STATISTICS AND PROBABILITY

CHAPTER 3 STATISTICS AND PROBABILITY (A) Man Concepts and Results Statstcs Measures of Central Tendency (a) Mean of Grouped Data () () () To fnd the mean of grouped data, t s assumed that the frequency of each class nterval s centred around ts md-pont. Drect Method Mean ( x ) = f x f, where the x (class mark) s the md-pont of the th class nterval and f s the correspondng frequency. Assumed Mean Method Mean ( x ) = a f d +, f a s the assumed mean and d = x a are the devatons of x from a for each.

54 EXEMPLAR PROBLEMS (v) Step-devaton Method Mean ( x ) = f u a+ h f, (v) where a s the assumed mean, h s the class sze and x a u =. h If the class szes are unequal, the formula n (v) can stll be appled by takng h to be a sutable dvsor of all the d s. (b) Mode of Grouped Data () () In a grouped frequency dstrbuton, t s not possble to determne the mode by lookng at the frequences. To fnd the mode of grouped data, locate the class wth the maxmum frequency. Ths class s known as the modal class. The mode of the data s a value nsde the modal class. Mode of the grouped data can be calculated by usng the formula Mode = l f f 2 f f f 0 + 0 2 h, where l s the lower lmt of the modal class, h s the sze of the class, f s frequency of the modal class and f 0 and f 2 are the frequences of the classes precedng and succeedng the modal class, respectvely. (c) Medan of Grouped Data () () Cumulatve frequency table the less than type and the more than type of the grouped frequency dstrbuton. If n s the total number of observatons, locate the class whose cumulatve frequency s greater than (and nearest to) 2 n. Ths class s called the medan class. () Medan of the grouped data can be calculated by usng the formula : n cf Medan = l 2 + h, f

STATISTICS AND PROBABILITY 55 where l s the lower lmt of the medan class, n s the number of observatons, h s the class sze, cf s the cumulatve frequency of the class precedng the medan class and f s the frequency of the medan class. (d) Graphcal Representaton of Cumulatve Frequency Dstrbuton (Ogve) Less than type and more than type. () () Probablty To fnd medan from the graph of cumulatve frequency dstrbuton (less than type) of a grouped data. To fnd medan from the graphs of cumulatve frequency dstrbutons (of less than type and more than type) as the abscssa of the pont of ntersecton of the graphs. Random experment, outcome of an experment, event, elementary events. Equally lkely outcomes. The theoretcal (or classcal) probablty of an event E [denoted by P(E)] s gven by P(E) = Number of outcomes favourable to E Number of all possble outcomes of the experment where the outcomes of the experment are equally lkely. The probablty of an event can be any number between 0 and. It can also be 0 or n some specal cases. The sum of the probabltes of all the elementary events of an experment s. For an event E, P(E) + P( E ) =, where E s the event not E. E s called the complement of the event E. Impossble event, sure or a certan event (B) Multple Choce Questons Choose the correct answer from the gven four optons: Sample Queston : Constructon of a cumulatve frequency table s useful n determnng the (A) mean (B) medan (C) mode (D) all the above three measures Soluton : Answer (B)

56 EXEMPLAR PROBLEMS Sample Queston 2 : In the followng dstrbuton : Monthly ncome range (n Rs) Number of famles Income more than Rs 0000 00 Income more than Rs 3000 85 Income more than Rs 6000 69 Income more than Rs 9000 50 Income more than Rs 22000 33 Income more than Rs 25000 5 the number of famles havng ncome range (n Rs) 6000 9000 s (A) 5 (B) 6 (C) 7 (D) 9 Soluton : Answer (D) Sample Queston 3 : Consder the followng frequency dstrbuton of the heghts of 60 students of a class : Heght (n cm) Number of students 50-55 5 55-60 3 60-65 0 65-70 8 70-75 9 75-80 5 The sum of the lower lmt of the modal class and upper lmt of the medan class s (A) 30 (B) 35 (C) 320 (D) 330 Soluton : Answer (B) Sample Queston 4 : Whch of the the followng can be the probablty of an event? (A) 0.04 (B).004 (C) 8 23 (D) 8 7 Soluton : Answer (C)

STATISTICS AND PROBABILITY 57 Sample Queston 5 : A card s selected at random from a well shuffled deck of 52 playng cards. The probablty of ts beng a face card s 3 4 6 9 (A) (B) (C) (D) 3 3 3 3 Soluton : Answer (A) Sample Queston 6 : A bag contans 3 red balls, 5 whte balls and 7 black balls. What s the probablty that a ball drawn from the bag at random wll be nether red nor black? (A) (B) 5 Soluton : Answer (B) 3 (C) EXERCISE 3. 7 5 Choose the correct answer from the gven four optons:. In the formula x = a f d +, f (D) for fndng the mean of grouped data d s are devatons from a of (A) (B) (C) (D) lower lmts of the classes upper lmts of the classes md ponts of the classes frequences of the class marks 2. Whle computng mean of grouped data, we assume that the frequences are (A) (B) (C) (D) evenly dstrbuted over all the classes centred at the classmarks of the classes centred at the upper lmts of the classes centred at the lower lmts of the classes 3. If x s are the md ponts of the class ntervals of grouped data, f s are the correspondng frequences and x s the mean, then ( f x x ) s equal to (A) 0 (B) (C) (D) 2 fu 4. In the formula x= a + h, for fndng the mean of grouped frequency f dstrbuton, u = (A) x + a h (B) h (x a) (C) x a h (D) 8 5 a x h

58 EXEMPLAR PROBLEMS 5. The abscssa of the pont of ntersecton of the less than type and of the more than type cumulatve frequency curves of a grouped data gves ts (A) mean (B) medan (C) mode (D) all the three above 6. For the followng dstrbuton : Class 0-5 5-0 0-5 5-20 20-25 Frequency 0 5 2 20 9 the sum of lower lmts of the medan class and modal class s (A) 5 (B) 25 (C) 30 (D) 35 7. Consder the followng frequency dstrbuton : Class 0-5 6-2-7 8-23 24-29 Frequency 3 0 5 8 The upper lmt of the medan class s (A) 7 (B) 7.5 (C) 8 (D) 8.5 8. For the followng dstrbuton : Marks the modal class s Number of students Below 0 3 Below 20 2 Below 30 27 Below 40 57 Below 50 75 Below 60 80 (A) 0-20 (B) 20-30 (C) 30-40 (D) 50-60 9. Consder the data : Class 65-85 85-05 05-25 25-45 45-65 65-85 85-205 Frequency 4 5 3 20 4 7 4

STATISTICS AND PROBABILITY 59 The dfference of the upper lmt of the medan class and the lower lmt of the modal class s (A) 0 (B) 9 (C) 20 (D) 38 0. The tmes, n seconds, taken by 50 atheletes to run a 0 m hurdle race are tabulated below : Class 3.8-4 4-4.2 4.2-4.4 4.4-4.6 4.6-4.8 4.8-5 Frequency 2 4 5 7 48 20 The number of atheletes who completed the race n less then 4.6 seconds s : (A) (B) 7 (C) 82 (D) 30. Consder the followng dstrbuton : Marks obtaned Number of students More than or equal to 0 63 More than or equal to 0 58 More than or equal to 20 55 More than or equal to 30 5 More than or equal to 40 48 More than or equal to 50 42 the frequency of the class 30-40 s (A) 3 (B) 4 (C) 48 (D) 5 2. If an event cannot occur, then ts probablty s (A) (B) 3 4 (C) 3. Whch of the followng cannot be the probablty of an event? 2 (D) 0 (A) 3 (B) 0. (C) 3% (D) 4. An event s very unlkely to happen. Its probablty s closest to (A) 0.000 (B) 0.00 (C) 0.0 (D) 0. 5. If the probablty of an event s p, the probablty of ts complementary event wll be 7 6 (A) p (B) p (C) p (D) p

60 EXEMPLAR PROBLEMS 6. The probablty expressed as a percentage of a partcular occurrence can never be (A) less than 00 (B) less than 0 (C) greater than (D) anythng but a whole number 7. If P(A) denotes the probablty of an event A, then (A) P(A) < 0 (B) P(A) > (C) 0 P(A) (D) P(A) 8. A card s selected from a deck of 52 cards. The probablty of ts beng a red face card s (A) 3 26 (B) 3 3 9. The probablty that a non leap year selected at random wll contan 53 sundays s (A) 7 (B) 2 7 20. When a de s thrown, the probablty of gettng an odd number less than 3 s (A) 6 (B) 3 (C) (C) (C) 2 3 3 7 2 (D) (D) 2 5 7 (D) 0 2. A card s drawn from a deck of 52 cards. The event E s that card s not an ace of hearts. The number of outcomes favourable to E s (A) 4 (B) 3 (C) 48 (D) 5 22. The probablty of gettng a bad egg n a lot of 400 s 0.035. The number of bad eggs n the lot s (A) 7 (B) 4 (C) 2 (D) 28 23. A grl calculates that the probablty of her wnnng the frst prze n a lottery s 0.08. If 6000 tckets are sold, how many tckets has she bought? (A) 40 (B) 240 (C) 480 (D) 750 24. One tcket s drawn at random from a bag contanng tckets numbered to 40. The probablty that the selected tcket has a number whch s a multple of 5 s (A) 5 (B) 3 5 25. Someone s asked to take a number from to 00. The probablty that t s a prme s (C) 4 5 (D) 3 (A) 5 (B) 6 25 (C) 4 (D) 3 50

STATISTICS AND PROBABILITY 6 26. A school has fve houses A, B, C, D and E. A class has 23 students, 4 from house A, 8 from house B, 5 from house C, 2 from house D and rest from house E. A sngle student s selected at random to be the class montor. The probablty that the selected student s not from A, B and C s (A) 4 23 (B) 6 23 (C) (C) Short Answer Questons wth Reasonng Sample Queston : The mean of ungrouped data and the mean calculated when the same data s grouped are always the same. Do you agree wth ths statement? Gve reason for your answer. Soluton : The statement s not true. The reason s that when we calculated mean of a grouped data, t s assumed that frequency of each class s centred at the md-pont of the class. Because of ths, two values of the mean, namely, those from ungrouped and grouped data are rarely the same. Sample Queston 2 : Is t correct to say that an ogve s a graphcal representaton of a frequency dstrbuton? Gve reason. Soluton : Graphcal representaton of a frequency dstrbuton may not be an ogve. It may be a hstogram. An ogve s a graphcal representaton of cumulatve frequency dstrbuton. Sample Queston 3 : In any stuaton that has only two possble outcomes, each 8 23 outcome wll have probablty. True or false? Why? 2 (D) 7 23 Soluton : False, because the probablty of each outcome wll be 2 only when the two outcomes are equally lkely otherwse not. EXERCISE 3.2. The medan of an ungrouped data and the medan calculated when the same data s grouped are always the same. Do you thnk that ths s a correct statement? Gve reason. 2. In calculatng the mean of grouped data, grouped n classes of equal wdth, we may use the formula x = a f d + f

62 EXEMPLAR PROBLEMS where a s the assumed mean. a must be one of the md-ponts of the classes. Is the last statement correct? Justfy your answer. 3. Is t true to say that the mean, mode and medan of grouped data wll always be dfferent? Justfy your answer. 4. Wll the medan class and modal class of grouped data always be dfferent? Justfy your answer. 5. In a famly havng three chldren, there may be no grl, one grl, two grls or three grls. So, the probablty of each s. Is ths correct? Justfy your answer. 4 6. A game conssts of spnnng an arrow whch comes to rest pontng at one of the regons (, 2 or 3) (Fg. 3.). Are the outcomes, 2 and 3 equally lkely to occur? Gve reasons. 7. Apoorv throws two dce once and computes the product of the numbers appearng on the dce. Peehu throws one de and squares the number that appears on t. Who has the better chance of gettng the number 36? Why? 8. When we toss a con, there are two possble outcomes - Head or Tal. Therefore, the probablty of each outcome s. Justfy your answer. 2 9. A student says that f you throw a de, t wll show up or not. Therefore, the probablty of gettng and the probablty of gettng not each s equal to ths correct? Gve reasons. 2. Is 0. I toss three cons together. The possble outcomes are no heads, head, 2 heads and 3 heads. So, I say that probablty of no heads s. What s wrong wth ths 4 concluson?. If you toss a con 6 tmes and t comes down heads on each occason. Can you say that the probablty of gettng a head s? Gve reasons.

STATISTICS AND PROBABILITY 63 2. Sushma tosses a con 3 tmes and gets tal each tme. Do you thnk that the outcome of next toss wll be a tal? Gve reasons. 3. If I toss a con 3 tmes and get head each tme, should I expect a tal to have a hgher chance n the 4 th toss? Gve reason n support of your answer. 4. A bag contans slps numbered from to 00. If Fatma chooses a slp at random from the bag, t wll ether be an odd number or an even number. Snce ths stuaton has only two possble outcomes, so, the probablty of each s 2. Justfy. (D) Short Answer Questons Sample Queston : Construct the cumulatve frequency dstrbuton of the followng dstrbuton : Class 2.5-7.5 7.5-22.5 22.5-27.5 27.5-32.5 32.5-37.5 Frequency 2 22 9 4 3 Soluton : The requred cumulatve frequency dstrbuton of the gven dstrbuton s gven below : Class Frequency Cumulatve frequency 2.5-7.5 2 2 7.5-22.5 22 24 22.5-27.5 9 43 27.5-32.5 4 57 32.5-37.5 3 70 Sample Queston 2 : Daly wages of 0 workers, obtaned n a survey, are tabulated below : Daly wages (n Rs) Number of workers 00-20 0 20-40 5 40-60 20 60-80 22 80-200 8 200-220 2 220-240 3 Compute the mean daly wages of these workers.

64 EXEMPLAR PROBLEMS Soluton : We frst fnd the classmark, x, of each class and then proceed as follows: Daly wages Class marks Number of workers f x (n Rs) (x ) (f ) Classes 00-20 0 0 00 20-40 30 5 950 40-60 50 20 3000 60-80 70 22 3740 80-200 90 8 3420 200-220 20 2 2520 220-240 230 3 2990 f = 0, f x = 8720 Therefore, Mean = x = f x f = 8720 0 = 70.20 Hence, the mean daly wages of the workers s Rs 70.20. Note : Mean daly wages can also be calculated by the assumed mean method or step devaton method. Sample Queston 3 : The percentage of marks obtaned by 00 students n an examnaton are gven below: Marks 30-35 35-40 40-45 45-50 50-55 55-60 60-65 Frequency 4 6 8 23 8 8 3 Determne the medan percentage of marks. Soluton : Marks Number of Students Cumulatve frequency (Class) (Frequency) 30-35 4 4 35-40 6 30 40-45 8 48 45-50 23 7 Medan class 50-55 8 89 55-60 8 97 60-65 3 00

STATISTICS AND PROBABILITY 65 Here, n = 00. n Therefore, = 50, Ths observaton les n the class 45-50. 2 l (the lower lmt of the medan class) = 45 cf (the cumulatve frequency of the class precedng the medan class) = 48 f (the frequency of the medan class) = 23 h (the class sze) = 5 Medan = l + n cf 2 h f = 50 48 45 + 5 23 = 0 45 + = 45.4 23 So, the medan percentage of marks s 45.4. Sample Queston 4 : The frequency dstrbuton table of agrcultural holdngs n a vllage s gven below : Area of land (n hectares) -3 3-5 5-7 7-9 9- -3 Number of famles 20 45 80 55 40 2 Fnd the modal agrcultural holdngs of the vllage. Soluton : Here the maxmum class frequency s 80, and the class correspondng to ths frequency s 5-7. So, the modal class s 5-7. l ( lower lmt of modal class) = 5 f (frequency of the modal class) = 80

66 EXEMPLAR PROBLEMS f 0 (frequency of the class precedng the modal class) = 45 f 2 (frequency of the class succeedng the modal class) = 55 h (class sze) = 2 Mode = l + f f0 2 f f f 0 2 h = 80 45 5 + 2 2(80) 45 55 = 5 + 35 2 60 = 5 + 35 30 = 5 +.2 = 6.2 Hence, the modal agrcultural holdngs of the vllage s 6.2 hectares. EXERCISE 3.3. Fnd the mean of the dstrbuton : Class -3 3-5 5-7 7-0 Frequency 9 22 27 7 2. Calculate the mean of the scores of 20 students n a mathematcs test : Marks 0-20 20-30 30-40 40-50 50-60 Number of students 2 4 7 6 3. Calculate the mean of the followng data : Class 4 7 8 2 5 6 9 Frequency 5 4 9 0

STATISTICS AND PROBABILITY 67 4. The followng table gves the number of pages wrtten by Sarka for completng her own book for 30 days : Number of pages wrtten per day 6-8 9-2 22-24 25-27 28-30 Number of days 3 4 9 3 Fnd the mean number of pages wrtten per day. 5. The daly ncome of a sample of 50 employees are tabulated as follows : Income (n Rs) -200 20-400 40-600 60-800 Number of employees 4 5 4 7 Fnd the mean daly ncome of employees. 6. An arcraft has 20 passenger seats. The number of seats occuped durng 00 flghts s gven n the followng table : Number of seats 00-04 04-08 08-2 2-6 6-20 Frequency 5 20 32 8 5 Determne the mean number of seats occuped over the flghts. 7. The weghts (n kg) of 50 wrestlers are recorded n the followng table : Weght (n kg) 00-0 0-20 20-30 30-40 40-50 Number of wrestlers 4 4 2 8 3 Fnd the mean weght of the wrestlers. 8. The mleage (km per ltre) of 50 cars of the same model was tested by a manufacturer and detals are tabulated as gven below :

68 EXEMPLAR PROBLEMS Mleage (km/l) 0-2 2-4 4-6 6-8 Number of cars 7 2 8 3 Fnd the mean mleage. The manufacturer clamed that the mleage of the model was 6 km/ltre. Do you agree wth ths clam? 9. The followng s the dstrbuton of weghts (n kg) of 40 persons : Weght (n kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 Number of persons 4 4 3 5 6 5 2 Construct a cumulatve frequency dstrbuton (of the less than type) table for the data above. 0. The followng table shows the cumulatve frequency dstrbuton of marks of 800 students n an examnaton: Marks Number of students Below 0 0 Below 20 50 Below 30 30 Below 40 270 Below 50 440 Below 60 570 Below 70 670 Below 80 740 Below 90 780 Below 00 800 Construct a frequency dstrbuton table for the data above.

STATISTICS AND PROBABILITY 69. Form the frequency dstrbuton table from the followng data : Marks (out of 90) Number of canddates More than or equal to 80 4 More than or equal to 70 6 More than or equal to 60 More than or equal to 50 7 More than or equal to 40 23 More than or equal to 30 27 More than or equal to 20 30 More than or equal to 0 32 More than or equal to 0 34 2. Fnd the unknown entres a, b, c, d, e, f n the followng dstrbuton of heghts of students n a class : Heght Frequency Cumulatve frequency (n cm) 50-55 2 a 55-60 b 25 60-65 0 c 65-70 d 43 70-75 e 48 75-80 2 f Total 50 3. The followng are the ages of 300 patents gettng medcal treatment n a hosptal on a partcular day : Age (n years) 0-20 20-30 30-40 40-50 50-60 60-70 Number of patents 60 42 55 70 53 20

70 EXEMPLAR PROBLEMS Form: () () Less than type cumulatve frequency dstrbuton. More than type cumulatve frequency dstrbuton. 4. Gven below s a cumulatve frequency dstrbuton showng the marks secured by 50 students of a class : Marks Below 20 Below 40 Below 60 Below 80 Below 00 Number of students 7 22 29 37 50 Form the frequency dstrbuton table for the data. 5. Weekly ncome of 600 famles s tabulated below : Weekly ncome (n Rs) Number of famles 0-000 250 000-2000 90 2000-3000 00 3000-4000 40 4000-5000 5 5000-6000 5 Total 600 Compute the medan ncome. 6. The maxmum bowlng speeds, n km per hour, of 33 players at a crcket coachng centre are gven as follows : Speed (km/h) 85-00 00-5 5-30 30-45 Number of players 9 8 5 Calculate the medan bowlng speed.

STATISTICS AND PROBABILITY 7 7. The monthly ncome of 00 famles are gven as below : Income (n Rs) Number of famles 0-5000 8 5000-0000 26 0000-5000 4 5000-20000 6 20000-25000 3 25000-30000 3 30000-35000 2 35000-40000 Calculate the modal ncome. 8. The weght of coffee n 70 packets are shown n the followng table : Weght (n g) Number of packets 200-20 2 20-202 26 202-203 20 203-204 9 204-205 2 205-206 Determne the modal weght. 9. Two dce are thrown at the same tme. Fnd the probablty of gettng () () same number on both dce. dfferent numbers on both dce. 20. Two dce are thrown smultaneously. What s the probablty that the sum of the numbers appearng on the dce s () 7? () a prme number? ()?

72 EXEMPLAR PROBLEMS 2. Two dce are thrown together. Fnd the probablty that the product of the numbers on the top of the dce s () 6 () 2 () 7 22. Two dce are thrown at the same tme and the product of numbers appearng on them s noted. Fnd the probablty that the product s less than 9. 23. Two dce are numbered, 2, 3, 4, 5, 6 and,, 2, 2, 3, 3, respectvely. They are thrown and the sum of the numbers on them s noted. Fnd the probablty of gettng each sum from 2 to 9 separately. 24. A con s tossed two tmes. Fnd the probablty of gettng at most one head. 25. A con s tossed 3 tmes. Lst the possble outcomes. Fnd the probablty of gettng () all heads () at least 2 heads 26. Two dce are thrown at the same tme. Determne the probabty that the dfference of the numbers on the two dce s 2. 27. A bag contans 0 red, 5 blue and 7 green balls. A ball s drawn at random. Fnd the probablty of ths ball beng a () red ball () green ball () not a blue ball 28. The kng, queen and jack of clubs are removed from a deck of 52 playng cards and then well shuffled. Now one card s drawn at random from the remanng cards. Determne the probablty that the card s () a heart () a kng 29. Refer to Q.28. What s the probablty that the card s () a club () 0 of hearts 30. All the jacks, queens and kngs are removed from a deck of 52 playng cards. The remanng cards are well shuffled and then one card s drawn at random. Gvng ace a value smlar value for other cards, fnd the probablty that the card has a value () 7 () greater than 7 () less than 7 3. An nteger s chosen between 0 and 00. What s the probablty that t s () dvsble by 7? () not dvsble by 7? 32. Cards wth numbers 2 to 0 are placed n a box. A card s selected at random. Fnd the probablty that the card has () an even number () a square number

STATISTICS AND PROBABILITY 73 33. A letter of Englsh alphabets s chosen at random. Determne the probablty that the letter s a consonant. 34. There are 000 sealed envelopes n a box, 0 of them contan a cash prze of Rs 00 each, 00 of them contan a cash prze of Rs 50 each and 200 of them contan a cash prze of Rs 0 each and rest do not contan any cash prze. If they are well shuffled and an envelope s pcked up out, what s the probablty that t contans no cash prze? 35. Box A contans 25 slps of whch 9 are marked Re and other are marked Rs 5 each. Box B contans 50 slps of whch 45 are marked Re each and others are marked Rs 3 each. Slps of both boxes are poured nto a thrd box and resuffled. A slp s drawn at random. What s the probablty that t s marked other than Re? 36. A carton of 24 bulbs contan 6 defectve bulbs. One bulbs s drawn at random. What s the probablty that the bulb s not defectve? If the bulb selected s defectve and t s not replaced and a second bulb s selected at random from the rest, what s the probablty that the second bulb s defectve? 37. A chld s game has 8 trangles of whch 3 are blue and rest are red, and 0 squares of whch 6 are blue and rest are red. One pece s lost at random. Fnd the probablty that t s a () trangle () square () square of blue colour (v) trangle of red colour 38. In a game, the entry fee s Rs 5. The game conssts of a tossng a con 3 tmes. If one or two heads show, Sweta gets her entry fee back. If she throws 3 heads, she receves double the entry fees. Otherwse she wll lose. For tossng a con three tmes, fnd the probablty that she () () () loses the entry fee. gets double entry fee. just gets her entry fee. 39. A de has ts sx faces marked 0,,,, 6, 6. Two such dce are thrown together and the total score s recorded. () How many dfferent scores are possble? () What s the probablty of gettng a total of 7? 40. A lot conssts of 48 moble phones of whch 42 are good, 3 have only mnor defects and 3 have major defects. Varnka wll buy a phone f t s good but the trader wll only buy a moble f t has no major defect. One phone s selected at random from the lot. What s the probablty that t s

74 EXEMPLAR PROBLEMS () () acceptable to Varnka? acceptable to the trader? 4. A bag contans 24 balls of whch x are red, 2x are whte and 3x are blue. A ball s selected at random. What s the probablty that t s () not red? () whte? 42. At a fete, cards bearng numbers to 000, one number on one card, are put n a box. Each player selects one card at random and that card s not replaced. If the selected card has a perfect square greater than 500, the player wns a prze. What s the probablty that () () the frst player wns a prze? the second player wns a prze, f the frst has won? (E) Long Answer Questons Sample Queston : The followng s the cumulatve frequency dstrbuton (of less than type) of 000 persons each of age 20 years and above. Determne the mean age. Age below (n years) 30 40 50 60 70 80 Number of persons 00 220 350 750 950 000 Soluton : Frst, we make the frequency dstrbuton of the gven data and then proceed to calculate mean by computng class marks (x ), u s and f u s as follows : Class Frequency Class mark (f ) (x ) u = x 45 0 f u 20-30 00 25 2 200 30-40 20 35 20 40-50 30 45 0 0 50-60 400 55 400 60-70 200 65 2 400 70-80 50 75 3 50 f = 000 f u 630 =

STATISTICS AND PROBABILITY 75 We have taken assumed mean (a) = 45. Here, h = class sze = 0 Usng the formula Mean = x = a + h fu f = 45 + 0 630 000 = 45 + 6.3 = 5.3 Thus, the mean age s 5.3 years. Sample Queston 2: The mean of the followng dstrbuton s 8. The frequency f n the class nterval 9-2 s mssng. Determne f. Soluton : Class nterval -3 3-5 5-7 7-9 9-2 2-23 23-25 Frequency 3 6 9 3 f 5 4 Class Md-pont Frequency x 8 u = 2 nterval (x ) (f ) f u -3 2 3 3 9 3-5 4 6 2 2 5-7 6 9 9 7-9 8 3 0 0 9-2 20 f f 2-23 22 5 2 0 23-25 24 4 3 2 f = 40 + f fu f 8 Let us take assumed mean (a) = 8. Here h = 2 Mean = x = a + h fu f

76 EXEMPLAR PROBLEMS = 8 + 2 f 40 + 8 f x = 8 (Gven) So, 8 = 8 + or f = 8 ( f ) 2 8 Hence, the frequency of the class nterval 9-2 s 8. 40 + f Sample Queston 3 : The medan of the dstrbuton gven below s 4.4. Fnd the values of x and y, f the total frequency s 20. Soluton : Class nterval 0-6 6-2 2-8 8-24 24-30 Frequency 4 x 5 y Class Frequency Cumulatve frequency nterval 0-6 4 4 6-2 x 4 + x 2-8 5 9 + x 8-24 y 9 + x + y 24-30 0 + x + y It s gven that n = 20. So, 0 + x + y = 20,.e., x + y = 0 () It s also gven that medan = 4.4 whch les n the class nterval 2-8.

STATISTICS AND PROBABILITY 77 So, l = 2, f = 5, cf = 4 + x, h = 6 Usng the formula Medan = n cf l 2 + h f 0 (4 + x) we get, 4.4 = 2+ 6 5 or 6 x 4.4 = 2+ 6 5 or x = 4 (2) From () and (2), y = 6. EXERCISE 3.4. Fnd the mean marks of students for the followng dstrbuton : Marks Number of students 0 and above 80 0 and above 77 20 and above 72 30 and above 65 40 and above 55 50 and above 43 60 and above 28 70 and above 6 80 and above 0 90 and above 8 00 and above 0

78 EXEMPLAR PROBLEMS 2. Determne the mean of the followng dstrbuton : Marks Number of students Below 0 5 Below 20 9 Below 30 7 Below 40 29 Below 50 45 Below 60 60 Below 70 70 Below 80 78 Below 90 83 Below 00 85 3. Fnd the mean age of 00 resdents of a town from the followng data : Age equal and above (n years) 0 0 20 30 40 50 60 70 Number of Persons 00 90 75 50 25 5 5 0 4. The weghts of tea n 70 packets are shown n the followng table : Weght (n gram) Number of packets 200-20 3 20-202 27 202-203 8 203-204 0 204-205 205-206 Fnd the mean weght of packets. 5. Refer to Q.4 above. Draw the less than type ogve for ths data and use t to fnd the medan weght.

STATISTICS AND PROBABILITY 79 6. Refer to Q.4 above. Draw the less than type and more than type ogves for the data and use them to fnd the medan weght. 7. The table below shows the salares of 280 persons. Salary (n thousand (Rs)) Number of persons 5-0 49 0-5 33 5-20 63 20-25 5 25-30 6 30-35 7 35-40 4 40-45 2 45-50 Calculate the medan and mode of the data. 8. The mean of the followng frequency dstrbuton s 50, but the frequences f and f 2 n classes 20-40 and 60-80, respectvely are not known. Fnd these frequences, f the sum of all the frequences s 20. Class 0-20 20-40 40-60 60-80 80-00 Frequency 7 f 32 f 2 9 9. The medan of the followng data s 50. Fnd the values of p and q, f the sum of all the frequences s 90. Marks Frequency 20-30 p 30-40 5 40-50 25 50-60 20 60-70 q 70-80 8 80-90 0

80 EXEMPLAR PROBLEMS 0. The dstrbuton of heghts (n cm) of 96 chldren s gven below : Heght (n cm) Number of chldren 24-28 5 28-32 8 32-36 7 36-40 24 40-44 6 44-48 2 48-52 6 52-56 4 56-60 3 60-64 Draw a less than type cumulatve frequency curve for ths data and use t to compute medan heght of the chldren.. Sze of agrcultural holdngs n a survey of 200 famles s gven n the followng table: Sze of agrcultural holdngs (n ha) Number of famles 0-5 0 5-0 5 0-5 30 5-20 80 20-25 40 25-30 20 30-35 5 Compute medan and mode sze of the holdngs.

STATISTICS AND PROBABILITY 8 2. The annual ranfall record of a cty for 66 days s gven n the followng table. Ranfall (n cm) 0-0 0-20 20-30 30-40 40-50 50-60 Number of days 22 0 8 5 5 6 Calculate the medan ranfall usng ogves (of more than type and of less than type) 3. The followng s the frequency dstrbuton of duraton for00 calls made on a moble phone : Duraton (n seconds) Number of calls 95-25 4 25-55 22 55-85 28 85-25 2 25-245 5 Calculate the average duraton (n sec) of a call and also fnd the medan from a cumulatve frequency curve. 4. 50 students enter for a school javeln throw competton. The dstance (n metres) thrown are recorded below : Dstance (n m) 0-20 20-40 40-60 60-80 80-00 Number of students 6 7 2 4 () Construct a cumulatve frequency table. () Draw a cumulatve frequency curve (less than type) and calculate the medan dstance thrown by usng ths curve. () Calculate the medan dstance by usng the formula for medan. (v) Are the medan dstance calculated n () and () same?