 2006, in thousands of euros (K e ) and rounded off to the unit, are given below : Mois Janvier Février mars Avril Mai Juin Chiffre d’affaires 330 K e 342 K e 360 K e 372 K e 375 K e 385 K e Mois Juillet Aout Septembre Octobre Chiffre d’affaires 415 K e 410 K e 424 K e 430 K e https://chingmath.fr sacados/152 Antilles-Guyane - Septembre 2005 - 11 points 1234567ABCDEAnnéeNombredepaysdel’UE(enmillionsd’habitants)Populationdel’UE(enmillionsd’habitants)Augmentationdelapopulationdel’UE(en%)Super∏ciedel’UE(enkm2)19576210,7123510319739279,715888291981102901720455198612341,92317515199515364,131501742004254393888717 sacados/1899
Mois Novembre Décembre Année 2006 Chiffre d’affaires 391 K e 350 K e 4 584 K e a What is the percentage change in the station’s annual sales, rounded to the thousand euro, between the years 2005 and 2006? To the nearest tenth? b Determine the minimum, maximum, median, first quartile and third quartile of the statistical series for the station’s monthly sales in 2006. c Construct, on the graph given in Appendix 2, the box plot of the station’s monthly sales statistical series in 2006. d What conclusions can the station manager draw from the comparison of the two box plots shown? E.2029 parts 1 and 2 are independent Part 1 The table - incomplete - below, gives the various results for cinema attendance in France, for the years 2004 and 2005. Numbers of admissions in millions are rounded to 0.01 and percentages to 0.1 % . Fréquentation totale (millions d’entrées) 2004 2005 Evolution 2005/2004 en % Janvier 15.18 14.30 -5.8 Février 19.94 -16.0 Mars 15.34 14.17 -7.6 Avril 17.40 15.51 -10.8 Mai 13.77 -9.6 Juin 18.83 12.36 -34.4 Juillet 16.26 14.50 -10.8 Août 14.98 12.73 Septembre 9.83 8.29 -15.7 Octobre 17.17 14.93 -13.0 Novembre 15.10 14.85 -1.7 Décembre 20.07 23.49 +17.1 Année 195.33 175.65 -10.1 Source: Centre National de la Cinématographie http ://www.cnc.fr According to the latest estimates from the Research Depart-ment, cinema attendance reached 23.49 million admissions in December 2005, 17.1 % more than in December 2004. During 2005, cinemas achieved 175.65 million admissions, or 10.1 % fewer than in 2004. Search for missing data. 1 Verify that the number of entries in 2005 for the month of February is 16.75 million. 2 Calculate the percentage drop in attendance between Au-gust 2004 and August 2005. 3 Calculate the number of admissions in May 2004. Part 2 A survey was carried out among a population of schoolchil-dren to find out their preference for the versions of foreign films they see at the cinema. The results are recorded in the following table : Garçons Filles Total Version doublée en français 40.45 % 23 % 63.45 % Version orginale sous-titrée 15.2 % 14.1 % 29.3 % Sans préférence 4.35 % 2.9 % 7.25 % Total 60 % 40 % 100 % This shows that 63.45 % of the students surveyed prefer a ver-sion dubbed in French. On the other hand, the number of girls who prefer the original version with subtitles represents 14.1 % of the total popula-tion of students surveyed. Clearly justifying with the necessary calculations, say whether the following statements are true or false : 1 ˇThe proportion of girls surveyed who have no preference for the version of foreign films seen at the cinema is the same as that of boys surveyed who have no preference for the vers ion of foreign films seen at the cinema.ı 2 ˇAmong the students surveyed who prefer the orginal ver-sion with subtitles, there are more 50 % boys.ı https://chingmath.fr sacados/2029 France - Septembre 2007 - 8 points
E.194 At the end of the deliberations of an examination comprising three tests, a teacher records the results of his 30 students in tests n o 1 , n o 2 and n o 3 . These notes are grouped together in the following table : Notes on 20 Effectifs Proof n o 1 Proof n o 2 Proof n o 3 5 0 3 0 6 6 0 0 7 5 5 2 8 8 0 1 9 1 8 6 10 3 0 3 11 0 3 5 12 2 4 0 13 0 0 2 14 1 1 6 15 2 4 3 16 2 2 2 1 In this question, we are interested in the statistical series E 1 formed by the scores of the test n o 1 . a Determine, for this statistical series, the minimum and maximum. b Determine the median. Justify. c Determine the 1 er and 3 e quartiles. Justify. d Draw the box plot corresponding to this series E 1 , on the sheet provided in the appendix, with the minimum and maximum for extreme values. 2 We are now interested in the statistical series E 2 formed by the test scores n o 2 . a Draw up the box plot corresponding to this series, on the attached sheet, with the minimum and maximum as extreme values. Specify the values used. b Calculate the arithmetic mean of the series E 2 . c Give the value of the standard deviation of the series E 2 . 3 What comments can you make comparing the two box plots corresponding to the E 1 and E 2 series? 4 We note E 3 the statistical series formed by scores on the n o 3 test. The standard deviation of the series E 3 is assumed to be 2.7. a Calculate the arithmetic mean of the series E 3 . b Calculate the percentage of students with a mark less than or equal to 9 in the test n o 3 . c What comments can you make comparing the results of the n o 2 test with those of the n o 3 test? 5 Knowing that the arithmetic mean in the n o 1 test is 9.13 and that this test n o 1 is assigned coefficient 3 and the tests n o 2 and n o 3 coefficient 1, what is the arithmetic mean, out of 20, of the 30 students’ marks in this exam? Statistical series E 1 - box diagram Statistical series E 2 - box diagram E.2022 The green alga Caulerpa taxifolia , native to tropical seas, was introduced to the Mediterranean in the early 1980s. Its adaptations make it highly competitive with Mediterranean species. Caulerpa taxifolia has not only managed to survive a new environment (conditions different from those in trop-ical waters) , but it is proliferating and developing to the point of raising some concerns about the consequences of its expansion. Sources : GIF. Posidonia Part A. Study of the evolution of the surface covered The following table shows the surface area covered by algae during recent measurements in the Mediterranean : Année 1989 1992 1993 1996 1997 Surface (en ha) 1 427 1300 3052 4630 Is the growth in the area covered by Caulerpa exponential? Justify. Part B. Study of the evolution of the number of col-onized sites The spreadsheet in Appendix 2 presents a table listing the number of sites colonized by the alga in the Mediterranean between 1989 and 1996. 1 In row 3 of the spreadsheet, we want to show the per-centage increase in the number of colonized sites from one year to the next. The cells are in the format ˇpour-centageı. What formula should be written in cell D3 so that it can be copied to the right? Complete the docu-ment with the calculated values, rounded to 1 % . 2 Is the growth in the number of sites colonized by the Caulerpa exponential? Justify using line 3 of the spread-sheet. 3 In cell C4 , we have written the formula shown on the worksheet in Appendix 2 and then copied it to the right. The cells in this row are in the format ˇpourcentageı. What is the formula written in cell F4 ? 4 Complete line 4 of the spreadsheet in Appendix 2, to be returned with the copy. The results will be rounded to 0.1 % . What do these results represent? Part C. Study of the evolution of algal size In the summer of 1996, the size of a stolon (long, rooting stem) of Caulerpa was measured. Its size on July 15 was 85 cm , its size on August 24 was 137 cm . During this forty-day period, stolon growth was found to be linear. Estimate by interpola-tion the size, to the nearest centimeter, of this stolon at 1 er August 1996. Number of sites colonized by the green alga Caulerpa taxifolia , in the Mediterranean from 1989 to 1996 https://chingmath.fr sacados/194 Centres Etrangers - Juin 2004 - 10 points 01020 01020 sacados/2022
E.155 On May 29, 2005, during the French referendum on the European Constitution, a polling firm an-alyzed exit polls in a small town. In this town, 3,062 people are registered to vote. Among these people, we distinguish between voters and non-voters. Among the votes cast by vot-ers, we consider ˇYESı votes, ˇNOı votes, and invalid or blank votes. Throughout this exercise, all percentages will be rounded to 0 ; 1% . Part A 1 Of the 3,062 registered voters, 1,048 did not vote. The referendum turnout rate is the percentage of voters out of the total number of registered voters. Determine this turnout rate. 2 During the vote, 2,000 people reported having voted ˇYESı or ˇNOı in the referendum. Their percentage dis-tribution is given in the following table : Percentage distribution by age group Âge YES NO 18 - 24 years 7,1% 8,9% 25 - 34 ans 10,4% 12,7% 35 - 44 ans 11,0% 16,8% 45 - 59 ans 5,3% 8,7% 60 - 69 ans 6,3% 5,0% 70 years and older 4,4% 3,4% a Find the percentage of people under 25 who voted ˇYESı b Determine the percentage of people between the ages of 18 and 24 c Determine the percentage of people who voted ˇYES.ı d Determine the number of people who voted ˇYES.ı 3 Fill in the data in the table below : 4 Among registered voters, determine the percentage of people who voted ˇNO.ı Part B Based on information obtained from the polling station on May 29, 2005, the institute also compiled the results presented in the table below : Table (Frequencies by row) Percentage Distribution of Registered Voters by Age Group Âge Voters Non-voters Total 18 - 24 years 100% 25 - 34 ans 55,0% 45,0% 100% 35 - 44 ans 68,0% 32,0% 100% 45 - 59 ans 77,3% 22,7% 100% 60 - 69 ans 89,8% 10,2% 100% 70 years and older 70,0% 30,0% 100% The results are given as a percentage of registered voters in each age group. 1 Among the 550 registered voters aged 18 to 24, there are 229 non-voters. What is the abstention rate in this age group? 2 In the table above, what does the number 77 ; 3% at the intersection of the 45-59 -year-old row and the ˇvotersı column mean? 3 Of all people aged 25 to 34, 378 did not vote. How many people in this age group are registered at this polling place? 10. Unclassified financial years E.121 Here is a double-entry table showing the language chosen as an option in a class of 34 students : 1 a Complete the following table : Allemand Espagnol Total Boys 10 17 Girl Total 19 https://chingmath.fr chapExoCorrec/155 sacados/155 Les inscrits......Les abstentionnistes......Les votants......Personnes ayantvoté ˇOUIı : ......Personnes ayantvoté ˇNonı :1110Personnes ayant votéblanc ou nul:...... sacados/121
b Determine the frequency of boys practicing German relative to the class as a whole. 2 a Determine among boys, the frequency of students practicing Spanish. b Complete the following frequency table : Allemand Espagnol Boys Girl Total 100 % 100 % c Complete the following frequency tree : d Find, from the frequency tree, the frequency of boys practicing German (question 1 d ) 3 a Determine among practitioners of German, the fre-quency of boys. b Complete the following table : Allemand Espagnol Total Boys 100 % Girl 100 % c Complete the following frequency tree : d Find the frequency, from this tree, of boys practicing German relative to the whole class (question 1 d ) E.160 A manufacturer of chocolate bars has printed, in large quantities, the same number of images of three singers Miss Pinson, Miss Rossignol and Miss Déci-bel. Mlle Pinson’s image is n o 1 Mlle Rossignol’s is n o 2 , and Mlle Décibel’s is n o 3 . A machine randomly inserts a picture into each chocolate bar made. There are as many chocolate bars containing the image of each singer. Every day, Aline buys a chocolate bar. She would like to have the complete collection of all three singers and wonders how many days it will take her to get it. Part A Aline has used a tree to list the different images that can be obtained over three days. This tree, partially completed, can be found oppo-site. For example, the 3 e possibility 1 1 3 means that on the first day, the choco-late bar contains the image of Mll Pinson, on the second day it contains that of Miss Pinson, and on the third day that of Miss Decibel. 1 Of these 27 possibilities, how many are there that result in a complete collection? https://chingmath.fr allemand%garçons%∏lles%espagnol%garçons%∏lles% garçons%allemand%espagnol%∏lles%allemand%espagnol% sacados/160 Inde - Avril 2005 - 8 points 1111 1 121 1 231 1 3211 2 121 2 231 2 3311 3 121 3 231 3 32112 1 122 1 232 1 3212 2 122 2 232 2 3312 3 122 3 232 3 33113 1 123 1 233 1 3213 2 123 2 233 2 3313 3 123 3 233 3 3
2 Are there more than 25 % cases in which a complete col-lection is obtained? Justify. Aline wants to obtain the complete collection. As her pocket money is limited, she would like to estimate the number of days she can expect to get the full collection. To do this, she’s going to run some simulations. Part B She runs a simulation by having her calculator display a ran-dom list of numbers, so that each of the numbers 1, 2 and 3 has the same chance of appearing. Here’s the list she gets : 1-1-1-1-1-3-1-2-1-3. According to this simulation, for the first five days, Aline dis-covers in her chocolate bar the image of Mlle Pinson, on 6 e day, miss Décibel’s, 7 e day, Miss Pinson’s, 8 e day Miss Rossig-nol’s ; on 9 e day that of Mlle Pinson, and on 10 e that of Mlle Décibel. Aline is therefore in possession of the complete col-lection at 8 e day. Imagine, on the previous model, a list of 9 numbers leading to the complete collection obtained at 5 e day. Part C To get a clearer idea, Aline runs 1000 simulations using her calculator. The results are shown in the following table : pt Number of jours nécessaires at obtention de the collection complète 3 4 5 6 7 8 9 10 Effectifs 227 203 179 126 99 56 40 25 pt Numbers from jours nécessaires to obtention de the collection complète 11 12 13 14 15 16 17 18 Effectifs 18 12 4 3 3 2 1 2 This means, for example, that of the 1000 simulations, 203 are situations for which the complete collection of images is obtained by 4 e day. 1 Determine the median, first and third quartile of this statistical series. 2 Aline makes two remarks while observing these simulated results. Remark 1 : In at least 50 % of the simulated situations the complete collection is obtained no later than the . . . day. Note 2 : In . . . % simulated situations, the complete im-age collection is obtained by 7 e day. Complete these remarks. 3 Aline says : ˇAfter 18 days, I’m sure I’ll get the full col-lectionı. What do you think of this statement? https://chingmath.fr