CHAPTER 4 STATISTICS AND PROBABILITY (A) Mai Cocepts ad Results Statistics Meaig of statistics, Primary ad secodary data, Raw/ugrouped data, Rage of data, Grouped data-class itervals, Class marks, Presetatio of data - frequecy distributio table, Discrete frequecy distributio ad cotiuous frequecy distributio. Graphical represetatio of data : (i) Bar graphs (ii) Histograms of uiform width ad of varyig widths (iii) Frequecy polygos Measures of Cetral tedecy (a) (i) Mea Mea of raw data xi x + x2 +... + x i= Mea = x = = where x, x 2,..., x are observatios.
30 EXEMPLAR PROBLEMS (ii) Mea of ugrouped data x (b) Media f x i = f i i where f i s are frequecies of x i s. A media is the value of the observatio which divides the data ito two equal parts, whe the data is arraged i ascedig (or descedig) order. Calculatio of Media Whe the ugrouped data is arraged i ascedig (or descedig) order, the media of data is calculated as follows : (i) (ii) (c) Mode Whe the umber of observatios () is odd, the media is the value of the + 2 th observatio. Whe the umber of observatios () is eve, the media is the average or mea of the 2 th ad + 2 th observatios. The observatio that occurs most frequetly, i.e., the observatio with maximum frequecy is called mode. Mode of ugrouped data ca be determied by observatio/ ispectio. Probability Radom experimet or simply a experimet Outcomes of a experimet Meaig of a trial of a experimet The experimetal (or empirical) probability of a evet E (deoted by P(E)) is give by P(E) = Number of trials i which the evet has happeed Total umber of trials The probability of a evet E ca be ay umber from 0 to. It ca also be 0 or i some special cases.
STATISTICS AND PROBABILITY 3 (B)Multiple Choice Questios Write the correct aswer i each of the followig : Sample Questio : The marks obtaied by 7 studets i a mathematics test (out of 00) are give below : 9, 82, 00, 00, 96, 65, 82, 76, 79, 90, 46, 64, 72, 68, 66, 48, 49. The rage of the data is : (A) 46 (B) 54 (C) 90 (D) 00 Solutio : Aswer (B) Sample Questio 2: The class-mark of the class 30-50 is : (A) 30 (B) 35 (C) 40 (D) 45 Solutio : Aswer (C) Sample Questio 3 : A die is throw 000 times ad the outcomes were recorded as follows : Outcome 2 3 4 5 6 Frequecy 80 50 60 70 50 90 If the die is throw oce more, the the probability that it shows 5 is : (A) 9 50 Solutio : Aswer (B) (B) 3 20 (C) EXERCISE 4. Write the correct aswer i each of the followig :. The class mark of the class 90-20 is : (A) 90 (B) 05 (C) 5 (D) 20 2. The rage of the data : 25, 8, 20, 22, 6, 6, 7, 5, 2, 30, 32, 0, 9, 8,, 20 is 4 25 (D) (A) 0 (B) 5 (C) 8 (D) 26 3. I a frequecy distributio, the mid value of a class is 0 ad the width of the class is 6. The lower limit of the class is : (A) 6 (B) 7 (C) 8 (D) 2 7 25
32 EXEMPLAR PROBLEMS 4. The width of each of five cotiuous classes i a frequecy distributio is 5 ad the lower class-limit of the lowest class is 0. The upper class-limit of the highest class is: (A) 5 (B) 25 (C) 35 (D) 40 5. Let m be the mid-poit ad l be the upper class limit of a class i a cotiuous frequecy distributio. The lower class limit of the class is : (A) 2m + l (B) 2m l (C) m l (D) m 2l 6. The class marks of a frequecy distributio are give as follows : 5, 20, 25,... The class correspodig to the class mark 20 is : (A) 2.5 7.5 (B) 7.5 22.5 (C) 8.5 2.5 (D) 9.5 20.5 7. I the class itervals 0-20, 20-30, the umber 20 is icluded i : (A) 0-20 (B) 20-30 (C) both the itervals (D) oe of these itervals 8. A grouped frequecy table with class itervals of equal sizes usig 250-270 (270 ot icluded i this iterval) as oe of the class iterval is costructed for the followig data : 268, 220, 368, 258, 242, 30, 272, 342, 30, 290, 300, 320, 39, 304, 402, 38, 406, 292, 354, 278, 20, 240, 330, 36, 406, 25, 258, 236. The frequecy of the class 30-330 is: (A) 4 (B) 5 (C) 6 (D) 7 9. A grouped frequecy distributio table with classes of equal sizes usig 63-72 (72 icluded) as oe of the class is costructed for the followig data : 30, 32, 45, 54, 74, 78, 08, 2, 66, 76, 88, 40, 4, 20, 5, 35, 44, 66, 75, 84, 95, 96, 02, 0, 88, 74, 2, 4, 34, 44. The umber of classes i the distributio will be : (A) 9 (B) 0 (C) (D) 2 0. To draw a histogram to represet the followig frequecy distributio : Class iterval 5-0 0-5 5-25 25-45 45-75 Frequecy 6 2 0 8 5
STATISTICS AND PROBABILITY 33 the adjusted frequecy for the class 25-45 is : (A) 6 (B) 5 (C) 3 (D) 2. The mea of five umbers is 30. If oe umber is excluded, their mea becomes 28. The excluded umber is : (A) 28 (B) 30 (C) 35 (D) 38 2. If the mea of the observatios : x, x + 3, x + 5, x + 7, x + 0 is 9, the mea of the last three observatios is (A) 0 3 (B) 2 0 3 (C) 3 (D) 2 3 3. If x represets the mea of observatios x, x 2,..., x, the value of ( xi x) is: (A) (B) 0 (C) (D) 4. If each observatio of the data is icreased by 5, the their mea (A) remais the same (B) becomes 5 times the origial mea (C) is decreased by 5 (D) is icreased by 5 5. Let x be the mea of x, x 2,..., x ad y the mea of y, y 2,..., y. If z is the mea of x, x 2,..., x, y, y 2,..., y, the z is equal to (A) x + y (B) x + y 2 (C) x + y (D) i = x + y 2 6. If x is the mea of x, x 2,..., x, the for a 0, the mea of ax, ax 2,..., ax, x a, x 2 a,..., x a is (A) a + x a (B) x a + a 2 (C) x a + a (D) a + x a 2 7. If x, x 2, x 3,..., x are the meas of groups with, 2,..., umber of observatios respectively, the the mea x of all the groups take together is give by :
34 EXEMPLAR PROBLEMS (A) i = x i i (B) x i i i= 2 (C) i i i= i= x i i x i= (D) 2 i 8. The mea of 00 observatios is 50. If oe of the observatios which was 50 is replaced by 50, the resultig mea will be : (A) 50.5 (B) 5 (C) 5.5 (D) 52 9. There are 50 umbers. Each umber is subtracted from 53 ad the mea of the umbers so obtaied is foud to be 3.5. The mea of the give umbers is : (A) 46.5 (B) 49.5 (C) 53.5 (D) 56.5 20. The mea of 25 observatios is 36. Out of these observatios if the mea of first 3 observatios is 32 ad that of the last 3 observatios is 40, the 3 th observatio is : (A) 23 (B) 36 (C) 38 (D) 40 2. The media of the data 78, 56, 22, 34, 45, 54, 39, 68, 54, 84 is (A) 45 (B) 49.5 (C) 54 (D) 56 22. For drawig a frequecy polygo of a cotious frequecy distributio, we plot the poits whose ordiates are the frequecies of the respective classes ad abcissae are respectively : (A) upper limits of the classes (B) lower limits of the classes (C) class marks of the classes (D) upper limits of perceedig classes 23. Media of the followig umbers : 4, 4, 5, 7, 6, 7, 7, 2, 3 is (A) 4 (B) 5 (C) 6 (D) 7 24. Mode of the data 5, 4, 9, 20, 4, 5, 6, 4, 5, 8, 4, 9, 5, 7, 5 is (A) 4 (B) 5 (C) 6 (D) 7 25. I a sample study of 642 people, it was foud that 54 people have a high school certificate. If a perso is selected at radom, the probability that the perso has a high school certificate is : (A) 0.5 (B) 0.6 (C) 0.7 (D) 0.8
STATISTICS AND PROBABILITY 35 26. I a survey of 364 childre aged 9-36 moths, it was foud that 9 liked to eat potato chips. If a child is selected at radom, the probability that he/she does ot like to eat potato chips is: (A) 0.25 (B) 0.50 (C) 0.75 (D) 0.80 27. I a medical examiatio of studets of a class, the followig blood groups are recorded: Blood group A AB B O Number of studets 0 3 2 5 A studet is selected at radom from the class. The probability that he/she has blood group B, is: 3 3 (A) (B) (C) (D) 4 40 0 8 28. Two cois are tossed 000 times ad the outcomes are recorded as below : Number of heads 2 0 Frequecy 200 550 250 Based o this iformatio, the probability for at most oe head is (A) 5 (B) 4 29. 80 bulbs are selected at radom from a lot ad their life time (i hrs) is recorded i the form of a frequecy table give below : (C) 4 5 (D) Life time (i hours) 300 500 700 900 00 Frequecy 0 2 23 25 0 Oe bulb is selected at radom from the lot. The probability that its life is 50 hours, is (A) 80 (B) 7 6 (C) 0 (D) 3 4
36 EXEMPLAR PROBLEMS 30. Refer to Q.29 above : The probability that bulbs selected radomly from the lot has life less tha 900 hours is : (A) 40 (B) 5 6 (C) 7 6 (D) 9 6 (C) Short Aswer Questios with Reasoig Sample Questio : The mea of the data : 2, 8, 6, 5, 4, 5, 6, 3, 6, 4, 9,, 5, 6, 5 is give to be 5. Based o this iformatio, is it correct to say that the mea of the data: 0, 2, 0, 2, 8, 8, 2, 6, 2, 0, 8, 0, 2, 6, 4 is 0? Give reaso. Solutio : It is correct. Sice the 2d data is obtaied by multiplyig each observatio of st data by 2, therefore, the mea will be 2 times the mea of the st data. Sample Questio 2 : I a histogram, the areas of the rectagles are proportioal to the frequecies. Ca we say that the legths of the rectagles are also proportioal to the frequecies? Solutio: No. It is true oly whe the class sizes are the same. Sample Quetio 3 : Cosider the data : 2, 3, 9, 6, 9, 3, 9. Sice 6 is the highest value i the observatios, is it correct to say that it is the mode of the data? Give reaso. Solutio : 6 is ot the mode of the data. The mode of a give data is the observatio with highest frequecy ad ot the observatio with highest value.. The frequecy distributio : EXERCISE 4.2 Marks 0-20 20-40 40-60 60-00 Number of Studets 0 5 20 25 has bee represeted graphically as follows :
STATISTICS AND PROBABILITY 37 Fig. 4. Do you thik this represetatio is correct? Why? 2. I a diagostic test i mathematics give to studets, the followig marks (out of 00) are recorded: 46, 52, 48,, 4, 62, 54, 53, 96, 40, 98, 44 Which average will be a good represetative of the above data ad why? 3. A child says that the media of 3, 4, 8, 20, 5 is 8. What does t the child uderstad about fidig the media? 4. A football player scored the followig umber of goals i the 0 matches :, 3, 2, 5, 8, 6,, 4, 7, 9 Sice the umber of matches is 0 (a eve umber), therefore, the media = th 5 observatio + 6 observatio 2 th = 8 + 6 = 7 2 Is it the correct aswer ad why? 5. Is it correct to say that i a histogram, the area of each rectagle is proportioal to the class size of the correspodig class iterval? If ot, correct the statemet. 6. The class marks of a cotiuous distributio are :.04,.4,.24,.34,.44,.54 ad.64 Is it correct to say that the last iterval will be.55 -.73? Justify your aswer.
38 EXEMPLAR PROBLEMS 7. 30 childre were asked about the umber of hours they watched TV programmes last week. The results are recorded as uder : Number of hours 0-5 5-0 0-5 5-20 Frequecy 8 6 4 2 Ca we say that the umber of childre who watched TV for 0 or more hours a week is 22? Justify your aswer. 8. Ca the experimetal probability of a evet be a egative umber? If ot, why? 9. Ca the experimetal probability of a evet be greater tha? Justify your awer. 0. As the umber of tosses of a coi icreases, the ratio of the umber of heads to the total umber of tosses will be. Is it correct? If ot, write the correct oe. 2 (D) Short Aswer Questios Sample Questio : Heights (i cm) of 30 girls of Class IX are give below: 40, 40, 60, 39, 53, 53, 46, 50, 48, 50, 52, 46, 54, 50, 60, 48, 50, 48, 40, 48, 53, 38, 52, 50, 48, 38, 52, 40, 46, 48. Prepare a frequecy distributio table for this data. Solutio : Frequecy distributio of heights of 30 girls Height Tally Marks Frequecy (i cm) 38 2 39 40 4 46 3 48 6 50 5 52 3 53 3 54 60 2 Total 30
STATISTICS AND PROBABILITY 39 Sample Questio 2 : The followig observatios are arraged i ascedig order : 26, 29, 42, 53, x, x + 2, 70, 75, 82, 93 If the media is 65, fid the value of x. Solutio : Number of observatios () = 0, which is eve. Therefore, media is the mea of 2 Here, th ad + 2 th observatio, i.e., 5 th ad 6 th observatio. 5 th observatio = x 6 th observatio = x + 2 Now, Media = x + ( x + 2) = x + 2 x + = 65 (Give) Therefore, x = 64 Thus, the value of x is 64. Sample Questio 3 : Here is a extract from a mortality table. (i) (ii) Solutio : (i) Age (i years) Number of persos survivig out of a sample of oe millio 60 6090 6 490 62 802 63 5448 64 3607 65 2320 Based o this iformatio, what is the probability of a perso aged 60 of dyig withi a year? What is the probability that a perso aged 6 will live for 4 years? We see that 6090 persos aged 60, (6090-490), i.e., 4600 died before reachig their 6 st birthday. Therefore, P(a perso aged 60 die withi a year) = 4600 6090 = 460 609
40 EXEMPLAR PROBLEMS (ii) Number of persos aged 6 years = 490 Number of persos survivig for 4 years = 2320 P(a perso aged 6 will live for 4 years) = 2320 490 = 232 49 EXERCISE 4.3. The blood groups of 30 studets are recorded as follows: A, B, O, A, AB, O, A, O, B, A, O, B, A, AB, B, A, AB, B, A, A, O, A, AB, B, A, O, B, A, B, A Prepare a frequecy distributio table for the data. 2. The value of π upto 35 decimal places is give below: 3. 459265358979323846264338327950288 Make a frequecy distributio of the digits 0 to 9 after the decimal poit. 3. The scores (out of 00) obtaied by 33 studets i a mathematics test are as follows: 69, 48, 84, 58, 48, 73, 83, 48, 66, 58, 84 000 66, 64, 7, 64, 66, 69, 66, 83, 66, 69, 7 8, 7, 73, 69, 66, 66, 64, 58, 64, 69, 69 Represet this data i the form of a frequecy distributio. 4. Prepare a cotiuous grouped frequecy distributio from the followig data: Mid-poit Frequecy 5 4 5 8 25 3 35 2 45 6 Also fid the size of class itervals. 5. Covert the give frequecy distributio ito a cotiuous grouped frequecy distributio:
STATISTICS AND PROBABILITY 4 Class iterval Frequecy 50-53 7 54-57 7 58-6 5 62-65 0 66-69 5 70-73 6 I which itervals would 53.5 ad 57.5 be icluded? 6. The expediture of a family o differet heads i a moth is give below: Head Food Educatio Clothig House Ret Others Savigs Expediture 4000 2500 000 3500 2500 500 (i Rs) Draw a bar graph to represet the data above. 7. Expediture o Educatio of a coutry durig a five year period (2002-2006), i crores of rupees, is give below: Elemetary educatio 240 Secodary Educatio 20 Uiversity Educatio 90 Teacher s Traiig 20 Social Educatio 0 Other Educatioal Programmes 5 Cultural programmes 25 Techical Educatio 25 Represet the iformatio above by a bar graph. 8. The followig table gives the frequecies of most commoly used letters a, e, i, o, r, t, u from a page of a book : Letters a e i o r t u Frequecy 75 25 80 70 80 95 75 Represet the iformatio above by a bar graph.
42 EXEMPLAR PROBLEMS 9. If the mea of the followig data is 20.2, fid the value of p: x 0 5 20 25 30 f 6 8 p 0 6 0. Obtai the mea of the followig distributio: Frequecy Variable 4 4 8 6 4 8 0 3 2. A class cosists of 50 studets out of which 30 are girls. The mea of marks scored by girls i a test is 73 (out of 00) ad that of boys is 7. Determie the mea score of the whole class. 2. Mea of 50 observatios was foud to be 80.4. But later o, it was discovered that 96 was misread as 69 at oe place. Fid the correct mea. 3. Te observatios 6, 4, 5, 7, x +, 2x 3, 30, 32, 34, 43 are writte i a ascedig order. The media of the data is 24. Fid the value of x. 4. The poits scored by a basket ball team i a series of matches are as follows: 7, 2, 7, 27, 25, 5, 4, 8, 0, 24, 48, 0, 8, 7, 0, 28 Fid the media ad mode for the data. 5. I Fig. 4.2, there is a histogram depictig daily wages of workers i a factory. Costruct the frequecy distributio table. Fig. 4.2
STATISTICS AND PROBABILITY 43 6. A compay selected 4000 households at radom ad surveyed them to fid out a relatioship betwee icome level ad the umber of televisio sets i a home. The iformatio so obtaied is listed i the followig table: Fid the probability: (i) (ii) (iii) Mothly icome Number of Televisios/household (i Rs) 0 2 Above 2 < 0000 20 80 0 0 0000-4999 0 240 60 0 5000-9999 0 380 20 30 20000-24999 0 520 370 80 25000 ad above 0 00 760 220 of a household earig Rs 0000 Rs 4999 per year ad havig exactly oe televisio. of a household earig Rs 25000 ad more per year ad owig 2 televisios. of a household ot havig ay televisio. 7. Two dice are throw simultaeously 500 times. Each time the sum of two umbers appearig o their tops is oted ad recorded as give i the followig table: Sum Frequecy 2 4 3 30 4 42 5 55 6 72 7 75 8 70 9 53 0 46 28 2 5
44 EXEMPLAR PROBLEMS If the dice are throw oce more, what is the probability of gettig a sum (i) 3? (ii) more tha 0? (iii) less tha or equal to 5? (iv) betwee 8 ad 2? 8. Bulbs are packed i cartos each cotaiig 40 bulbs. Seve hudred cartos were examied for defective bulbs ad the results are give i the followig table: Number of defective bulbs 0 2 3 4 5 6 more tha 6 Frequecy 400 80 48 4 8 8 3 2 Oe carto was selected at radom. What is the probability that it has (i) o defective bulb? (ii) defective bulbs from 2 to 6? (iii) defective bulbs less tha 4? 9. Over the past 200 workig days, the umber of defective parts produced by a machie is give i the followig table: Number of 0 2 3 4 5 6 7 8 9 0 2 3 defective parts Days 50 32 22 8 2 2 0 0 0 8 6 6 2 2 Determie the probability that tomorrow s output will have (i) (ii) (iii) (iv) o defective part atleast oe defective part ot more tha 5 defective parts more tha 3 defective parts 20. A recet survey foud that the ages of workers i a factory is distributed as follows: Age (i years) 20-29 30-39 40-49 50-59 60 ad above Number of workers 38 27 86 46 3 If a perso is selected at radom, fid the probability that the perso is: (i) (ii) 40 years or more uder 40 years
STATISTICS AND PROBABILITY 45 (iii) (iv) havig age from 30 to 39 years uder 60 but over 39 years (E) Log Aswer Questios Sample Questio : Followig is the frequecy distributio of total marks obtaied by the studets of differet sectios of Class VIII. Marks 00-50 50-200 200-300 300-500 500-800 Number of studets 60 00 00 80 80 Draw a histogram for the distributio above. Solutio: I the give frequecy distributio, the class itervals are ot of equal width. Therefore, we would make modificatios i the legths of the rectagles i the histogram so that the areas of rectagles are proportioal to the frequecies. Thus, we have: Marks Frequecy Width of the class Legth of the rectagle 00-50 60 50 50-200 00 50 200-300 00 00 300-500 80 200 50 60 60 50 = 50 00 00 50 = 50 00 50 00 = 50 80 20 200 = 500-800 80 300 50 80 30 300 = Now, we draw rectagles with legths as give i the last colum. The histogram of the data is give below :
46 EXEMPLAR PROBLEMS Fig. 4.3 Sample Questio 2 : Two sectios of Class IX havig 30 studets each appeared for mathematics olympiad. The marks obtaied by them are show below: 46 3 74 68 42 54 4 6 83 48 37 26 8 64 57 93 72 53 59 38 6 88 75 56 46 66 45 6 54 27 27 44 63 58 43 8 64 67 36 49 50 76 38 47 55 77 62 53 40 7 60 58 45 42 34 46 40 59 42 29 Costruct a group frequecy distributio of the data above usig the classes 0-9, 0-9 etc., ad hece fid the umber of studets who secured more tha 49 marks. Solutio : Class Tally Marks Frequecy 0-9 0-9 2 20-29 4 30-39 6 40-49 5 50-59 2 60-69 0 70-79 6 80-89 3 90-99 Total 60
STATISTICS AND PROBABILITY 47 From the table above, we fid that the umber of studets who secure more tha 49 marks is (2 + 0 + 6 + 3 + ), i.e., 32. EXERCISE 4.4. The followig are the marks (out of 00) of 60 studets i mathematics. 6, 3, 5, 80, 86, 7, 5, 48, 24, 56, 70, 9, 6, 7, 6, 36, 34, 42, 34, 35, 72, 55, 75, 3, 52, 28,72, 97, 74, 45, 62, 68, 86, 35, 85, 36, 8, 75, 55, 26, 95, 3, 7, 78, 92, 62, 52, 56, 5, 63,25, 36, 54, 44, 47, 27, 72, 7, 4, 30. Costruct a grouped frequecy distributio table with width 0 of each class startig from 0-9. 2. Refer to Q above. Costruct a grouped frequecy distributio table with width 0 of each class, i such a way that oe of the classes is 0-20 (20 ot icluded). 3. Draw a histogram of the followig distributio : Heights (i cm) Number of studets 50-53 7 53-56 8 56-59 4 59-62 0 62-65 6 65-68 5 4. Draw a histogram to represet the followig grouped frequecy distributio : Ages (i years) Number of teachers 20-24 0 25-29 28 30-34 32 35-39 48 40-44 50 45-49 35 50-54 2
48 EXEMPLAR PROBLEMS 5. The legths of 62 leaves of a plat are measured i millimetres ad the data is represeted i the followig table : Legth (i mm) Number of leaves 8-26 8 27-35 0 36-44 2 45-53 7 54-62 7 63-7 5 72-80 3 Draw a histogram to represet the data above. 6. The marks obtaied (out of 00) by a class of 80 studets are give below : Marks Number of studets 0-20 6 20-30 7 30-50 5 50-70 6 70-00 26 Costruct a histogram to represet the data above. 7. Followig table shows a frequecy distributio for the speed of cars passig through at a particular spot o a high way : Class iterval (km/h) Frequecy 30-40 3 40-50 6 50-60 25 60-70 65 70-80 50 80-90 28 90-00 4 Draw a histogram ad frequecy polygo represetig the data above.
STATISTICS AND PROBABILITY 49 8. Refer to Q. 7 : Draw the frequecy polygo represetig the above data without drawig the histogram. 9. Followig table gives the distributio of studets of sectios A ad B of a class accordig to the marks obtaied by them. Sectio A Sectio B Marks Frequecy Marks Frequecy 0-5 5 0-5 3 5-30 2 5-30 6 30-45 28 30-45 25 45-60 30 45-60 27 60-75 35 60-75 40 75-90 3 75-90 0 Represet the marks of the studets of both the sectios o the same graph by two frequecy polygos.what do you observe? 0. The mea of the followig distributio is 50. x f 0 7 30 5a + 3 50 32 70 7a 90 9 Fid the value of a ad hece the frequecies of 30 ad 70.. The mea marks (out of 00) of boys ad girls i a examiatio are 70 ad 73, respectively. If the mea marks of all the studets i that examiatio is 7, fid the ratio of the umber of boys to the umber of girls. 2. A total of 25 patiets admitted to a hospital are tested for levels of blood sugar, (mg/dl) ad the results obtaied were as follows : 87 7 83 67 85 77 69 76 65 85 85 54 70 68 80 73 78 68 85 73 8 78 8 77 75 Fid mea, media ad mode (mg/dl) of the above data.