Quantitative Techniques

Unit II:

1) “Statistics are numerical statement of facts but all facts numerically stated are not statistics “. Clarify this statement and point out briefly which numerical statement of facts are statistics.2) Define statistics. Discuss its scope and limitations

3) What are the reasons of distrust in statistics? How can be wrong use of statistics be avoided?

4) Define statistics and explain the functions and importance of statistics.

5) Distinguish between primary and secondary data and discuss the various methods used in the collection of primary data.

6) Distinguish between a census and a sample inquiry and discuss briefly their comparative advantages. Explain the conditions under which each of these methods may be used with advantage.7) Define a statistical error. Discuss the various types of statistical errors and suggest how they can be avoided or corrected?8) Arrange the following series in ascending and descending orders.

(i) Marks (out of 50) : 25 18 42 50 35 11 27 19 46

(ii) Income (in Rs) : 22 16 11 37 23 42 60

No. of workers : 2 8 6 9 16 3 8

(iii) Age Group : 16-24 40-48 48-56 21-32 32-40

No. of persons : 2 7 13 9 4

9) The length of life, measured in hours, of 40 light bulbs appears below. Construct exclusive class intervals using Sturge’s Rule:

318 252 428 508 560 248 304 392

314 400 101 430 108 460 499 387

300 426 208 457 246 438 308 513

403 472 402 589 461 222 491 546

410 202 489 545 504 363 160 481

10) Convert the following into exclusive series:

(i) Total Income(Rs) : 6010-7000 7010-8000 8010-9000 9010-10,000

No. of Assesses : 2132 5585 7519 6201

(ii) Classes : 0-9 10-19 20-29 30-39 40-49 50-59

Frequency : 5 15 13 11 10 9

(iii) Income in Rs : 15-19.99 20-24.99 25-29.99 30-34.99

No of persons : 8 11 22 30

11) Convert the following unequal class-interval series into equal class interval series

Class Frequency Classes Frequency

(i) 0-2 2 14-15 13

2-5 4 15-17 3

5-8 7 17-20 2

8-10 8 20-25 1

10-14 10

(ii) Classes : 20-23 23-25 25-30

Frequency : 8 10 30

Classes : 30-35 35-40 40-50

Frequency : 33 38 45

(iii) Classes : 0-10 10-20 20-50 50-100 100-200 200-250

Frequency : 6 10 17 27 37 57

(iv) Income (Rs) : 0-100 100-200 200-500 500-1000 1000-2000 2000-5000

No. of person : 20 50 100 80 10 2

12) Convert the following series into equal intervals:

(i) Central Value: 10 14 18 22 26 30

Frequency : 2 8 10 12 16 20

(ii) Central Size : 8 12 16 20 24

Frequency : 2 8 14 20 30

13) Convert the following series into unequal intervals:

(i) Mid. Value : 10 20 40 80 120 150

Frequency : 5 12 20 15 8 7

(ii) Mid. Value: 5 10 20 40 100 200

Frequency: 10 18 27 13 10 6

14) Change the following into exclusive continuous series:

Mid Value : 8.5 13.5 18.5 24.5 34.0 47.5 60.0

No. of persons : 4 8 15 25 12 6 4

15) Calculate the mid values from the following values:

(i) Classes : 0-9.5 9.5-19.5 19.5-29.5 29.5-39.5 39.5-49.5

Freq : 10 24 32 40 18

(ii) Groups : 0-5 5-10 10-20 20-30 30-50 50-80

Freq : 5 8 8 15 10 8

16) Compute the mid values from the following series:

(i) Below : 50 40 30 20 10

No. of students : 12 8 5 4 2

(ii) Marks over : 0 40 60 80 100

No. of students : 50 43 25 17 7

17) (i)Convert the following series into cumulative frequency:

Marks : 0-10 10-20 20-30 30-40 40-50 50-60

No. of Students : 2 5 10 12 15 25

(ii)Classes : 50 40 30 20 10

No. of students : 26 16 13 10 5

18) Convert the following into an ordinary series:

Wages in Rs No. of WorkersUnder 25 7Under 30 10Under 35 1535-40 2240-45 2545-50 3050-55 3855-60 40

19) Convert the following into an ordinary series:

Income in Rs No. of PersonsLess than 30 16Less than 40 36Less than 50 6150-60 15Below 70 87Below 80 9580 & above 5

20) From the following information, prepare a table showing class and residence-wise number of students in the faculty of commerce of a college.

Total number of students is 700. Class-wise distribution:

Pre-university 100

B.Com Pt. 1 300

B.Com Pt.2 200

B.Com Pt.3 100

Residence-wise distribution

Pre-University B.Com Pt. 1 B.Com Pt.2 B.Com Pt.3Hostellers 80 200 150 70Day Scholars 20 100 50 30

21) The position of different examination results of a college are as follows:

There were three faculties,viz, Arts, Commerce and Science and three classes in each one of them, viz , B.Com Pt.I , B.Com Pt.II and B.Com Pt. III. The total number of students appeared in Arts 300, in commerce 400 and in science 250. The class wise strength and pass percentage being:

I A 120 60%II A 105 67%III A 75 90%I C 160 70%II C 140 70%III C 100 85%I S 100 40%II S 90 60%III S 60 65%

Tabulate the above information in a suitable form showing actual number of students, passed class as well as faculty wise and also the total.

22) Calculate arithmetic mean from the following data:

Profits: (in ‘000)

35-40 25-30 20-25 15-20 10-15 5-10

No. of shops

7 10 21 27 18 9

23) Find out the mean from the following if the step deviations have been taken from assumed mean 35 and magnitude of class interval is 10:Step Deviation

-3 -2 -1 0 +1 +2

Frequency 5 10 25 30 20 10

24) From the following data, calculate the mean.

Income in Rs No. of PersonsBelow 30 16Below 40 36Below 50 61Below 60 76Below 70 87Below 80 9580 & above 5

25) Calculate missing value from the following data if arithmetic mean is 66.5 years.

Price 15 20 45 47 55 ? 75 82 95Freq. 1 2 3 5 7 12 16 10 4

26) The following marks have been obtained in three papers of statistics in an examination by 12 students. In which paper is the general level of the knowledge of the students highest? Give Reasons.Students 1 2 3 4 5 6 7 8 9 10 11 12Paper A 36 56 41 46 54 59 55 51 52 44 37 59Paper B 58 54 21 51 59 46 65 31 68 41 70 36Paper C 65 55 26 40 30 74 45 29 85 32 80 39

27) Calculate median and quartile from the following data.

No. of Flowers

5 6 7 8 9 10 11

No. of Plants

15 18 25 30 12 8 7

28) Calculate median and quartile from the following

Size 0-8 8-16 16-24 24-32 32-40Freq 6 10 12 9 3

29) Calculate median and range of income of middle 80% of workers :

Wages in Rs. No. of workersBelow 10 5Below 20 8Below 30 1530-40 2040-50 3050-60 3360-70 35

30) Find out median and quartile from the following data :

Central Value

115 125 135 145 155 165 175 185 195

Frequency 6 25 48 72 116 60 38 22 3

31) The median of the following incomplete distribution is 48. Find out the missing frequency and then find out arithmetic mean :

Size Frequency0-10 1010-20 1520-30 2030-40 ?40-50 5050-60 6060-70 ?70-80 1580-90 10

N = 250

32) Size of collors of 200 students of a college is given in the following statement. Calculate modal size:

Collar Size (cm)

32 33 34 35 36 37 38 39 40 41

No. of students

7 14 30 28 35 34 16 14 36 16

33) Calculate arithmetic mean , median and mode from the following data:

Age in years No. of personsLess than 25 8Less than 30 23Less than 35 51Less than 40 81Less than 45 103Less than 50 113Less than 55 117Less than 60 120

34) From the following data, find out the missing frequency . If value of mode is 22.55 N = 50

Mid-value 5 15 25 35 45Frequency 5 ? 20 ? 3

35) If the median of following frequency distribution is 18.83 and mode is 19.24. Find out missing Frequencies and then calculate mean also.Central Size

5 10 15 20 25 30 35 40

Frequency 14 8 7 ? ? 10 9 3

36) Find out the mean, mode and median from the following information:

Central size

1 2 3 4 5 6 7 8

Frequency 5 50 45 30 20 10 15 5

37) From the following data, find the (i) average salary (ii) median salary and (iii) the most common salary :Salary in Rs. No. of personsMore than 100 100More than 98 98More than 93 93More than 83 83More than 300 43More than 350 23More than 400 12More than 450 5

38) From the following series calculate mean , median and mode:

Value (Rs.) Frequency10-20 410-30 1610-40 5610-50 9710-60 12410-70 13710-80 14610-90 150

39)

Central Size No. of persons1 32 53 124 205 326 187 98 49 310 2

(i) Find out the range of marks within which middle 50% of the students lie;

(ii) Find out the marks of a student whose marks are greater than that of 50% of the students.

40) The working results in 31 March, 2009 of 50 branches of a concern are given below. Find out mean, median and mode.

Loss(Rs.) No. of Branches Profit(Rs.) No. of Branches2000-3000 3 0-1000 121000-2000 5 1000-2000 160-1000 6 2000-3000 8

41) Calculate mean, median, quartile and mode from the following series:

M.V. 4 10 18 28 38 46.5Frequency 8 6 9 12 7 8

42) Calculate mean, median and quartile from the following data:

Income in Rs. No. of personsBelow 30 16Below 40 36Below 50 6150-60 15Below 70 87Below 80 9580 and above 5

43) From the following data, calculate mean , median & mode:

Wages in Rs. No. of PersonsUnder 8 5Under 16 128-24 2924 & over 3132-40 840 & over 1948 & over 5

44) Calculate mean, median & mode from the following data:

Wages in Rs. No. of PersonsBelow 13 313-15 515-17 6Below 19 21Below 21 2621-23 4Below 25 33

45) Calculate mean, median and mode from the following data :

Wages in Rs No. of WorkersLess than 5 7Less than 10 205-15 3815 and above 5520-25 2025 and above 0530 and above 110

Unit III:

46) The monthly Profit/Loss for six months of a company is as under. Calculate range and its coefficient:

Months April May June July August SeptemberProfit/Loss 700 400 -1300 1800 5000 -200

47) Find out the range and its coefficient from the following:

Marks below

10 20 S 40 50

No. of students

20 40 80 89 95

48) The daily wages of 500 workers in a factory are given below. Calculate coefficient of Quartile deviationWages( in Rs)

90-100 80-90 70-80 60-70 50-60 40-50

No. of workers

185 120 92 60 30 13

49) Find out the coefficient of quartile deviation from the following

Central value (in Kg)

11 13 15 17 19 21 23

No. of boxes

2 9 20 25 24 15 5

50) Calculate dispersion of the following series by using coefficient from mean

Weekly wages (in Rs.) Factory ALess than 5 205-10 1810-15 3015-20 2520-25 2025-30 15

51) Calculate mean and standard deviation from of all laborers from following distribution relating to salary of skilled & unskilled laborers of a factory:Salary Skilled Unskilled20-30 4 630-40 3 840-50 8 450-60 4 960-70 3 470-80 2 5

52) The following is the record of goals scored by team ‘A’ in a football season:

No. of goals scored

0 1 2 3 4

No. of matches

1 9 7 5 3

For team B the average number of goals scored per match was 2.5 with S.D. of 1.25 goals. Which team may be considered more consistent.

53) Goals scored by two teams A and B in a football match during season were as follows:

No. of goals scored in match

Number of matches

Team A Team B

0 15 271 10 092 9 073 7 4 4 4 3Total 45 50

Which team may be considered more consistent?

Unit IV:

54) Calculate the coefficient of correlation for the ages of husband and wife:

Age of husband

23 27 28 28 29 30 31 33 35 36

Age of wife

18 20 22 27 21 29 27 29 28 29

55) Calculate Karl Pearson’s coefficient of correlation between age and passing of examination & its probable error.

Age in years

17 18 19 20 21 22 23

Pass Percentage

61 60 57 66 64 61 51

56) From the following table. Calculate the coefficient of correlation between X and Y series:

Y Series X Series

10-13 6-9 2-5 Total

20-24 2 1 --- 3 15-19 1 3 --- 4 10-14 --- 2 2 4 5-9 --- 1 1 2 0-4 --- 1 1 2

57) Find out the coefficient of correlation from the following bivariate table

x/y -3 -2 -1 0 1 2 3 Total-3 --- --- --- --- --- --- 10 10-2 --- --- --- --- 16 6 8 30-1 --- --- --- 18 14 8 --- 400 --- --- 4 10 18 6 --- 381 --- 4 6 12 --- --- --- 222 6 6 --- --- --- --- --- 123 8 --- --- --- --- --- --- 8Total 14 10 10 40 48 20 18 160

58) The competitors in a beauty contest are ranked by two judges in the following order:

First Judge

1 6 5 10 3 2 4 9 7 8

Second Judge

2 8 4 1 6 9 5 3 7 10

Calculate the coefficient of rank correlation.

59) Calculate the coefficient of concurrent deviation from the following data:

Years 2002 2003 2004 2005 2006 2007 2008Supply 150 154 160 172 160 165 180Price 200 180 170 160 190 180 172

60) The following are the marks obtained by a group of 9 students in accountancy and statistics out of 100 in each paper:Marks in accountancy

70 55 40 60 25 50 35 80 85

Marks in statistics

50 40 60 70 30 45 55 65 80

61) The heights of fathers and sons are given in the following table:

Height of father(inches)

65 66 67 67 68 69 71 73

Height of son (inches)

67 68 64 68 72 70 69 70

62) From the following data, find the two regression equation:

X 1 2 3 4 5 6 7Y 2 4 7 6 5 6 5

63) Two random variables have least square regression lines with regression equations:

2X + 3Y - 26 = 0 X + 6Y - 31 = 0 σX

2 = 9

Find out mean σ y / σx of both the variables and value of r also.

64) For two correlated series of X and Y:

= 36; =40; σx = 6; σ y = 8; r = 4/5 Find the two lines of regression. Predict Y when X = 21c and predict X when Y = 45.

65) Given the following data:

Variance of X = 9, Regression equations:

4X – 5Y + 33 = 0

20X -9Y – 107 = 0

Find (i) the mean values of X and Y , (ii) the S.D. of Y, and (iii) coefficient of correlation between X & Y.

66) (a) A bag contains 8 white and 4 red balls. 5 balls are drawn at random from the bag. What is the probability that 2 are red and 3 are white balls?

(b) How many different groups of 3 males and 3 females can be formed out of 12 males and 16 females?

67) A card is drawn at random out of pack of 52 cars. What is the probability that the card is

(i) A queen (ii) A black cards (iii) An ace of spade?

68) (A) A bag contains 25 balls, numbered from 1 to 25. One is to be drawn, find the probability that the number of the drawn ball will be a multiple of 3 or 5. (B) A bag contains 9 red and 14 white balls, 2 balls are drawn from the bag. Find the probability that both are of the same color.

69) (a)A bag contains 5 white and 4 black balls. One ball is drawn from the bag and then replaced . Another ball is drawn after the replacement. Find the probability that both drawings are of white balls. (b) A bag contains 7 red and 11 white balls. If two balls are drawn from the bag, find the probability that (i) both are red, (ii) one ball is red and one is white.

70) The odds against an event is 5:3 and in favor of another independent event is 7:5. What is the probability of happening at least one event?

71) 7 coins are tossed 256 times. Find the probability and expected frequencies of the following outcomes: (i)5 heads (ii) 5 or more heads (iii) less than 3 heads (iv) at least 6 heads (v) no head

72) If the proportion of defectives in a lot is 5%. Find the probability of not more than 2 defectives in a sample of 10 units. (e-5 = 0.6065 ) 73) The following table gives the number of automobile accidents and number of days for a period of 50 days. Obtain Poisson distribution from the given data:

No. of accidents

0 1 2 3 4

No. of days 21 18 7 3 1

74) The Delhi Municipal Corporation installed 10,000 electric bulbs in the streets of the city. If these bulbs have an average life of 1850 hours with a standard deviation of 200 hours, what number of bulbs may be expected to burn less than 1600 hours?