TRIBHUVAN UNIVERSITYfR

Faculty Of Management

Office Of The Dean

BBS / 4 Years Programme / First year / MGMT

Business Statistics ( MGT 202 )

Full Marks :100

Pass Marks : 35

Time : 3 hrs.

MODEL QUESTION – 2020 Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.

Group ‘A’

Brief Answer Questions.

Attempt All Questions. (10X2=20) 1) The mean of 200 items was 50. Later on it was found that two items were wrongly taken

as 92 and 8 instead of 192 and 88. Find the correct mean. 2) In a batch of 15 students, 3 students failed in an examination. The marks of passed 12

students were 9, 6, 7, 8, 4, 5, 8, 10, 9, 7, 5, 7. What was the median mark of all 15 students? 3) In a moderately skewed frequency distribution, the mean is 10 and its median is 9, if the

coefficient of variation is 20%. Find the Pearson’s coefficient of skewness of the

distribution.

4) List out the various methods of collecting primary data and secondary data.

5) Calculate the lower and upper quartiles from the following marks distribution: Marks Below 25 25-29 30-34 35-39 40-44 Above 44

Students 5 12 22 25 17 9 6) If regression coefficient of y on x (byx) = - 0.61 and regression coefficient x on y (bxy) =

- 0.53, calculate the coefficient of correlation and interpret the result. 7) The standard deviation of symmetrical distribution is 5. What must be the value of

fourth moment about mean in order that the distribution be mesokurtic ? 8) A bag contains 20 balls numbered from 1 to 20. One ball is drawn at random. Find the

probability that the number of the drawn ball be multiple of (i) 3 or 7 (ii) 3 or 5. 9) Calculate the price index number from the following data by simple aggregative method.

Commodities A B C D E

Price in 2075 125 105 260 150 250

price in 2076 125 155 250 160 300 10) The following table shows the pay-off matrix related to the demand and

strategy of a business person. What should be the decision if he / she uses I)

maximax criterion ii) maximin criterion?

Demand

Strategy D1 D2 D3

S1 300 100 80

S2 200 120 60

S3 80 60 20

Group ‘B’

Descriptive Answer Questions.

Attempt any FIVE Questions.. (5X10=50) 11) List the five number summary and prepare a box-and-whisker plot from the following

information. Also, comment on the nature of frequency distribution. Class Size 20-30 30-40 40-50 50-60 60-70 70-80

Frequency 10 12 25 35 40 50

12) Solve the following system of equations by using determinant method. 2x+5y-z = -3, 4x+3y+2z = 1 & x+2y+3z = -5

13) Calculate Fisher’s ideal index number for the following data and show that it satisfies (i) Time Reversal Test and (ii) Factor Reversal Test.

Commodity 2018 2019

Price (Rs) Quantity(Units) Price Quantity(Units)

(Rs)

A 8 60 12 58

B 4 110 4 125

C 6 70 8 70

D 12 40 14 38

14) Fit straight line trend by the method of least square to the data given below. Also, find

the trend values and predict the sales for the year 2022. Year 2013 2014 2015 2016 2017 2018 2019

Sales(000 units) 15 16 17 16 19 23 25

15) A factory manufactures three types of white sheet of papers I, II, and III and distributes them in two markets A and B. The sales of papers during one year are given below:

Markets Papers

I II III

A 12,000 5,000 23,000

B 14,000 22,000 13,000 a) If unit sale prices of papers I, II and III are Rs 5, Rs 4 and Rs 3 respectively, find

the total revenue in each market with the help of matrix algebra.

b) If the unit costs of the above papers are Rs 3, Rs 2 and Rs 1 respectively, find the gross profit.

16) The following is the net profits of two companies in millions of rupees, find which

company shows the greater consistency in the net profit. Justify your answer with

statistical evidence.

Company A 18 19 23 19 25 23 27

Company B 17 18 22 23 24 25 25

Group ‘C’

Analytical Answer Questions.

Attempt any TWO Questions. (2X15=30) 17) A family income and its percentage expenditure on food for 100 families gave the

following bivariate frequency distribution. Find out if there exists any relationship

between family income and expenditure on food and interpret the result. Also test the

significance of the result. Estimate the percentage expenditure on food when family

income = Rs 90,000.

Food expenditure (%) Family income (000Rs)

20-40 40-60 60-80 80-100 100-120

15 - - - 3 7

20 - 4 9 4 3

25 7 6 12 5 -

30 3 10 19 8 -

18) The following distribution shows the frequency distribution of weekly expenditure on

foods of students in certain locality of Kathmandu Metropolitan City. Describe the various

characteristic features of the frequency distribution. Also, comment the nature of the

distribution.

Expenditure 30-40 40-50 50-60 60-70 70-80 80-90 90-100 100-110

(Rs.00)

Number of 10 15 28 33 40 35 25 14

students

19) What do you mean by linear programming model? A person requires minimum 10, 12

and 12 units of chemicals A, B and C respectively. A liquid product contains 5, 2 and 1

units of A, B and C per jar. A dry product contains 1, 2 and 4 units of A, B and C per

carton. If the liquid product costs Rs 30 per jar and dry product costs Rs 20 per carton.

Formulate this problem in linear programming model. How many of each product should

be purchased in order to minimize the cost to meet the requirements. Also find the

minimum cost.

MGT 202: Business Statistics

BBS 1st Year

Model Question

Full Marks: 100

Pass Marks: 35

Time: 3 hours

Candidates are required to give their answer in their own words as far as practicable.

The figures in the margin indicate full marks.

Attempt All Questions

Group 'A'

Brief Questions Answer [10 x 2 = 20]

1. The mean of 200 items was found to be 80, later it was found that 61 and 45 were

misread as 16 and 15. Find correct mean.

2. The following results

were obtained

Coefficient of

variation = 50%

Karl Pearson's coefficient of

skewness = 0.5 Standard

deviation = 2

Find mean and mode.

3. In a single throw of two dice, find the probability that sum of two faces is 7 or 11.

4. Differentiate bet ween probability and non-probability sampling.

5. The year of origin of the following straight line trend equation of profits in

lakhs of rupees is 2008.

y=35+2x

Estimate profit for the year 2015.

6. Prepare regret table from the given conditional profit table.

Demanded Units Decision Alternatives

15 16 17 18

15 150 120 90 90

16 150 160 130 100

17 150 160 170 140

18 150 160 170 180

7. The following calculations were based on the life of refrigerators of two companies.

Company A Company B

Average life 8 years 6 years

Standard deviation 12 years 8 years

Which company's refrigerator shows greater consistency in terms of life?

8. On the basis of the given information find the regression coefficient of X on Y.

∑XY = 750 ∑X2 = 2085 ∑Y2 = 285

∑X = 135 ∑Y = 45 N = 9

9. The coefficient of correlation between 10 pairs of values of demand and supply was

found to be

0.8. Test the significance of the result.

10. The first four moments about mean of a distribution are

µ1 = 0 µ2 = 16 µ3 = - 30 µ4 = 40

Test for the normality of the distribution.

Group 'B'

Descriptive Answer Questions (attempt any five) [5 x 10 = 50]

11. a) Two merchants M1 and M2 had the following units of three commodities

and the prices of these commodities in three different cities of the country.

Supply

Matrix

Commoditie

s

X Y Z

M1 50 60 90 A =

M2 40 50 70

Price

Matrix

Commoditie

s

Cities X Y Z

C1 18 20 16

P = C2 16 22 14

C3 20 24 16

To which city each merchant should supply the commodities in order to get

the maximum receipt?

b) An aeroplane has 40 seats for passengers. Passengers travelling in economy

class can take 20 Kgs of baggage each and business class can take 60 Kgs of

baggage each. The aircraft can carry only 1200 Kgs of baggage. Find the

number of passengers of each kind by using Cramer's rule.

12. The following distribution represents yearly income of 2500 employees of an

industrial concern in thousand of rupees. Employees earning 2 lakhs or more

have to pay 15% tax to the government. Find the average income of the

Income (In '000' Rs.) 70-100 100-130 130-160 160-190 190-220 220-250 250-280

employees and the amount of tax to be paid to the government.

No. of Employees 50 150 200 400 900 500 300

13. The following is the net profits of two companies in millions of rupees, which

companys shows greater consistency in net profit. Justify.

Year 2006 2007 2008 2009 2010 2011 2012

Company A 15 16 20 16 22 20 24

Company B 14 15 19 20 21 22 22

14. a) A company has two plants to manufacture scooters. Plant I manufactures

80% of scooters and plant II manufactures 20%. At plant I, 85 out of 100

scooters are rated Standard Quality. At plant II, only 65 out of 100 scooters

are rated Standard Quality. What is the probability that scooter came from

plant -II if it is known that the scooter is of Standard Quality.

b) In a certain distribution, the first four moments about an arbitrary point were

1,3,7 and 21. Test the Skewness of the distribution.

15. a) Why is it necessary to analyze time series data? Discuss various components of time

seri es.

b) Calculate moving averages for the following data, assuming the length of business cycle as 3 year s.

Year 2004 2005 2006 2007 2008 2009 2010 2011 2012

Sales In '000' 35 50 60 65 70 80 90 92 95

16. a) What do you understand by classification of data? What are its objectives?

Classify the given data using Sturge's rule.

110 175 161 157 155 108 164 128 114 178 165 133 195 151 71 94 97

42 30 62 138 156 167 124 164 146 116 149 104 141 103 150 162 149

79 113 69 121 93 143 140 144 187 184 197 87 40 122 203 148

b) What do you understand by sampling distribution? Differentiate between

point and interval estimate.

Group 'C'

Analytical Answer Questions (attempt any two) [

2 x 15 = 30]

17. From the following bi-variate table find out whether there exists any

relationship between security prices and dividends and test the significance of the

result. Also estimate the amount of dividend when price of security is Rs. 150.

Security Prices

(in Rs.)

Annual Dividends (in Rs.)

6-8 8-10 10-12 12-14 14-16 16-18

70-80 2 1 - - - -

80-90 3 1 - 1 - -

90-100 2 2 1 1 - -

100-110 - 2 2 3 - -

110-120 - 1 3 2 2 -

120-130 - 1 3 3 3 2

130-140 - - 3 1 4 1

1. A firm manufacturers two types of electrical items E 1 and E2. The profit

contribution per unit of E1 and E2 are Rs. 1600 and Rs. 2400 respectively. Both

E1 and E2 make use of two essential components, a motor and a transformer.

Each unit of E 1 requires 3 motors and 2 transformers and each unit of E2

requires 2 motors and 4 transformers. The total supply of components per

month is restricted to 210 motors and 300 transformers. E 2 is an export

model requiring a voltage stabilizer, which has supply restriction to 65 units.

Formulate the above problem in a mathematical form describing the objective and

limitations of the problem. Solve the formulated problem by graphic method

with an objective of maximization of profit.

2. What are Index numbers? Why are they called economic barometers? You are

required to prove from the following data that Fisher index number is an ideal

index number.

Commodities

2010

Price / units Expenditure

2011

Price / units Expenditure

A 50 600 60 840

B 40 840 80 2400

C 30 900 30 1500

D 20 600 20 1000

E 60 360 70 2100

F 80 640 80 2400

TRIBHUVAN UNIVERSITY

Faculty Of Management

Office Of The Dean

BBS / 4 Years Programme / I year / MGMT Full Marks :100

Business Statistics ( MGT 202 ) Pass Marks : 35

MODEL QUESTION – 2020

Time : 3 hrs.

Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.

Group – A

Brief Answer Questions.

Attempt ALL Questions. [ 10 x 2 = 20 ] 1. The mean of marks in Statistics of 100 students in a class was 72. The mean

of marks of 70 boys was 75. Find out the mean marks of girls in the class. 2. The difference between the upper quartile and the lower quartile of a certain

frequency distribution is 4 and their sum is 16. Calculate the quartile deviation

and its coefficient. 3. In a single throw of two dice, what is the probability of getting the same

numbers on both dice ? 4. The personnel director for Nepal Drug Limited recorded the average percentage

absentee rates for each quarter for a 4 years period are 55, 67.5, 62.5 and 53,find

the seasonal indices. 5. The coefficient of variation of a symmetrical distribution is 9 % and mean of

the distribution is 40. Find the value of standard deviation and variance. 6. What do you mean by five number summary?What is its application in statistics ?

7. Reconstruct the following index number by shifting the base year as 2053.

Year 2049 2050 2051 2052 2053 2054 2o55

Index Number 100 115 126 134 147 155 163 8. From the following pay- off table, find the best strategy if (i) Maximax criteria is

applied (ii) Maximin criteria is applied. . PAY- OFF TABLE :

N1 N2 N3

S1 200 50 40

S2 100 60 30

S3 40 30 10 9. For∑ eight=156pairs,∑of observations=132,∑ on two variables∑ sales ( X )

∑and Pricing ( Y ) , the following results were obtained.

= 4162 , = 2434 , = 2884 Find out1 if −there2 exists

any relationship between sales and pricing.

10. Find the3 adjoint7 matrix of the matrix given below.

Group – B

[ 5 × = ]

Descriptive Answer Questions.

Attempt any FIVE questions. 11. The average weekly wages, standard deviations and number of workers of

two factories are given below. Factory A Factory B

Average weekly wage Rs. 4600 Rs.4900

Standard Deviations Rs.50 Rs.40

Number of workers 100 80 Calculate the mean and variance of weekly wage of all workers taken together.

Which factory has greater variability in the distribution of weekly wages? Justify your result with appropriate Statistical tool.

12. Differentiate between “Census” and “Sampling” method of data collection.

Why sampling method is suitable to collect data from large population?

13. (a) Solve the following linear programming problem using graphical method. x + ≤ 30

Maximize Z = 30 x + 50 y x , y ≤ 40

Subject to constraints: x + y

≥ 0 2y

(b) A manufacturing company has 1,000 employees. 10 % of the employees earn less than Rs. 500 per day , 200 earn between Rs. 500 and Rs. 999 , 30 % earn between Rs. 1000 and Rs. 1,499 , 250 employees earn between Rs. 1,500 and Rs. 1,999 and rest earn Rs. 2,000 and above. Calculate the suitable average wage. Also give the reason for your choice of average.

14. Calculate the index number by using suitable formula for 1985 on the basis of

1980 from the following information :

Year Product X Product Y Product Z

Price Quantity Price Quantity Price Quantity

1980 4 54 3 10 2 5 1 1

10 1

40 8

8 4 5 1985 15. (a) Prove the following by using properties of determinants.

= ( a- b)( b- c )(c- a )( a+ b +c )

( b ) Solve the following equations by using Matrix method. -x + 3y = 5 2 x – 4 y = 0

16. From the following data compute Bowley’s coefficient of skewness and

interpret your result. Income(00 Rs.) Below 200 200-400 400-600 600-800 800-1000 1000 & above

No. of families 25 40 80 75 20 16

Group – C

[ 2 × = ]

Analytical Answer Questions.

Attempt any TWO questions. 17. The following table represents the annual trend of net profit of two different

companies seeking investment for their development project. As an investment advisor, in which company would you suggest to invest money ? Justify your answer by using necessary statistical tools.

Year Net Profit (in million Rs.)

Company - A Company - B

2007 16 16

2008 32 16

2009 40 22

2010 24 36

2011 40 40

2012 32 44

2013 88 48 18. From the following bi-variate frequency tabl , find out if there exists any

relationship between advertisement expenditure (in 00 Rs.) and sales revenue (

in 000 Rs.) and test the significance of the result. Also estimate sales revenue

when advertisement expenditure is Rs. 40,000.

Advertisement Sales Revenue ( in 000 Rs. )

Expenditure (in 00 Rs.) 0-50 50 - 100 100 - 150 150 - 200 200 - 250

0-40 12 6 8 - -

40 - 80 2 18 4 5 1

80 - 120 - 8 10 2 4

120 - 160 - 1 10 2 1

160 - 200 - - 1 2 3 19. Under an employment promotion programme , it is proposed to allow sale of

newspapers on the business during peak hours. A newspaper boy has the following probability of selling a magazine.

No. of copies sold 10 11 12 13 14

Probability 0.10 0.15 0.20 0.25 0.30 Cost per copy of magazine is Rs. 30 and sale price per copy is Rs. 50. He cannot return unsold copies where salvage value is zero.

a. Calculate the expected monetary value ( EMV ) for each strategy.

b. How many copy should be ordered ? c. Compute expected profit with perfect information ( EPPI ).

d. Also calculate expected value of perfect information ( EVPI ).

Chapter 1 Introduction to Statistics

1. Distinguish between primary data and secondary data with examples.[5][2076][2073][2071][2063][2061][2056]

2. Discuss the problems of data collection.[5][2073]

3. Explain primary data and secondary data with examples.[5][2072] Discuss the methods and problems of data collection.[5][2072][2071][10][2067][2060][2058]

4. Define statistics. Discuss its limitations.[5][2071][2057]

5. What is collection of data? Describe different methods of collecting data.[10][2068][2063]

6. What is questionnaire method of data collection? State the points that should be kept in mind while drafting questionnaire.[5][2065]

7. Explain indirect personal interview method of data collection. In what situation would you select this method, point out precautions while selecting this method.[5][2064]

8. Explain the important applications of statistics together with the limitations statitics.[10][2061]

9. Explain the Mailed questionnaire method. Point out the area of applicability and discuss limitations[10][2061]

10. What are the characteristics of statistical data? Explain.[10][2059]

11. Explain use of statistics in business.[140][2058]

12. State the sources of secondary data. What precautions are to be observed when such data are to be used for any investigation?[10][2056]

13. What method would you employ in collection of data regarding statistical enquiryabout Graduates unemployment in Kathmandu considering accuracy time and cost enquiry? [10][2055]

Chapter 2 Classification and Presentation of Data

1. What is the meaning of classification of data?[2][2072][2064][2064]

2. Differentiate between exclusive class and inclusive class. How can you change inclusive classes to exclusive? [2][2071]

3. Explain the types of classification of data. Write down its objectives.[10][2074][2066][2064]

4. Discuss the importance of classification and tabulation of data in statistical analysis. [10][2070]

5. What are the basic principles you should follow while preparing the grouped frequency distribution? [5][2065][2061][2058][2055]

6. Why is it necessary for data analysis?[5][2064]

7. What is false base line? Explain its utility in graphic representation of statistical data.[5][2056]

8. Locate the mode by using histogram[2][2075]

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

No. of Students 5 12 20 16 12

9. Following are the temperature distribution of hundred days. Find the number of days when the temperature is less than 18 degrees graphically.[2][2074]

Temp in degrees

-10 to 0 0-10 10-20 20-30 30-40

No. of days 18 25 40 10 7

10. Draw the pie diagram from the following data related to the monthly expenditure of two families:[10][2074]

Monthly expenditure (00Rs)

Family A Family B

Food 40 30

Clothing 30 10

Education 50 20

Fuel 20 20

Misc. 60 20

11. Represent the following data by histogram, frequency and polygon[10][2072]

Salaries(Rs) 300-310 310-320 320*330 330-350 350-390

No. of workers

8 16 20 18 12

12. Prepare a histogram and frequency polygon from the following data: [10][2072]

Height (in cm) 0-5 5-10 10-15 15-25 25-30 30-35

No. of plant 8 16 30 40 24 4

13. Construct suitable diagram for the following data.[5][2071]

Items food clothes fuel rent Misc.

Expenditure in Rs 300 180 120 600 240

14. From the following income distribution, construct an ogive and find the number of persons having income between Rs 20,000 and Rs50,000 from Ogive curve.[5][2059][48]

Income(Rs000) 0-10 10-20 20-30 30-40 40-50 50-60

No. of p[ersons 5 10 18 23 7 6

15. Draw Ogives and locate the value of median from the following:[10][2068][79,062.5 and 15]

Wages in 000 Rs No.of employees

50or more 65

60 or more 57

70 or more 47

80 or more 31

90 or more 17

100 or more 7

110 or more 2

120 or more 0

16. Draw Ogives for the following data and locate the value of median. Also find the number of students scoring less than 35 marks and the number of students having marks 45 or more. [10][20678][Md = 35, 16, 9]

Marks 0-10 10-20

20-30 30-40 40-50 50-60 60-70

No. of students 2 4 6 8 6 4 2

17. Represent the data given below by constructing Pie-diagram.[10][2066]

Items Expenditure(in Rs)

Family A Family B

FOOD 1500 1200

CLOTHING 1000 800

RENT 600 800

EDUCATION 900 200

FUEL 800 400

Misc. 600 200

5400 3600

18. Draw the frequency polygon from the following data.[5][2065]

Variable Frequency

Less than 10 4

Less than 20 12

Less than 30 20

Less than 40 32

Less than 50 60

Less than 60 66

Less than 70 66

19. Represent the following data by Multiple Bar Diagram. [5][2064]

Types of man power (in health service)

Number of personnel in thousand

2031 2041 2051

Doctor 2.7 4.5 7.3

Nurse 6.5 15.9 22.3

Health Assistant 6.4 10.2 21.8

Health Worker 15.5 31.2 16.5

20. Represent the following percentage sub divided bar diagram.[10][2064]

Items Expenditures

Family A Family B

FOOD 4000 6000

CLOTHING 2000 1500

EDUCATION 1500 2500

RENT 1500 1000

FUEL 600 600

Misc. 400 400

10000 12000

21. The number of tourist recorded in Kathmandu Airport is as follows:[2063]

No. opf Tourist 2062 2063

India 70 75

USA 50 35

Europe 45 40

Domestic 35 30

Others 50 70

total 250 250

Draw pie-diagram to represent the above data 22. Drawing Ogives and locate the value of median[2063][Rs 175]

0-50 15

50-100 20

100-150 25

150-200 30

200-250 25

250-300 20

300-350 15

23. Construct the histogram of the distribution of students by marks given by the

following table. [2062] Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-80 80-100

STUDENTS 3 5 9 14 18 26 20 6

24. Construct the pie diagram for the following data: [2061]

Components Districts Population

1991 2001

Dolpa 25013 22071

Jumla 75964 69226

Mugu 36364 31465

25. Draw the cumulative frequency curve for the following data:[2061]

Age No of person in Group A No of person in Group B

30-35 10 15

35-40 15 25

40-45 25 30

45-50 40 50

50-60 30 20

60-70 10 10

26. Construct a pie-diagram for the following:[2061]

TYPES OF OPERATION No of cases

General SURGERY 90

Thoracic Surgery 20

Prostate surgery 60

Urologic surgery 80

Surgery of abdomen 110

Total 360

27. Construct a multiple bar diagram for import and export for last three years as given

below:[2060]

Year Import (million Rs) Export (million Rs)

1998 15 12

1999 18 10

2000 22 15

28. The elements of cost, sale proceeds and loss per chair for 1992 are given below.

[2060][2057]

Material Painting and Finishing

Wages Total Cost Sale price Loss

RS 32 Rs 10 Rs 22 Rs 64 Rs 60 Rs 4

29. Construct a pie diagram for the following data.[2058]

Items food cloths fuel rent misc Total

Expenditure(Rs) 500 150 200 130 100 1080

30. Draw Histogram and Frequency Polygon from the following data:[2056][2055] Daily Wages in Rs 10-15 15-20 20-25 25-30 30-40 40-60

No. of Workers 8 18 25 15 12 12

31. The asset holdings of commercial banks at 1990 are as follows. REPRESENT THE

DATA INTO PIE CHART.[2054]

Commercial NBL RBB NABIL INDOS GRND ADB

Assets Holding (Rs in

Billions)

129 96 13 8 12 46

Chapter 3 Measures of Central Tendency

1. State the ideal properties for a good measure of central tendency. Examine merits

and demerits of arithmetic mean. [5][2064][2061]

2. Examine merits and demerits of standard deviation in the light of ideal

properties.[5][2061]

3. If the arithmetic mean of the following data set is 33 find the value of x:x +10, 30,

x+20, 4x+5, 40 [2][2076]Ans 10

4. The mean monthly salary of 80 male and 20 female employees of a company are Rs

1560 and Rs 1260 respectively What is the mean salary of all

employees?[2][2075][Ans Rs 1500]

5. A person drives a car between the two places x and y. On his outward journey he

consumes 8 km per liter of petrol. On his return journey he covers 12 km per liter.

Find the average of his mileage assuming the distance between the two places to be

100 kms.[2][2074][Ans 9.60km/liter]

6. Find the most repeated value by empirical relationship if X= 42.2 and median = 41.9.

[2][2073][Ans Mn = 41.3]

7. A man travelled by car for 3 days . he covered 480 km each day. On the first day, he

dove for 10 hours at 48 km/hr. on the second day, he drove for 12 hours at 40 km/hr

and on last day he drove for 15 hours at 32km/hr. What was his average speed?

[2][2073][38.92km/hr]

8. The average monthly salary of 10 male staffs and 5 female staffs of a manufacturing

company are Rs 20,000 and Rs 18,000 respectively. FIND THE AVERAGE

MONTHLY SALARY OF ALL STAFFS TAKEN TOGETHER. [2][2072][Rs 19, 333.33]

9. From a batch of 13 students who had appeared in an examination, 4 students were

failed. The marks of passed students were 43,57, 45, 61,75,64,53,50 and 40.

Calculate median marks of all students. [2][2072][Ans 45]

10. If mean = 40 and mode = 30 then find median.[2][2072]Ans 36.67

11. What will be the value of median of a moderately asymmetrical distribution, if the

mean and mode are 30 and 24 respectively?[2][2071][28]

12. Classify the following scores in class interval 0-9, 10-19, 20-29…etc. Compute the

modal value of the constructed frequency table. [10][2075]Ans 23.94

24,12,37,49,30,6,28,10,13,18,12,19,2,33,4,25,42,22,16,29,2,21,23,31,21,34,20,27,2

3,35.

13. Define ARITHMETIC Mean and its properties. Following are the income distribution

in thousands of 50 families.

63 52 65 78 79 52 81 41 77 70 96 104 94 76

78 79 40 76 69 67 50 59 1115 49 56 82 74 65

50 110 61 9 69 49 81 72 73 103 66 64 60 72

86 84 53 51 80 68 42 90

Finance ministry wants to know the limit of the income of middle 60% observed families. [10][2074][Ans Rs 57,500 and Rs 84, 285.71]

14. The daily expenditure of 100 students in mobile phone is given below:

[10][2074][ANS 20 and 15]

Expenditure(in Rs) 0-20 20-40 40-60 60-80 80-100

Number of students 15 - 30 20 -

15. The medium expenditure of 1000 families were found to be Rs 650 and the

distribution was as follows: [10][2073][Ans 225, 175 and Rs 655]

Expenditure(RS) 400-500 500-600 600-700 700-800 800-900

No. of families 50 - 450 - 100

16. A frequency distribution of marks of 100 students is given below. Frequencies

corresponding to two groups are missing from the table. The median is known to be

49.5 marks.[10][2072][Ans 23 and 21. 24.72 marks and 75.69 marks[20

MARKS 0-19 20-39 40-59 60-79 80-99

No. of students 14 ? 26 ? 16

17. The following table shows the distribution of 100 families according to their

expenditure per week. If median = 25 find mean and mode.[10][2072]Ans �̅� = 25, Mo

= 24[2072]

Expenditure 0-10 10-20 20-30 30-40 40-50

No. of families 14 - 27 - 15

18. Find geometric mean and harmonic mean from following data: [10] [2072] Ans GM =

31.19 HM= 29.05[2072]

Class interval 10-20 20-30 30-40 40-50 50-60

Frequencies 30 75 70 60 15

19. The weekly expenditure of 1,000 families is given below:

Expenditure(Rs.`00’) 40-59 60-79 80-99 100-119 120-139

No. of families 50 ? 500 ? 50

The median for the distribution is Rs.8,700. Calculate the missing frequencies. Calculate the mode of the distribution. Ans. 263 and 137, M0 =Rs. 8,740[2072]

20. A manufacturing company has 1000 employees. 10% of the employees earn less

than Rs.500 per day, 200 earn between Rs.500 and Rs. 999;30% earn between

Rs.1000 and Rs. 1,499; 250 employees earn between Rs.1,500 and 1,999 and rest

earn Rs.2,000 and above. Calculate the suitable average wage. Give the reason for

your choice of average. Ans. 1332.83 per day[2071]

21. From the following distribution of marks of 250 students of a campus, find the

minimum pass mark if only 20% of the students has failed and also find the minimum

marks obtained by the top 25% of the students. Ans. 30 marks; 58.33 marks[2071]

Marks 0-20 20-40 40-50 50-60 60-80 80-100

No. of students 25 50 75 45 30 25

22. From the following income distribution:[2070]

Find: 1. Lowest income of richest 10% of the people

2.highest income of poorest 40% of people

3.Range of income of middle 60% of the people

Income(Rs 000) 0-4 5-9 10-14 15-19 20-24

No. of persons 160 200 430 140 70

Ans.1.Rs. 18,428.60 2. Rs. 9,965.10 3. Rs. 9,375.10 23. Calculate the appropriate average marks from the following marks distribution. Give

the reason for your choice. Ans. Md=43.28 marks; P25=33.46; P75=51[2069]

Marks upto 15 25 35 45 55 65 75

No. of students 7 15 28 57 92 101 104

Also calculate the limits of marks obtained by middle 50% of the students. 24. If the median income of 150 workers of a factory is Rs. 8000 find the missing

frequencies and compute mean income of the workers. Ans.15,25and

Rs.7,000[2068]

Income in `000 Rs. No. of workers

0-2 10

2-4 -

4-6 25

6-8 50

8-10 -

10-12 15

12-14 10

25. From the following distribution of income of 1500 persons, find[2067]

1. Limits of income of central 60% of the persons.

2. Lowest income of richest 60% of the persons.

3. Highest income of poorest 60% of the persons.

Income in Rs.`000

0-5 5-10 10-1 15-20 20-25 25-30 30-35

No. of persons 100 150 400 550 200 75 25

Ans. 1. Rs.10,625 and Rs. 20,000 2. Rs.14,375 3. Rs. 17,273 26. From the following frequency distribution find the range of income of middle 70% of

the employees and median income. [2066]

Income in Rs. 500-600 600-700 700-800 800-900 900-1000

No. of employees 150 300 500 200 50

Ans.P15= Rs. 610; P85= Rs. 835; Range= 835-610= Rs. 225; Md= Rs. 730 27. A toy factory has assigned a group of four workers to complete an order of 1400 toys

of a certain type. The productive rates for the four workers are respectively 4, 6, 10

and 15 minutes per toy. Calculate the average minutes per toy by the group of

workers by using Harmonic mean. Ans: HM 6.856 minutes/toy [2065]

28. Find the appropriate average wage of the workers from the following wage

distribution: [2064]

Wages No. of workers

Less than 500 30

Less than 1000 60

Less than 1500 125

Less than 2000 240

Less than 2500 300

Less than 3000 445

Less than 3500 500

Ans: Md=2,083.33 29. The following table represents the marks of 100 students:[2063]

Marks 0-20 20-40 40-60 60-80 80-100

No. of students 14 - 27 - 15

If the mode value is 48 find the missing frequencies and the mean marks of all 100 students. Ans: 23,21; Mean 50

30. Find the missing frequencies of the following distribution, given mean and variance of

the distribution as 2.9 and 2.49 respectively.[2062]

Variable (X) 0 1 2 3 4 5 6 Total

Frequency (f) 2 4 ? 8 ? 3 2 30

Ans: 6 and 5 31. Calculate average income from the given income distribution of 1400 workers of a

factory.[2062]

Income in Rs. No. of workers

Below 1500 200

1500-2000 225

2000-2500 275

2500-3000 250

3000-3500 180

3500-4000 150

4000 or above 120

N=1400

Ans: Average income is Rs. 2,576.79 or Md= Rs. 2,500 32. The postal service handles six basic types of letters and cards. The mail volume

during 2002/03 is given in the table.[2061]

Type of mailing Ounces delivered (in million)

Price per ounce (Rs.Z)

Air mail 1800 0.18

First class 77000 0.15

Second class 24100 0.10

Third class 16000 0.05

Registered 1000 0.40

Certified 500 0.45

Find out the average revenue per ounce for these services for the year 2002/03. Ans; Rs.0.13

33. From the following distribution, find the missing frequencies if the median wages is

Rs. 42.50 and total number of workers in the factory are 120.[2060]

Daily wages in Rs 20-25 25-30 30-40 40-45 45-50 50-55 55-60 60-70 70-80

No. of workers 6 12 ? 30 10 ? 8 5 2

Ans 27 and 20

34. The mark distribution if 104 students are given below:[2059]

Central rank of group 10 20 30 40 50 60 70

No. of students 7 8 13 29 35 9 3

Find the pass marks if 78 students passed the examination Ans [33.46] 35. From the following distribution of marks of 500 students of a campus, find the

minimum pass mark if only 20% of the students has failed and also find the minimum

marks obtained by the top 25% of the students.

Marks 0-20 20-40 40-50 50-60 60-80 80-100

No. of students 50 100 150 90 60 50

36. Mr. Shrestha is the director of the students financial Aid office at a campus. Her has

used available data on the summer earnings of all students who have applied to his

office for financial aid. The frequency distribution is given below:[2057]

Summer of earnings No. of families

0-500 231

500-1000 304

1000-1500 400

1500-2000 296

2000-2500 123

2500-3000 68

3000 or more 23

Find the model value for Shrestha’s data.

If students’ aid is restricted to those whose summer earnings were at least modal summer earnings, how many of the applicants qualify? Ans Mn=RS 1240; 727

37. Calculate appropriate measures of central tendency from the following distribution

and support for your choice of measure.[2055][Md= Rs 2134.64

Monthly Income (Rs ) in Locality X No. of Families

Below 1000 50

1000-1999 500

2000-2999 555

3000-3999 100

4000-4999 30

5000 and above 15

38. Before and after the implementation of an economic program to uplift the economic

condition of a commodity following information were found. Give your answer on the

basis of statistical analysis of the available data.[2061]

Monthly Income(in Rs. 00) Prior for the plan After the plan

No. of families No. of families

4-6 10 8

6-8 70 65

8-10 35 37

10-12 20 15

12-14 10 15

14-16 3 5

16-18 2 5

Is there any improvement in the community in terms of average monthly income?

Indicate the percent of amount changed in the highest income of the poorest 40% of the population before and after the plan.

Indicate the percent of amount changed in lowest income of richest 40% of the population before and after plan.

Obtain the limits of income of middle 50% of families before and after the plan and comment on the results.

Chapter 4 Measures of Dispersion

1. What are the absolute and relative measures of variation, discuss in what

situation they are used.[5][2065]

2. The coefficient of variation of a symmetrical distribution is 9% and mean of that

distribution is 40. Find the value of variance. [2][2076[12.96]

3. Find the coefficient of variation from the following informations.[2][2076][17.20%]

4. From the following distribution, determine the quartile deviation. [2][2075][5

inches]

Height(inches) 30 35 40 45 50

No of students 5 15 20 12 8

5. The mean and standard deviation of a series of 20 items are found to be 15 and

10 respectively. If an additional item of 5 included in the series, find the revised

values of mean and standard deviation.[2][2074][14.52 and 9.99]

6. From the data relating to the performance of 2 workers, state who is more

consistent workers?[2][2073][CV(X)=20%>CV(Y)14.29% Worker Y

Workers X Worker Y

MEAN TIME TAKEN TO COMPLETE A JOB

20 HRS 35Hrs

STANDARAD DEVIATION 4 5

7. The mean and coefficient of variation of a certain data set are 12 and 25%

respectively. Calculate the value of standard deviation and variance of the data.

[2][2072] ANS 3 AND 9 RESPECTIVELY.

8. The difference between the upper quartile and lower quartile of a certain

frequency distribution is 4 and their sum is 16. Calculate the quartile deviation

and its coefficient.[2][2071][2 units and 0.25]

9. The average price and standard deviation of price of Mansuli rice per kg for the

last seven days in two markets M1 and M2 are recorded below: [2][2071][Since

CV(M1) = 7.78% > CV(M2) = 6.38% the market M1 shows greater variation in

price.

Average price (Rs) S.D of price

Market M1 90 7

Market M2 94 6

Which market shows greater variation in price? 10. Two samples (each of size 10) of ages of students of BBS programme and MBS

program of Tribhuvan University are described below:[10] [2076]

Age of students of BBS (in years)

24 30 28 23 25 22 26 27 28 25

Age of students of MBS (in year)

26 33 29 28 27 29 33 34 27 28

If homogeneity in age of the class is a positive factor in teaching and learning process, which of the two groups will be easier to teach? Ans: C.V. (X1)=9.15%;

C.V.(X2)=9.29%; Morning MBS will be easier to teach 11. The average weekly wages, standard deviations and number of workers of two

factories are given below:[10] [2074]

Factory A Factory B

Average weekly wage Rs. 4600 Rs. 4900

Standard deviations Rs. 50 Rs.40

Number of workers 100 80

Calculate combined standard deviation. Also, explain which factory has greater variability in the distribution of weekly wages? Ans: Rs. 155.96, C.V.(A) = 1.09%, C.V.(B)= 0.82% and Factory A

12. A buyer obtained samples of electric fan from two companies A and B. He got

these samples tested in his laboratory for length of life in number of hours. The

following are the results of these tests. [10] [2073]

Length of life ( hours) Number of electric fans

Company A Company B

600-800 20 6

800-1000 32 20

1000-1200 52

1200-1400 20

1400-1600 16

What would you conclude as to which supplier’s fan are more uniform for length of life?

Ans: CV(A)= 21.92%>CV(B)=14.65%, Company B 13. The following is the wage distribution of workers of a company working in two

different shifts. [10] [2073]

Morning Shift Day shift

No. of workers 3,000 200

Average wage 8,000 9,000

Variance of wage distribution 100 121

Find: 1. Average wage of all workers of the company.

2.Standard deviation of wage of all workers of the company.

3.Coefficient of variation of all workers of th company.

Ans: 1. 8062.50 2. 242.27 3. 3.0049% 14. The mean and variance of the marks in statics obtained by all the 50 students of

a certain college was computed as 60 and 100 respectively. Later on it was

discovered that the scored 76 was wrongly taken as 676. Find the mean and

standard deviation of scores when wrong value is omitted? Also calculate the

coefficient of variation of marks after ignoring wrong value? [10] [2072] Ans:

Mean= 59.86 marks, σ= 10.03 marks, C.V. 16.76%

15. Two brands of tyres are tested for their life and following results were found: [10]

[2072]

Life (000km) 20-24 24-28 28-32 32-36 36-40

Brand X 8 15 12 8 7

Brand Y 6 20 14 5 5

Both the brands are offering same price and advertising in favour of their bramds saying that the brand has consistent life. If you are required to decide to purchase tyre of one of these two brands, which one do you prefer and why? Ans: Since C.V(X)= 17.43%, I prefer to purchase tyre of brand Y.

16. For a group of 200 candidates, the ,ean and standard deviation were found to be

40 and 15 respectively. Later on it was discovered that the scored 43 and 35

ewre misread as 34 and 35 respectively. Find correct mean and standard

deviation. [10] [2072] Ans: Mean= 39.955; σ=14.9714

17. The combined mean and variance of salaries of 250 workers of city A and city B

are 560 and 5497 respectively. The mean and variance of the salaries of 150

workers of city B are 500 and 81 respectively. Find the variance of salary of

workers of city A.[10] [2071] Ans:121

18. If the mode for the following distribution is Rs.24, find the mean deviation from

mean. [10] [2071]

Expenditure in Rs. 0-10 10-20 20-30 30-40 40-50

No. of families 14 23 27 ? 15

Ans: Rs. 10.20 19. The mean weight of 150 students is 60kg. The mean weight of boys is 70 kg with

a standard deviation of 10 kg. For the girls, the mean weight is 55 kg and

standard deviation is 15 kg. Find no. of boys and girls in a class and combined

standard deviation. [10] [2071] Ans: 100, 50, 15.28

20. From the prices (in 00 rupees) of the shares ‘P’ and ‘Q’ given below, state which

share is more stable in value and why? [10] [2069] Ans: CV(P)=5%;

CV(Q)=1.90%; Share Q

P 55 54 52 53 56 58 52 50 51 49

Q 108 107 105 105 106 107 104 103 104 101

21. The mean and standard deviation of 100 observations were found to be 20 and 3

respectively. After the calculation were made it was found that three of the

observations were incorrect, which were as 20,22 and 18. Find mean and

standard deviation if the incorrect observations are omitted. [10] [2068] Ans: 20,

3.032

22. A sample of 60 cars of two makes P and Q, is taken and their average running

life in years is recorded as follows:[10] [2067]

Life (years) 0-2 2-4 4-6 6-8 8-10

0Make P 8 12 22 14 4

Make Q 10 14 19 12 5

1. Find the mean life of each make.

2. Which make shows greater consistency in performance and why?

Ans: 1.4.8 years, 43.6 years 2.CV(P)= 46.04%; CV(Q)= 51.30%; Make P 23. An analysis of monthly wages paid to workers in two firm A and B, belonging to

the same industry, gives the following results:[10] [2066]

Firm A Firm B

No. of worker 500 600

Average wage 1500 1550

Variance of distribution of wage 100 121

Find 1.In which firm is there greater variability in individual wages? 2.The average wage and variance of all the workers of firm A and B taken together. Ans:1.CV(A)=0.67%; CV(B)=0.71%; Firm B 2.1527.27,731.29

24. The mean and standard deviation of a set of 100 observations were worked out

as 40 and 5 respectively by a computer but later on it was found that by mistake

the computer read the value 50 in place of 40 for the observation. Find the

correct mean and deviation.[10] [2065]

Ans:Correct Mean 39.9; Correct S.D. 4.898 25. From the following data find out the missing frequencies and compute the

standard deviation of the distribution of the average number of tables to cure

fever was 20.96. [10] [2064]

No. of tablets No. of persons cured

4-8 10

8-12 12

12-16 14

16-20 16

20-24 ?

24-28 16

28-32 ?

32-36 10

36-40 5

Total 115

Ans: 20,12,8.8752 26. A sample of 5oo cars of each of two makes X and Y is taken and average

running life years is recorded.

Life (No. of years) No. of cars

Make X Make Y

0-2 80 60

2-4 120 100

4-6 170 200

6-8 100 120

8-10 30 20

If price of car is same, which makes car should be preferred by the buyer? Ans: CVx=49.78%, CVy=43.37%, Since CVy<CVx, make Y should be preferred by

the buyer. 27. The average BBS CMAT of 200 students of Private Campuses with valley is

observed to be 70 with S.D. 30 and the average CMAT score of 300 students of

constituent campus is observed to be 60 with S.D. 30. What is the combined

average and combined standard deviation?

Ans: Combined average = 64, Combined S.D = 30.40 28. An analysis of monthly wages paid to workers of two firms A and B belonging to

the same industry were as follows:

Firm A Firm B

No. of workers 400 600

Average wages 5000 4500

Standard Deviation of wages distribution 11 10

Find the coefficient of variation and variance of 1000 workers of the industry.

Ans: C.V = 5.22%, Variance = 60108.4 29. An analysis of monthly income of workers of industry A and B are as follows:

Industry A Industry B

No. of workers 500 600

Average monthly income 4200 4000

Standard Deviation 9 8

Find the coefficient of variation and variance of all 1100 workers of the industry A and B taken together.

Ans: Combined Variance = 9989.08, combined CV = 2.44% 30. The following table represents the annual income distribution of lower middle

class families of city A and city B. Show graphically in which city prevails more

inequality in distribution of income and why.

Income in Rs. 9000 10000 11000 12000 13000 14000 15000 16000

City ‘A’ 33 30 24 21 18 12 9 3

City ‘B’ 32 22 15 12 10 6 2 1

Ans: since the curve of city B is farther than City A, rom the line of equal distribution, there is greater inequality in distribution of income in City B.

31. Compute the combined mean and standard deviation for the following data:

No. of administrative staffs 75

No. of teaching staffs 50

Average monthly salary of administrative staffs Rs. 4500

Average monthly salary of teaching staffs Rs. 6000

Standard deviation of the salary of the administrative staff Rs. 800

Standard deviation of the salary of the teaching staff Rs. 900

Ans: �̅� = 𝑅𝑠. 5100, σ = Rs 1117.14 32. The following table gives the monthly income of the works in Kathmandu and

Jhapa represent the data graphically in order to bring out inequality of the

distribution of monthly income. Also indicate in which city greater inequality exist

in distribution of monthly wage.

Monthly income (Rs.) No. of persons (000)

Kathmandu Jhapa

450 5 32

500 15 22

550 20 15

600 20 12

650 12 10

700 10 6

750 9 2

800 9 1

Ans: Jhapa 33. One hundred and twenty (120) students appeared in English test, the marks

distribution is reported as below:

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70

No. of Students 5 7 12 13 40 28 15

Compute coefficient of variation and interpret the result.

Ans : 44.74% 34. Following two samples describes the age of the students in regular morning MBS

program and evening MBS program of SDC. If homogeneity of the class is a

positive factor for learning which of the two programs will be easier to teach.

Evening MBS 24 30 28 23 25 22 26 27 28 25

Morning MBS 28 27 34 33 29 27 28 29 22 26

Ans: C.V. for Evening = 9.15%, C.V. for Morning = 11.41%; Evening MBS 35. Given below is the distribution of profit (‘000 Rs.) earned by Book Store in

Kathmandu Valley Profit (in Rs.

‘000) Below 20 Below 30 Below 40 Below 50 Below 60 Below 70 Below 80 Below 90 Below 100

No of persons.

5 14 27 48 68 83 91 94 100

Compute coefficient of quartile deviation.

Ans: 0.25 36. The running capacity of two horses is given below, state which consistent and

why?

Horse A 250 255 280 290 295 300

Horse B 280 282 290 295 298 295

Ans: CVA = 6.93 and CVB = 2.35; B 37. A student obtained the mean and standard deviation of 100 observations as 40

and 5.1 respectively. It was later found that one observation was wrongly copied

as 40, the correct figure being 50. Find the correct mean and standard deviation.

Ans: Correct Mean = 40.1, Correct Standard Deviation = 5.2 38. For a group of 200 candidates, the mean and standard deviation were found to

be 40 and 15. Later on it was discovered that the score of 53 was misread as 35.

Find the correct mean and standard deviation corresponding to the corrected

figures.

Ans: Correct Mean = 40.09 and Correct Standard Deviation = 15.02 39. The mean and standard deviation of a set of 100 observations were found to be

40 and 12 respectively. On checking, it was found that two observations were

wrongly taken as 23 and 15 instead of 43 and 18. Calculate the correct mean and

standard deviation.

Ans: Correct Mean = 40.23 and Correct Standard Deviation = 11.82 40. The following are the date regarding the wealth (in millions of Rs.) of families in

Ward No. 1 and No. 2 in Kathmandu Mahanagar Palika. Show which ward has

greater inequality in wealth distribution. Give your decision.

Wealth (in millions Rs.) Below 2 2-5 5-10 10-20 20-30

No. of Families

Ward No. 1 650 100 120 80 50

Ward No. 2 100 170 130 60 40

Ans: CV1 = 143.47% and CV2 = 93.96%, Ward No. 1 41. Students age in the regular daytime MBA program and the morning program of

National University are described by two samples. If the homogeneity in age of

the class is a positive factor in learning make suggestion, with reason, which of

the two groups will be easier to teach?

Regular MBA Program Morning MBA Program

Age No. of Students Age No. of Students

23 9 27 10

29 2 31 8

28 5 30 5

22 10 29 4

30 1 28 6

21 4 33 5

25 11 34 5

26 6 35 11

27 3 36 2

24 9 32 4

Total 60 Total 60

Ans: CVX = 9.16% and CVY = 9.52%; Regular MBA Program. 42. Monthly income (in’000 Rs.) distribution of 400 families of a certain town are

given below:

Income, below (‘000 Rs.) 80 120 160 200 240 280

No. of families 10 35 165 310 380 400

a. Calculate the appropriate measure of central tendency from the above

distribution and justify for your choice of measure.

b. Calculate the most suitable measure of dispersion and support for your

choice.

c. Find the lowest income of richest 20% of the families.

d. Find the highest income of the poorest 20% of the families.

e. Find the limits of incomes of middle 50% families.

Ans: (a) Md = Rs 169655.20 (b) Q.D = Rs 28620.70 (c) P80 = Rs 205714.30 (d) P20 = Rs 133846.20 (e) The incomes of middle 50% families lies between Rs 140000 and Rs 197241.4

43. What do you understand by measures of dispersion and distinguish between

absolute & relative measure of dispersion? A sample of 50 cars, each of two

makes X & Y, is taken and their average running life in years is recorded.

Life (no of years) 0-5 5-10 10-15 15-20 20-25

Make X 6 10 20 12 2

Make Y 8 12 17 10 3

Which of these two makes car would you prefer to buy if the costs of both Makes are same regarding consistency?

Ans: X̅ = 11.90 years; σx = 5.16 years; CV(X) = 43.36%; �̅� = 11.30 years; σY =

years; CV(Y) = 49.73%; Make X

44. Two companies ONIDA and BALTRA of home appliances profit distribution are

given as follows:

Profit (million Rs.)

0-2 2-4 4-6 6-8 8-10 10-12

ONDIA 2 7 11 20 7 3

BALTRA 5 4 9 13 15 4

a. An investor wants to invest according to the profit of the company, in which

company it will be better to invest?

b. Also find the average profit and variances of profit of both the companies

together.

Ans: (a) CV(X1) = 37.58% and CV(X2) = 42.47%, company ONIDA (b) Rs 6.46 million and 6.7934

45. The following two samples describes the age of students in Model Campus and Galaxy campus BBS programme.[20][2072]

Model 25 31 29 24 26 23 27 28 29 26

Galaxy 29 28 35 34 30 28 29 30 34 27

a. Calculate the mean and standard deviation of age of students of each campus. [Model Campus mean = 26.80, standard deviation = 2.36, Galaxy Campus mean = 30.40, standard deviation = 2.73]

b. If homogeneity in age of the students is a positive factor, for learning which of the two campuses will be easier to teach? [Model campus CV (X1) = 8.81%, Galaxy Campus CV(X2) = 8.98% Model campus will be easier to teach.

c. Calculate the mean variance and coefficient of variation of age of students of both campuses taken together.[Mean12=28.60, standarddeviation12

2 = 9.75125, CV(all) = 10.91%

d. Which campus’s students are more intelligent and why?[cannot be determined on the basis of age of students]

46. The following two samples describes the age of the students in Morning MBS and Evening MBS program of Public Campus.. [20][2072]

Morning 24 30 28 23 25 22 26 27 28 25

Evening 28 27 34 33 29 27 28 29 33 26

a. Calculate the mean and standard deviation of age of students of each program.

b. If homogeneity in age of the students is a positive factor for learning which of the two programs will be easier to teach?

c. Calculate the mean variance and coefficient of variation of age of students of both program taken together.

d. Which program’s students are more intelligent and why?

a.

Ans Morning MBS Evening MBS

Mean age 25.80 29.40

S.D. of age 2.36 2.73

CV Of age 9.15% 9.29%

b. Morning MBS program, 27.60, 9.75125 and c. 11.30% d. cannot be determined.

47. The following are the weekly production in units (output) of 60 workers ofa factory.

72 23 48 51 64 82 12 33 50 39

57 35 88 77 25 39 52 48 64 49

52 41 72 62 49 32 54 67 46 55

57 82 44 75 56 51 63 59 69 53

42 75 85 68 55 52 45 40 57 20

75 46 51 50 16 62 56 54 40 55

The management has decided to give bonus of Rs 5,000: 6,000: 7,000: 8,000 and 9,000 to each worker in the respective output group 40 to 50; 50 to 60 and so on.

Find: a. Mean output of all the workers b. Average bonus received by the workers c. standard deviation of bonus. [Ans a. 53.83 units, b. Rs 6,400, c. Rs 1,200.]

Chapter 5. Skewness, Kurtosis and Moments

1. Define Skewness and kurtosis of a distribution. Also explain how skewness is measured.[10][2069]

2. In a moderately skewed frequency distribution, the mean, median and standard deviation are 10, 8.50, and 2 respectively. Find the Karl Pearson’s coefficient of skewness. [2][2076][2.25, positive skewed]

3. The first four moments about mean of a certain distribution are 0, 16, -30, and 40 respectively. Calculate the coefficient of Kurtosis and interpret the

result.[2][2076][𝛽2=0.15625, platykurtic]

4. For a group of 10 items ∑ 𝑋 =452, ∑ 𝑋2 =24,270 and mode = 43.7. Find Pearson’s coefficient of skewness.[2][2075][0.08, right skewed]

5. The first four moments about the arbitrary value 5 are -0.1, 2.82, 0.2 and 20.58. Find the second and third moment. [2][2074][2081 and 1.044]

6. The first four moments about mean are 0,3,9 and 21. Test the symmetry. [2][2073] Ans 𝛽1= 3; positive or right skewed

7. List the five number summary from the following daily sales data (in Rs.000) of seven different shops: [2][2072][10,20,50,70,80]

Sales(Rs000) 50 20 80 10 60 30 70

Shops A B C D E F G

8. Test for the normality of the distribution on the basis of the following information:[2][2072]

9. Lower Quartile (Q1) = 41.5 Upper Quartile (Q3) = 58.25 Tenth percentile(P10) = 31.428 90th percentile (P90) =70

10. The standard deviation of a symmetrical distributions is 7. What must be the value of the 4th moment about the mean in order that the distribution be mesokurtic? [2][2072][Ans 7,203]

11. The following table gives the distribution of daily wages in a company.

Wages(Rs) No. of Workers

40-50 10

50-60 15

60-70 20

70-80 28

80-90 35

90-100 25

100-110 18

110-120 10

120-130 8

Calculate Karl Pearson’s Coefficient of Skewness. Is the distribution

symmetrical? [10][[2076][Ans -0.044≠0;distribution is not symmetrical. 12. The first four moments about the value 5 are 2, 20, 40, and 50 respectively.

Calculate mean,[7] variance,[16] coefficient of skewness[1] and coefficient of kurtosis [0.633]of the distribution.[10][2075]negatively skewed, platykurtic]

13. The data below represent the amount of grams of carbohydrates in a serving of breakfast cereal. 11, 15, 23, 29, 29, 22,21, 230,15, 25, 17. Construct a bix-and-whisker plot for the carbohydrate amounts.[10][2075][X1 = 11, Q1 = 15, Md= 21, Q3 = 25, XL = 29, left or negatively skewed.

14. Taking into account the following information comment on the nature of the distribution. [10][2074] [𝛽1= 0.0001, approximately symmetrical 𝛽2 = 2.05, plutykurtic]

N= 100, ∑ 𝑓𝑑𝑥 = -14, ∑ 𝑓𝑑2𝑥=154 ∑ 𝑓𝑑3𝑥 = -62 ∑ 𝑓𝑑4𝑥 = 490

15. From the following distribution ifnd the first four moments about mean, skewness and kurtosis of the distribution and interpret the result. [10][2073][𝜇1 = 0, 𝜇2 =

5.44, 𝜇3 = 0, 𝜇4 = 67.84, 𝛽1 =0; symmetrical; 𝛽2 = 2.29 platykurtic, K = 0.217 platykurtic

Profit in lakh(Rs)

0-2 2-4 4-6 6-8 8-10

No. of Companies

3 5 9 5 3

16. The first four moments of a distribution about the arbitrary value 5 are 2, 20, 40 and 50. Calculate the mean, standard deviation, skewness and kurtosis of the

distribution. [10][2072][�̅� =7, 𝜎 = 4, 𝛽1 =1, 𝛽2 = 0.633, PLATYKURTIC 17. A manufacturer of battery took a sample of 13batteries from a day’s production

and used them continuously until they were drawned. Numbers of hours they were used until failure were 342, 426, 317, 545, 264, 451, 1049, 631, 512, 266, 492, 562, 298. [10][2072][264, 307.5, 451, 535.5, 1049][Yes the data are right skewed]

a. List of five number summary b. Construct a bo-and –whisker plot for the data c. Are the data skewed? If so how?

18. Find the skewness based on moment from the following data:[10][2072][ 𝛽1 = 0.703125 or 𝛾1 = 0.835, positively skewed.

No of hours worked 1-3 3-5 5-7 7-9

No. of days 3 5 1 1

19. Compute mean, variance and skewness by using method of moments from the

following data: [10][2072][ �̅� = 25, 𝜇2 = 146.67, 𝛽1=0

Class interval

0-10 10-20 20-30 30-40 40-50

Frequencies 4 6 10 6 4

20. Find the skewness and kurtosis by the method of moments from the following

data: [10][2072][ 𝛽1=0.0486 or 𝛾1 =0.2204, positive skewed, 𝛽2 = 2.563 or 𝛾2 = -0.437, platykurtic]

Weight (kg) 0-20 20-40 40-60 60-80 80-100

Frequency 8 12 20 6 4

Also comment on the results obtained.

21. Pearson’s coefficient of skewness for a distribution is 0.4 and its coefficient of variation is 30%. If mode is 88, find mean and median. [5]

The first three moments of a distribution about the value 4 are 1,4 and 10. Find the mean and variance and also the 3rd moment about mean.[2][2071]

Ans 100,96 5,3,0

22. If the standard deviation of a symmetrical distribution is 4. What should be the value of fourth moment about mean in order that the distribution is a. platykurtic

b. Leptokurtic.[5][2070] Ans 𝜇4 <768 and 𝜇4> 768

23. The first four moments about an arbitrary point 4 are 1,3,7 and 21. Find mean and standard deviation of the distribution.[5][2070][Ans 5 and 1.41]

24. Karl Pearson’s coefficient of skewness of a distribution is 0.5. If median and mode of the distribution are respectively 42 and 36. Find coefficient of variation.[10][2068][40%]

25. The first four moments about the value 4 are -1.5, 17, -30 and 108. [10][2066][a.

Mean = 2.5 and S.D = 3.84 b. 𝛽1=0.49; r1 = 0.7, positively skewed; 𝛽2 = 0.65; r2 = -2.35 platykurtic]

a. Find mean and standard deviation of the distribution.

b. Test for skewness and kurtosis of the distribution. 26. Compute the Pearsonian measure of skewness for the following distribution and

interpret the result. [10][2065][Ans Mean 40.364 inches, S.D. 3.296 inches, Mode 40.304, SK = 0.018, Positive]

Size in inches No. of observations

30-33 3

33-36 5

36-39 26

39-42 46

42-45 20

45-48 10

27. In a certain distribution the first four moments about the arbitrary point 4 are -1.5,

17, -30 and 108. Find 𝛽2 and state whether the distribution is Leptokurtic,

Mesokurtic or Platykurtic. [10][2064][Ans 𝛽2 = 0.654,platykortic] 28. If the standard deviation of a symmetrical distribution is 11, what must be the

value of 4th moment for the distribution to be normal?[ 𝜇4 = 43923] If the first four central moments about the mean are 0,1.75, 39.75 and 152.31 find

𝛽1 and 𝛽2 and state the nature of the distribution. [𝛽1 = 0.49 positively skewed

𝛽2= 0.70 platykurtic]

29. The first four moments about an arbitrary point A=4 are 1,4,10 and 45. Find 𝛽1

and 𝛽2 and state the nature of the distribution. [2062][Ans 𝛽1 = 0, so the distribution is symmetrical and 𝛽2 = 2.89 hence the distribution is platykurtic]

30. The first four moments about the point 25 is given as 3, 90, -900 and 2100. Calculate Y1 and Y2 and state the nature of the curve as explained by them. [2062][Ans Y1=- 2.27 and Y2 = -0.30

31. Calculate coefficient of skewness on the basis of quartile values for the distribution and interpret the result. [2061][Ans SKB = 0.22 positively skewed

Daily wage(Rs) No.of workers Below 100

Below 100 20

100-150 81

150-300 120

300-500 150

500-1000 130

100-1500 30

1500-2000 10

2000 and Above 5

32. If the first four central moments about the mean are respectively 0, 2.8, -2, and 24.5. Calculate the coefficient of kurtosis and skewness and interpret the result.[2061][Ans Y1 = -0.43;negatively skewed, 𝛽2 = 3.125, leptokurtic

33. From the following grouped frequency distribution compute coefficient of Karl Pearson’s skewness [2058][Ans SKP = 0.08, Positively skewed]

Mid value of Income 150 250 350 450 550 650 750 850

No. of workers 80 105 120 165 100 90 60 40

34. The standard deviation of a symmetrical distribution is 5. What must be the value of the fourth moment about the eman in order that the distribution be a.

Leptokurtic, b. Mesokurtic c. Platykurtic [2056][ a. 𝜇4 >1,875 b. 𝜇4= 1,875 c. 𝜇4 < 1,875

35. Find Kurtosis of the following distribution by the method of moments, Comment on the result obtained.[2055][𝛽2 = 3.125, Leptokurtic

No. of hours worked 1-3 3-5 5-7 7-9

No. of days 3 5 1 1

36. Calculate the coefficient of skewness for the following distribution and hence comment on the type of frequency distribution[2054][Ans SKP = -0.18, Negatively skewed

X 20-30 30-40 40-50 50-60 60-70 Total

f 7 10 20 18 7 62

37. Calculate the skewness and kurtosis from the following wage distribution of a factory. Explain the charactyeristics of a factory. Explain the characteristics of

frequency distribution on the basis of skewness and kurtosis.[20][2074][𝛽1=

0.0155, negatively skewed; 𝜇3 is negative, 𝛽2 = 2.5921, platykurtic

Wage per hours (Rs) Number of workers

0-20 5

20-40 7

40-60 16

60-80 20

80-100 28

100-120 12

120-140 10

140-160 2

38. The manager of Bakery Café selected a random sample of 50 customers’ waiting time is recorded as follows:

29 28 51 43 24 40 52 72 41 23

25 30 22 34 19 31 32 29 45 24

60 48 19 47 54 68 17 43 23 56

39 40 43 48 56 42 21 36 24 65

60 31 50 31 47 43 30 32 35 39

a. Develop a frequency distribution using 7 classes. b. Comment on the nature of distribution of customer’s waiting time in the café.

[15][2073]Ans

Waiting time

10-20 20-30 30-40 40-50 50-60 60-70 70-80

No. of customers

3 11 12 13 6 4 1

𝛽1 = 0.1481; positive skewed; 𝛽2 = 2.595; platykurtic 39. The assest structure of listed companies in Nepal is given below:

[15][2071][0.114 and 2.36 49.48%]

Asset (Rs. In millions) No.of listed companies

0-5 20

5-10 25

10-15 50

15-20 40

20-25 20

25-30 15

Total 170

a. Compute the skewness and kurtosis of the asset distribution by using moment method.

b. Interpret the result of skewness and kurtosis. c. Also calculate the coefficient of variation of asset distribution.

40. The following is the daily wages of the workers of two factories.[2070]

Daily wages in Rs No. of workers

Factory X Factory Y

50-75 30 50

75-125 60 80

125-150 88 120

150-175 120 70

175-200 60 40

200-225 28 30

225-250 14 10

Find Mean and standard deviation of each group and state which group of workers has greater consistency in the wage. Test the normality of the distributions. Ans Factory X Mean =Rs 150, S.D. = Rs 42.22, CV = 28.15 % Factory Y Mean =Rs 138.125, S.D. = Rs 44.89, CV = 32.50 % Factory X K(X) = 0.203;K(Y) = 0.237 Both are Platykurtic Factory X is more flat.

41. You are given the position in a factory before and after settlement of an industrial dispute. Comment on the gains or losses from the points of view of workers and that of management. [20][2064]

Before After

No. of workers 300 250

Mean wages (Rs) 2000 2500

Median wages(Rs) 2200 2400

Modal wage (Rs) 2600 2200

Standard deviation (Rs) 105 115

Ans ∑ 𝑋 before = Rs 6,00,000 and ∑ 𝑋 after = RS 625,000. CV(Before) = 5.25% and CV (after) = 4.76%. Thus, after the settlement of dispute there are less disparities in wages and from management point it will result in greater satisfaction to the workers. Before dispute SK (P) = -5.71; After dispute, SK (P) = 2.61. Thus, the curve of distribution becomes positively skewed after the dispute settlement form negatively skewed before the dispute, which implies number of workers getting highet wages has decreased.

42. The asset structure of listed companies in Nepal is as follow:[2063]

Assets (Rs in millions) No. of listed companies No.of unlisted companies

0-5 20 20

5-10 25 30

10-15 50 40

15-20 40 50

20-25 20 20

25-30 15 10

170 170

Find the skewness of the asset distribution and make the comment on both listed and unlisted companies. Also, comment on the consistency of asset distribution between the companies. Ans Listed companies are positively skewed(y1 = 0.114)

whereas Unlisted companies is approximately symmetrical (𝛽1=0.00031) Distribution of unlisted companies has more constancy. CV = 49.48% , 48.74%

43. Find the mean, median and mode and skewness of the distribution by making

class of 25-50, 50-75 and so on.[�̅� =128.5, Md = 134.09, Mo = 133.33 (Using Grouping Method) SK (P) = -0.12, S.D. = 40.55

44. Compute first four moments about the mean and measure skewness and kurtosis by the method of moments and explain the characteristics of the following distribution by interpreting the computed values.

Daily wage in Rs No. of persons Daily wage in Rs No.of persons

0-200 50 800-1000 280

200-400 70 1000-1200 120

400-600 160 1200-1400 100

600-800 200 1400-1600 20

N= 1000

Ans Y1 = 0.124, 𝛽1=0.01549, Negatively skewed 𝛽2 = 2.5920; Platykurtic

45. The following are the weekly production of production X in units of 60 workers in a manufacturing company.

23 48 51 64 52 19 33 50 39 72 57 49

48 52 39 25 77 88 35 41 72 62 49 32

67 46 55 52 57 53 69 59 63 51 56 75

82 75 85 68 55 52 45 40 57 20 42 75

40 54 56 62 16 50 51 46 64 54 44 55

The management has decided to give bonus of Rs 5, 10, 20, 25,30 to each of the worker in the respective output group of 10 or over upto 70 weekly output. Construct the frequency distribution of bonus and describe the characteristics of the frequency distribution as regard to central tendency, dispersion, skewness

and kurtosis and draw conclusions. [2059][Ans �̅� = 21.9, 𝜎 = 6.32, 𝛾1 = -0.92 –

Negatively skewed, 𝛽2 = 3.41 Leptokurtic

46. A factory pays workers on piece rate basis and also a bonus to each worker on the basis of individual output in each quarter. The rate of bonus payable is as follows:

Output(in units) 70-74 75-79 80-84 85-89 90-94 95-99 100-104

Bonus (in Rs) 35 45 50 55 60 70 80

The individual output of a batch of 50 workers is given below:

94 83 78 76 88 86 93 80 91 82

89 97 92 84 92 80 85 93 98 103

87 88 88 81 95 86 99 81 87 90

94 97 80 75 93 101 82 82 89 72

85 83 75 72 83 98 77 87 71 80

Construct suitable frequency distribution and describe the characteristics of frequency distribution of bonus and draw conclusions as regards to central tendency, dispersion,

skewness and kurtosis. Ans �̅� = 55, 𝜎 =9.85, 𝛽1 = 0.21 –Positively skewed 𝛽2 = 3.46 – Lepokurtic

47. In which distribution do you think that mean is more representative and also test for normality of these two distributions and interpret the result on the basis of the information’s given below:

Daily wages in Rs No. of workers

Factory X Factory Y

20-30 15 25

30-40 30 40

40-50 44 60

50-60 60 35

60-70 30 20

70-80 14 15

80-90 7 5

Ans CVX 28.25% and CVV = 31.83% Factory X 𝛽2 = 2.65(for X) 2.66(forY): Both platykurtic Chapter 6 Simple Correlation and Regression Analysis

1. Clculate Karl Pearson’s coefficient of correlation from the following information.[2][2076] [Ans r=0.6030]

N=8, ∑ 𝑥1 = 544, ∑ 𝑥2= 552, ∑ 𝑥12= 37028, ∑ 𝑥2

2=38132 and ∑ 𝑋1𝑋2=37560.

2. If bxy = -0.87 and byx = -0.63, find the correlation coefficient. [2][2075][Ans -0.74]

3. Determine the number of pains of observation were ∑ 𝐷2= ∑(R1-R2)2 is 30 and rank correlation, R=0.75 [2][2074][Ans0.76]

4. The regression coefficient of y on x is 0.68 and correlation coefficient is 0.72, find the regression coefficient of x on y. [2][2073][.76]

5. For eight pairs of observation on two variables sales (X) and pricing (Y) the following results were obtained,[2][2072][0.5789, medium degree of positive relationship] ∑X = 156, ∑Y= 132, ∑X2 = 4162, ∑Y2 = 2434, ∑XY = 2844

6. A student calculated the value of r = 0.81 from 10 observations and conclude that r is highly significant. Approve statistically his conclusion.[2][2072][r=0.81, P.E.(r) = 0.4404, significant]

7. The rank correlation between marks in English and Math obtained by 10 students was found to be 0.7. Later on it was found that the difference in the rank in two subjects obtained by one student was wrongly taken as 5 instead of 8. Find correct rank correlation. [2][2072][0.4636]

8. The sum of the squares of the differences of ranks obtained from eight pairs of observations of Xand Y variable is calculated as 6. Calculate the rank correlation coefficient between the variables.[2][2071][0.929]

9. Calculate the Karl Pearson’s Correlation Coefficient from the following data. Also interpret the data on the basis of probable error. [10][2074][r=0.7453, P.E. (r) =0.1133 and significant

X 67 65 68 64 66 70 63

Y 69 65 69 65 67 67 64

10. Calculate the rank correlation coefficient from the following data and also interpret the result:[10][2072][r=0.917, high degree of positive correlation]

Marks in Stat

20 30 40 40 50 60 70 80

Marks in A/C

10 15 20 20 18 25 30 40

11. FROM THE FOLLOWING BI-VARIATE FREQUENCY DISTRIBUTION,

DETERMINE THE MOST PROBABLE AGE OF THE WIFE WHOSE

HUSBAND’S AGE IS 75 YEARS.[10][2072][Ans 𝑌 ̂=6505964 YEARS]

Age of wives (in yrs) Age of husband in years

20-30 30-40 40-50 50-60 60-70

15-25 5 9 3 - -

25-35 - 10 25 2 -

35-45 - 1 12 2 -

45-55 - - 4 16 5

55-65 - - - 4 2

12. The coefficient of rank correlation of the marks obtained by 10 students in English and statistics was found to be 0.8. It was later discovered difference in rank obtained by one student was wrongly taken as 7 instead of 9. Find the correct rank correlation.[5][2071][0.61]

13. Find whether there exists any relationship between the age of driver and the no of accidents from the following bi-variate table.[10][2070][r = 0.7073, high degree of positive relationship

No of accidents 20-22 22-24 24-26 26-28 28-30

0 5 7 13 - -

1 - 10 15 2 -

2 - 3 12 6 -

3 - - 4 12 5

4 - - - 4 2

14. Calculate the two regression coefficients and correlation coefficient between the heights of fathers and sons from the following bi-variate frequency distribution.[10][2069][the regression coefficients are -0.39 and -0.351 and r=-0.37

Height of fathers(inch)

Height of sons (inch)

62-64 64-66 66-68 68-70 70-72

60-62 1 2 - 1 3

62-64 - 1 - 2 1

64-66 2 3 2 1 -

66-68 - 1 1 - 1

68-70 1 2 - - -

15. From the data given below, find whether there is any relationship between the age of the drivers and the number of accidents made by them.[10][2067][r=-0.6914, medium degree of negative correlation]

Driver’s age

No. of accidents

0-2 2-4 4-6 6-8 8-10

25-30 - - 3 4 12

30-35 3 - 5 9 7

35-40 5 9 10 6 3

40-45 7 4 3 - 1

45-50 8 1 - - -

16. If the sum of the squares of the difference between two ranks is 288 and the

number of pairs of observation (n) is 7, find the coefficient of Rank correlation. Also give your opinion whether there is any agreement between order of two ranks.[5][2065][R=-4.14,t here is no agreement between order of two ranks]

17. Regression coefficient of Y on X and X on Y are given as -2.002 and -1.461. Find the value of correlation coefficient between X and Y. If mean of X and Y are 87.2 and 127.2 estimate X when Y = 133. [5][2065][r = -0.9607, X =

84.5262] 18. Find the most likely price in Kathmandu corresponding to the price of Rs 70 at

Birgunj from the following data: [5][2064][Ra 67.56

Average price in birgunj Rs 65

Average price in Kathmandu Rs 67

Standard deviation of price in Birgunj 2.5

Standard deviation of price in Kathmandu 3.5

Correlation coefficient between the price in two cities +0.08

19. In a research report following information are given[5][2064][r=0.933, positive]

No.of observation 9

Sum of the product between two variables income (X) and expenditure (Y)

731

Sum of the square of income (X) 285

Sum of the square of expenditure (Y) 2085

Mean income Rs5

Mean expenditure Rs 15

Check whether the relationship between income and expenditure id positive or negative

20. For a bi-variate data the mean value of X is 20 and mean value of Y = 45. The regression coefficient of Y on X is 4and that of X on Y is 1/9. Find a. the standard deviation of X if the standard deviation of Y is 12.[𝜎𝑦 = 2]

b. coefficient of correlation [r = 0.67] c. the two regression equations [ Y = 4X-35 and X = 1/9 Y + 15 d. estimate the value of X when Y = 25 [ X = 17.78]

21. Quotation of Index numbers of ordinary shares prices of a Joint Venture Bank and the price of preference share are as follows: [2063]

Year 1 2 3 4 5 6 7

Ordinary share price

100.5 102.6 100 98.5 99.6 105.1 101.9

Preferences share price

97.8 96.5 100 101.2 106.8 99.9 98.5

Find the rank correlation coefficient and make the interpretation. [-0.714, high degree of negative correlation]

22. The advertisement expenses and the sales of a new product are recorded as below: [2063]

Adv. Exp (Rs 000)

1 5 6 8 10

Sales(Rs 000)

50 60 80 100 110

Estimate the sales when the advertising expenses is Rs 15,000. [Rs 144.557] 23. You are given two regression equation 4x-5y+33 = 0 and 20x – 9y -107 = 0.

Find Mean of x and y[ 13 and 17] Two regression coefficients [byx = 4-5 and bxy = 9/20. Coefficient of correlation between x and y. [r = 0.6]

24. Find whether there exists any relationship between income and expenditure from the following and interpret the result. [2062][r=0.795,high degree of positive correlation between income and expenditure]

Income in ‘000’ Rs

Expenditure in ‘000’ Rs

0-4 4-8 8-12 12-16 16-20 Total

0-4 5 9 3 - - 17

4-8 - 10 25 2 - 37

8-12 - 1 12 2 - 15

12-16 - - 4 16 5 25

16-20 - - - 4 2 6

Total 5 20 44 24 7 100

25. You are given the following information about advertising and sales.

Advertising expenditure(lakh) Sales (in lakh)

Mean 10 90

Standard deviation

3 12

The coefficient of correlation between advertising and sales is equal to 0.8. Estimate the likely sales when the advertisement expenditure is Rs. 15 lakh. [Estimated likely sales is Rs 1.6 lakh][2062]

26. Find the rank correlation between following data:[2062][rank correlation coefficient is 0.952]

X 54 57 61 64 64 72 72 79 86 89

Y 31 36 50 60 83 93 90 85 99 122

27. Find the rank correlation between following data:[2062][rank correlation coefficient is 0.59]

X 58 54 65 40 54 70 65 30 45 54

Y 52 48 68 35 71 50 56 38 40 60

28. Following information’s given below are related with the ages of husband (x)

and wife(y) for married couples living together in a sample survey. Calculate the co-efficient of correlation between age of husband and that of his wife; test the significance of calculated r. [ r = 0.52, r is significant] N = ∑f = 72, ∑fx = 3560. ∑fx2 =196800. ∑fy = 3260, ∑fy2 = 168400, ∑fxy = 172000

29. Find rank correlation from following data of the age of wife and husband:[R =1, perfect positive correlation]

Age of husband 23 27 28 29 30 31 33 35 36 39

Age of wife 21 22 23 24 25 26 28 29 30 32

30. The table below gives the ages and blood pressure of 10 patients in a clinic. [

Ans 0.89]

AGE 56 42 36 47 49 42 60 72 63 55

b.p. 147 125 118 128 145 140 155 160 149 150

Find the correlation coefficient between age and blood pressure. Ans 0.89 31. Age and weight of newborn calf is taken at weekly intervals.[2061]

Age (week) 1 2 3 4 5 6 7 8 9 10

Weight (lbs) 50 51 53 55 57 58 62 70 75 108

Estimate the line of best fit and estimate the weight of a calf of age 15 weeks. [Ans Yc = 37.335+4.83x , 109.785 lbs]

32. Compute Spearman’s rank correlation for the given values. Ans R = 0.66

X 30 50 90 25 75 120 128 135 20 40

Y 40 60 70 20 8/0 125 130 140 90 85

33. Calculate the Karl Pearson’s coefficient from the biviarate sample of 140 pairs of X and Y as distributed below: State what conclusions can be drawn from the result. [r = 0.7044, high degree of positive correlation]

Y/X 10-20 20-30 30-40 40-50

10-20 20 26 - -

20-30 8 14 37 -

30-40 - 4 18 3

40-50 - - 4 6

34. The following data gives the experiences of machine operators inyears and

their performance as given by the number of good parts turned out per 100 pieces. Calculate the regression equation of performance ratings on experience and estimate the probable performance if an operator has 8 years experience. Ans Y = 69.96+1.133X,78.734

Operator 1 2 3 4 5 6 7 8

Experience (X)

16 12 18 4 3 10 5 12

Performance (Y)

87 88 89 68 78 80 75 83

35. Given the following data: Variance of X = 9, Regression equations: 4X -5Y +33 = 0, 20X -9Y-107 = 0 a. The mean value of X and Y [13 AND 17] b. The coefficient of correlation between Xand Y [r = 0.6] c. The standard deviation of Y [4]

36. The following table shows the number of motor registration and the sales of Gorakhali Motor tyres by a wholesale dealer in Kathmandu for the term of 5 years. [2055]

Year Motor Registration(000 Nos ) No. of Tyres sold(000 Nos)

1 60 125

2 63 110

3 72 130

4 75 135

5 80 150

Estimate the sale of tyres when expected motor registration in next year is 90,000. [159.870] Also calculate correlation coefficient and interpret. [r = 0.85, higher degree of positive correlation]

37. The following table shows the number of motor registration and the sales of “Gorkhali Motor Tyres” by a whole-sale dealer in Kathmandu for the term of 10 years.[15][2076]

Years 1 2 3 4 5 6 7 8 9 10

Motor registration in (000 nos)

60 62 65 70 48 53 73 65 82 72

No of tyre sold in(000 nos)

68 60 62 80 40 52 62 60 81 85

a. Find the coefficient of correlation between the motor registration and number of tyres to sold. [r = 0.8342]

b. Interpret your result. [r = 0.8342 .6P.E. (r) = 0.3894, coefficient of correlation is significant]

c. Compute the two regression coefficients.[byx = 0.1678; bxy = 0.5959 d. Estimate the number of tyres to be sold when the expected motor

registration is 92,000? [Yc = 96530.6 ≈96531 38. From the following bi-variate table, compute coefficient of correlation. Also

test the significance of the result. Estimate the sales revenue when advertisement cost is Rs 70 thousand. [15][2075] Ans r = 0.596, significant estimate = Rs 305,005

Sales revenue (000 Rs)

ADVERTISEMENT EXPENDITURES (000 Rs)

5-15 15-25 25-35 35-45 Total

75-125 4 1 - - 5

125-175 7 6 2 1 16

175-225 1 3 4 2 10

22-275 1 1 3 4 9

total 13 11 9 7 40

39. You are given the sales and advertisement expenditure in crores:

Sales Advertisement Expenditure

20-30 30-40 40-50 50-60 60-70 Total

10-15 - - - 3 7 10

15-20 - 4 9 4 3 20

20-25 7 6 12 5 - 30

25-30 3 10 19 8 - 40

Total 10 20 40 20 10 100

Is there significant relationship between advertisement expenditure and sales if significant relationship exists between given variables, estimate the sales when advertisement expenditure is 82 crores.[Ans let X = Advertisement

expenditure in crores, y = sales, r = -0.438, P.E. ® = 0.0545, SIGNIFICANT, �̂�

= 31.05-0.2 X and �̂� = 15.1 40. A researcher is interested in seeing how accurately a new job performance

index measures what is important for corporation. One way to check is to look at the relationship between job evaluation index and an employee’s salary is significant or not. A sample of eight employees was taken, and information about salary in thousand rupees and job performance index (1-10, 10 is best) was collected.

Job performance index 9 7 8 4 7 5 5 6

Salary ‘000’ Rs 36 25 33 15 28 19 20 22

Develop an estimating equation that best describes these data and estimate the salary of an employee whose job performance is 10 and 2. Also find whether there exists significant relationship between the given variables. [15][2073][Ans YC = -2.1130+ 4.2138X Rs 40,025.80, Rs 6,314.60, r = 0.9853, P.E. (r) = 0.007; significant

41. In a study, the following information were obtaine:[20][2073]

No. of observations 9

Sum of product between two variables income(X) and expenditure (Y)

731

Sum of squares on income (X) 2085

Sum of squares of expenditures (Y) 285

Mean income 15

Mean expenditure 5

Find a. Two regression coefficients[byx = 0.9333; bxy = 0.9333 b. whether there exists any relationship between income and expenditure [r=0.9333; positive correlation] c. The two regression lines [Y = -8.9995 + 0.9333 X and X= 10.3335+0.9333Y d. the most probable amount of expenditure when income is Rs200.[Rs 177.6605]

42. the data in sales and promotion expenditures on a newly launched product for 6 years are given below: [15][2072]

Year 2003 2004 2005 2006 2007 2008

Sales (in Rs. 00,000) 16 20 18 24 20 22

Promotion expenses(Rs ‘000)

4 4 6 10 10 12

a. Calculate the two regression coefficients from the above data of sales and expenses. [Let X = Promotion expenses(Rs 000) Y = sales (Rs ’00,000’) byx = 0.6067, bxy = 0.90, r = + 0.738, high degree of positive correlation,

b. Compute the correlation coefficient between sales and promotional expenditure

and interpret. [ nothing can be concluded]

c. Test the significance of correlation coefficient. [ nothing can be concluded]

d. Develop the estimating equation that describes the effect if promotional expenses on sales. Estimate the expected sales if the promotional expenses is Rs 20,000. [Y = 15.346611 + 0.6067 X , Rs 27,48061.10

e. Explain the meaning of each parameter of the equation ; in terms of above information. [a=15.346611 indicates the sales when the promotional expenses is zero and byx = 0.60607 indicates the rate of change of sales when the unit change in the promotional expenses; That is if the promotional expenses is increased by Rs 1,000 the sales is increased by Rs 60,670

43. The data on the sales and frequency of advertisement of the company available for the past 30 days were as follows: [2072][15]

Volumes of sales per day (unit in 000)

Frequency of advertisement in electronic media’ day

0 2 4 6 8

0-5 2 - - - -

5-10 - 4 5 3 4

10-15 - - - 4 6

15-20 - -- - - 2

a. Is there any significant relationship between frequency of advertisement on electronic media on sales? r = 0.6860, P.E. (r) = 0.065, sales is significant

b. Should be increase or decrease frequency of advertisement an electronic media to promote his sales. [ should increase]

c. If he found relationship is significant, what will be his sales per day when the frequency of advertisement per day is7? [Rs 10, 904]

44. From the following Bi-variate table, compute correlation coefficient between advertisement expenditure and sales revenue and test the significance of the result. ALSO ESTIMATE SALES REVENUE WHEN ADVERTISEMENT EXPENDITURE IS Rs 4,00,000. [2071]

Advertisement Sales revenue (in Rs ‘000)

0-50 50-100 100-150 150-200 200-250

0-40 12 6 8 - -

40-80 2 18 4 5 1

80-120 - 8 10 2 4

120-160 - 1 10 2 1

160-200 - - 1 2 3

Ans r = 0.572, significant Y = Rs 270456.5424(in’000)

45. From the following bi-variate table. Compute two regression coefficients, coefficient of variations and correlation coefficient. Estimate the sles revenue when advertisement cost is Rs 90 thousand. [20][2071]

Sales revenue in ‘000 Rs

Advertisement cost in ‘000 Rs.

5-15 15-25 25-35 35-45 45-55

75-125 4 1 - - -

125-175 7 6 2 1 3

175-225 1 3 4 2 2

225-275 1 1 3 4 -

275-325 3 1 1 - -

Ans X: adv. Cost Y = sales revenue byx = 0.713, bxy = 0.038, CVx = 54.06% CV Y = 30.24% r = 0.165 , Rs 236.6302 thousand

Chapter 7 Analysis of Time Series 1. The quarterly average sales of a certain product of four quarters for the year

1982 to 1986 are 75.25, 117, 96.8and 177.5 respectively. What are the seasonal indices of different quarters? [2][2076][64.52 ;100.31; 82.99;152.18]

2. The following figures are giving relating to the output in a factory. Find the trend values with the help of 3 yearly moving average method. [2][2075]

Year 2001 2002 2003 2004 2005 2006 2007

Output 5 8 10 7 12 11 14

ANS 7.67;8.33;9.67;10.33;12.33

3. THE PERSONNEL DIRECTOR FOR Nepal Drug Limited recorded the average percentage absentee rates for each quarter for a 4 years period are 55, 67.5, 62.5, and 53, find the seasonal index. [2][2074][59.50, 91.44, 113.45 and 89.09

4. The sales manager of Media home appliances has collected the data regarding unir sales during the last 5 years 2009 to 2013 and calculated the values as a =52.4 and b = 9.2 taking deviation 2011. Estimate the sales for the year 2015. Ans 89.2 units [2][2073]

5. Consider the following straight line trend equation obtained from the data of annual profit (in Rs 000) Y = 90+2X Interpret the meaning of coefficient of X of this model. Whay is monthly increase in profit? [2][2072][annual profit is increased by Rs 2,000 Rs 166.67]

6. The year of origin of the following straight line trend equation of profits in 000 Rs is 2008. The equation is y = 6.5+3x estimate profit for the year 2014.[2][2072][Rs 24,500]

7. In the following straight line trend equation of production (‘000’ tons) the year of origin is considered as 2004. Y = 20 +3.5 X Estimate the production for the year 2009. [2][2071][37,500 tons]

8. From the following annual data of sales (in 000 Rs) Find the trend values by using least square method. ALSO ESTIMATE THE SALES OF 2014.[10][2076][Ans Rs 103,000

Year 2004 2005 2006 2007 2008 2009 2010

Sales(000) 77 88 94 85 91 98 90

9. Find the seasonal indices using simple average method and comment on the result. [10][2074 old][110.34, 128.74, 87.36, 7356]

Year Spring Summer Fall Winter

2012 8 10 7 5

2013 9 10 7 6

2014 10 11 7 6

2015 10 12 8 7

2016 11 13 9 8

10. Bhat Bahteni Departmental store has been expanding market share during the past 7 years, posting the following gross sales in million rupees. [10][2073]

Year 2006 2007 2008 2009 2010 2011 2012

Sales 148 207 246 329 378 476 517

a. Find the linear estimating equation that best describes the data. [ 328.371+ 63.46 X]

b. Estimate the sales for the year 2014.[Rs 646.01 million c. Do these figures show a rising trend or falling trend? How do you arrive at

your conclusions? B=62.46>0 rising trend 11. Kathmandu Metropolitan Police is studying the number of traffic facilities

resulting from drunk driving during 2005-2011[10][2073]

Year 2005 2006 2007 2008 2009 2010 2011

No. of deaths

75 90 94 91 85 90 98

Compute trend values using least square method and estimate the no. of deaths in the year 2015. Ans

52.5713;84.7142;86.8571;89.0;91.1429;93.2858;95.4287; Y2016≈ 104 12. Compute the seasonal index for the following data, assuming that there is no

need to adjust the data for the trend.[10][2072] Ans 92.80;105.60;93.44;108.16

Quarter 1988 1989 1990 1991

I 3.5 3.5 3.5 4.0

II 3.9 4.1 3.9 4.6

III 3.4 3.7 3.7 3.8

IV 3.6 4.8 4.0 4.5

13. Compute the seasonal indices by using simple average method from the following data and state which quarter is seasonally high, justify your solution. [10][2072][93.24, 104.76, 92.73, 109.24, quarter IV]

Quarter 2008 2009 2010 2011

I - 3.5 3.5 4

II 3.9 4.1 3.9 4.6

III 3.4 3.7 3.7 3.8

IV 3.6 4.8 4.5 -

14. Calculate the four yearly centered moving averages and short time fluctuations for the following data: [10][2072]

Year 2001 2002 2003 2004 2005 2006 2007 2008

Profit (000)

506 620 673 588 696 1116 738 663

Also plot the trend values on graph. Ans --;620.50;706.25;776.375;793.875;-;- 15. From the data given below, fit a straight line trend by the method of least

square ALSO calculate the ternd values.[10][2071]

Year 2006 2007 2008 2009 2010

Profit(in Rs 000) 12 18 20 23 27

Also estimate the profit for the year 2012. Ans Y= 20+3.5 X ;Trend values are 13,16.5, 20,23.5, 27,Profit = Rs 34,000

16. Fit a straight line trend by method of least square and obtain trend values from following data: [10][2071]

Year 2005 2006 2007 2008 2009

Production in 000 tons

100 120 110 140 80

Also predict the sales for the year 2015. Ans Yc = 110-2X; 119, 112, 110, 108, 106, Y2015= 94

17. Assuming that trend is absent, determine the seasonality, if any in the data given below and state which quarter of the year is seasonally high. [10][2070]

Year 1st quarter 2nd quarter 3rd quarter 4th quarter

2006 172 168 180 170

2007 176 170 190 174

2008 174 166 300 180

2009 176 174 200 178

2010 178 174 250 182

Ans 93.89;91.32;120.04;94.75; 3rd quarter 18. The following are the annual profit in thousands in a certain business.

[15][2075]

Year 2001 2002 2003 2004 2005 2006 2007

Profit(000Rs) 30 72 75 65 80 85 97

a. Fit a straight line trend by the method of least square and tabulate the trend values and short term fluctuation.

Ans YC = 76.29+5.07X; Trenf values 61.08, 66.15, 71.22, 76.30, 81.38, 86.46, 91.54, Short term fluctuations; Multiplicative model= 97.69, 108.69, 105.43, 85.53, 98.94, 99.16, 104.88, Additive model = -1.42, 5.72, 3.86, -11, -1.86, -0.72, 4.42

b. What is the monthly increase in profit?[Rs 442.50] c. Estimate profit for 2014.[126,990]

19. The tourist industry is subject to enormous seasonal variation. The Fulbari Resort in Pokhara has recorded its occupancy rate percentage of total rooms for each quarter during the last 4 years. These data are shown in the following table.[15][2074]

Year Occupancy Rate(%)

Q1 Q2 Q3 Q4

2008 56 70 80 60

2009 57 73 86 62

2010 60 74 90 65

2011 66 83 87 87

Analyze the quarterly time series to determine the effects of seasonal components by ratio to moving average method using multiplicative model. Ans 84.79, 105.72, 121.71, 87.78

20. The following table represents the annual trend of net profit of two different companies seeking investment for their development project. In which company would you invest money. Justify your solution by using necessary statistical tool.[15][2072]

Year Net profit in millions Rs

Company A Company B

2001 16 16

2002 32 16

2003 40 22

2004 24 36

2005 40 40

2006 32 44

2007 88 48

Company A (based on the annual rate of growth of net profit) Company B (based on the consistency of the net profit)

Chapter 8 Index Numbers 1. Reconstruct the following index number by shifting base year as

2063.[2][2075]

Year 2059 2060 2061 2062 2063 2064 2065

Index No. 100 112 123 131 144 152 160

Ans 69.44;77.78;85.42;90.97;100;105.56;111.11

2. If during a period, the price index number goes up from 100 to 200 and the wages of a worker are raised from Rs 300 to Rs 500, does the worker really gain? [2][2074][No]

3. Reconstruct the following indices using 2010 as the base year: [2][2073]

Year 2007 2008 2009 2010 2011 2012 2013

Index 100 110 130 150 175 180 200

Ans 66.67; 73.33; 86.67; 100; 116.67; 120; 133.33

4. Given that ∑PW = 12610, ∑W =100, where P and W have their usual meaning. Calculate the cost of living index. A man of middle class family of a remote area of Nepal was earning Rs 50,000 in the base period. What should be his salary in the current period, if his living standard is to remain the same? [ Rs 63,050, 126.10][2][2072]

5. From the information given below, calculate the real wage index number. [2][2072]

Year 2010 2011 2012 2013

Wage 650 980 1200 1800

Index No. 100 120 130 150

Ans 100,125.64,142.01,184.62

6. From the information of prices and quantities of four commodities in the base year 1983 and current year 1984, the following results are obtained:

∑p0q1 = 184, ∑p1q0 = 1221, ∑p1q1 = 192, ∑p0q0 = 108, calculate the price index by Fisher’s formula. [2][2071][Ans 108.12]

Show that Fisher’s ideal index number satisfies both reversal test and factor reversal test from the following information’s.[10][2076]

Commodities 2010 2012

Price Expenditure Price Expenditure

W 4 32 5 50

X 5 50 6 72

Y 3 18 4 28

Z 8 40 10 40

Ans P01(F) x P10 (F) = 1; P01(F) X Q01(F) = 190

140 and V01 =

∑ 𝑝1𝑞1

∑ 𝑝𝑜𝑞𝑜=

190

140; P01(F)x

Q01(F) = V01. Hence, Fisher’s ideal index number also satisfies factor reversal rest.

7. Compute Fisher Index number. Show that fisher Index number satisfied time reversal test from the following table.[10][2075]

Commodities 2012 2013

Quantity Value Quantity Value

A 75 150 80 240

B 125 625 140 560

C 50 350 30 180

Ans = P01 (F) = 91.35

8. Define index number and explain the various types of index number. The Sajha Bus Provides transportation for Kathmandu and in addition it sells bus to different cities of Nepal. The company has collected in the following data in order to analyze its sales for year 2011 and 2012.[10][2074]

Cities Average selling price per bus in lakhs

Number of Buses sold

2011 2012 2011 2012

Kathmandu 24 25 17 20

Pokhara 20 20 14 18

Biratnagar 25 30 21 25

Compute the suitable price index number for the given data. Ans 109.98

9. From the data given below. Calculate the price,index number for 2017 taking 2016 as base year using (a) Laspeyre’s method (b) Passche’s method (c)Fisher’s ideal method. [10][2074 old][251.64; 252.17; 251.90]

Commodities 2016 2017

pRICE qUANTITY Price Quantity

A 10 100 20 120

B 11 150 25 180

C 12 180 30 210

D 14 200 40 250

10. Define index number and discuss the various problems that are faced in the construction

of an index number. The data given below related to the workers in a housing. [10][2073]

Groups Cost of living index for 2013 upon 2010

% of expenditure

Food 260 50

Fuel 140 5

Housing 150 15

Clothing 190 18

Other items 110

12

If the average monthly pay of the workers in 2010 was Rs 20,000. What should be thw average monthly pay in 2013 in order that the workers may be able to maintain the same standard of living as that of 2010.[10][2073][Rs 41,380] 11. The following information is related to budget of middle class families of Kathmandu.

Items Percentage of expenses

Price in 2010 Price in 2011

Food 35 1500 1650

Fuel 10 1650 1900

Clothing 20 1180 1200

Rent 15 1200 1500

Misc. 20 370 500

What is the changes in cost of living index in 2011 as compared to the prices of 2010?[10][2073][16.131%] 12. calculate the index number by using suitable formula for 1983 on the basis of 1982 from the following information:[10][2072]

Article-I Article-II Article-III

Year Price Exp Price Exp Price Exp

1982 5 50 8 48 6 18

1983 4 48 7 49 5 20

Ans 83.40 13. Prepare real wage index from the following data:[10][2072 old][Ans base =2000, 100;75;16;62.96,62.50;61.45;60.46;59.52]

Year 2000 2001 2002 2003 2004 2005 2006

Wages in Rs

180 230 340 360 365 370 375

Price indices

100 170 300 320 330 340 350

14. compute the suitable index number from the following data:[10][2072 old][Ans 249.74]

Year Product P Product Q Product R

Price Qty Price Qty Price Qty

2009 4 54 2 5 3 10

2010 10 40 4 5 8 8

15. Compute suitable index number for the year 2010 with 2008 as base year from the following data:[10][2071][Ans 250, Rs 25,000]

Year Product P Product Q Product R

Price Qty Price Qty Price Qty

2008 4 50 3 10 2 5

2010 10 40 8 8 4 4

If the income of a family in 2008 was 10,000 what should be the income of same family in 2010 so that the living standard would be same? 16. Construct real wage index numbers from the following: [5][2070][Rs 100;76.53;80.36;75.19;74 .83;81.17]

Year Wage in Rs Price index

2005 1400 100

2006 1500 140

2007 1800 160

2008 2000 190

2009 2200 210

2010 2500 220

17. Find the cost of living index from the following:[5][2070][145.75%]

Items Percentage of expenditure Price relatives

A 25 125

B 35 140

C - 160

D 110 130

E 15 190

Chapter 9 Probability

1. StaTe the addition theorem of probability in case of two events A and B.

[2][2076][P(A and B) = P(A)+ P (B)

2. When two dice are rolled, what is the probability of getting sum of two faces is 10?

[2][2075][1/12]

3. Given P(A) = 3/14, P(B)=1/6, P(C) =1/3 P(AC)=1/7, find the probabilities P(A/C) and

P(C/A) [2][2074][3/7 and 2/3]

4. Two events A and B are statistically dependent. If P(A) =0.35, P(B) = 0.20, P(A or B)

= 0.45 Find the probability that a. neither A nor B will occur. B. B will occur, given hat

A has occurred. [2][2073][0.55; 0.29]

5. The probability that a manufacturer will produce brand X product is 0.13, the

probability that he will produce brand Y product is 0.28 and the probability that he will

produce both brands is 0.06. What is the probability that the manufacturer who has

produced brand Y will also have produced brand X? [2][2072][0.2143]

6. A Box contains 4 black and 4 red balls. If 2 balls are drawn at random find the

probability that they are black.[2][2072][0.2143]

7. Out of 9 candidates 6 men and 3 women apply for two vacancies of a manufacturing

company, what is the probability that one man and one woman are selected in that

vacancies? [2][2071][0.5]

8. A bag contains 30 balls numbered from 1 to 30. One ball is drawn at random, what is

the probability that the ball drawn will be multiple of 3 or 7? [5][2075][13/30]

9. An urn contains 9 red , 7 white and 4 black balls. A ball is drawn at random. Find the

probability that the ball drawn is a. red ball and b. not red ball. [5][2071 old][0.45;

0.55]

10. A bag contains 10 white balls and 8 red balls. Two balls are drawn one after the other

with replacement. Find the probability that both balls drawn are white.

[5][2069][0.3084]

11. In a lot of 12 pens 3 pens are defective. If two pens are drawn randomly what is the

probability of getting a defective pen and a non-defective pen. [5][2065][9/22]

12. If the probability of happening two mutually exclusive events A and B are respectively

2/5 and 1/5. Find the probability of happening one of these events. [5][2064][3/5]

Chapter 10 Sampling and Estimation

1. What do you mean by sampling? Discuss the various types of sampling

techniques. [2][2073][10][2063 old][2060]

2. Differentiate between point and interval estimate. [2][2072]

3. Differentiate between census and sampling method of data collection. Why

sampling method is suitable to collect data from large population?

[10][2072][2060]

4. Discuss various sampling techniques with their merits and demerits. [10][2063

old]

5. Suggest suitable sampling method, if you are interested in studying the popularity

of the entertainment centre in Bhrikuti Mandap.[10][2061]

6. Write short note on systematic sampling vs. stratified sampling. [5][2055]

7. State and explain with examples the need and importance of sampling

techniques. [10][2054]

Chapter 11. Quantitative Analysis

1. Prepare regret (or loss) table from the following pay-off table. [2][2076][2075]

Demands(Units) Decision Alternatives

15 16 17 18

15 150 120 90 60

16 150 160 130 100

17 150 160 170 140

18 150 160 170 180

Ans Regret Table

Demanded Units Decision Alternatives

15 16 17 18

15 0 30 60 90

16 10 0 30 60

17 20 10 0 30

18 30 20 10 0

2. Following are the payoff table of rooms available in the hotel with different types

of rooms. [2][2074][R3 ; R1]

Strategies (Room) Demand

Low Medium High

R1 25 35 50

R2 10 40 70

R3 30 20 100

Find the best strategy if you were a. A complete optimistic

b. A complete pessimistic

3. From the following pay-off table, prepare regret table: [2][2071]

N1 N2 N3

S1 200 50 40

S2 100 60 30

S3 40 30 10

N1 N2 N3

S1 200-200=0 60-50=10 40-40=0

S2 200-100=100 60-60=0 40-30=10

S3 200-40=160 60-30=30 40-10=30

4. Given the following information: Prepare pay off table and regret table.

Units demanded: 12, 13, 14, 15, 16, 17

Cost price per unit = Rs 140.

Selling price per unit = Rs 200.

You are required to give decision on the basis of a. Maximax criteria b. Minimax regret criteria c. Maximin criteria [10][2073 old][Stock 17, 13, 12]

5. A coca-cola distributor buys the bottles for Rs 6 and sells them for Rs 10 each.

All the bottles leftover are worthless. His daily sales of cold drinks is never less

than 15 and not more than 20. Prepare pay-off table and loss table. What will be

the distributor’s decision if the criterion adopted be a. maxi-max and mini-max

regret? [10][2072 old]

6. From the following payoff matrix, give decision based on a. maximax approach b.

maximin approach.

Action Event A B C

1 80 50 40

2 100 50 30

3 40 50 110

Ans ACTION3, Action 1 or 3. 7. A merchant of Bishal Bazar Co. buys an article at the rate of Rs 300 and sells

them at the rate of Rs 50. The merchant knows that he cannot sell more than 50

articles in a day and minimum sales would not be less than 45. Assuming that

unsold articles do not have any salvage value, prepare pay off table and regret

table and give decision according to a. optimistic and pessimistic approaches, b.

Minimax regret criteria [10][2070]Ans Max Payoff Rs 10,000 at stock 50 and Min

Payoff Rs 9,000 at stock 45 b. Minmax of Rs 600 at stock 47

8. A beer distributor buys kegs for Rs. 30 each and sells them for Rs 50 each. All

the kegs left at the end of the day are worthless where salvage value is zero. The

following table gives the distribution of sales and corresponding

probabilities.[15][2076]

Kegs sold 10 11 12 13 14

Probability 0.10 0.15 0.20 0.25 0.30

a. Construct a pay-off table

b. Calculate the expected monetary value for each strategy

c. Find the optimal quantity that can maximize the expected profit.

d. Calculate expected profit with perfect information

e. Calculate the expected value of perfect information.

9. What do you mean by decision with uncertainty? What are the methods of

decision making under the situation of uncertainty? A grocery shopkeeper orders

and receives 1 liter Snow Fun Curd early in the morning each day. The cost of

curd per liter is Rs 100 and he sells it for Rs 130. Thus making profit of Rs 30 per

liter. Based on his previous year’s sales statistics, the grocery shopkeeper

estimates his daily sale of curd as follows: [15][2073]

Demand 15 16 17 18 19

Probabilities of demand

0.30 0.25 0.20 0.15 0.10

Any unsold curd have to be thrown away at the end of the day. How many curds of 1 liter should the shopkeeper stock each day as to maximize his profit in the long run? Also compute expected value for perfect information.[Max Profit=Rs 450, stock =15, EVPI = Rs 45

10. Under an employment promotion programme, it is proposed to allow sale of

newspapers on the buses during peak hours. A newspaper boy ha the following

probability of selling magazine.

No. of copies sold

10 11 12 13 14

Probability 0.10 0.15 0.20 0.25 0.30

Cost per copy of magazine is 30 and sale price per copy is Rs 50.

He cannot return unsold copies where salvage value is zero.

a. Construct a pay-off table.

b. Calculate the expected monetary value for each strategy.

c. How many copies should be ordered. [12]

d. Compute expected profit with perfect information. [250]

e. Also calculate expected value of perfect information and interpret. [27.50]

11. An ice-cream retailer buys ice-cream at a cost of Rs 5 per cup and sells it for R 8

per cup. Any remaining unsold at the end of the day are useless and thrown

away. Past sale have been ranged between 15 and 18 per day. Assuming the

sale history has the following probabilities. [15][2072]

Quantity sold 15 16 17 18

Probabilities 0.10 0.20 0.40 0.30

What quantity should be bought to maximize expected profit? [17]

What will be his maximum expected profit? If he has perfect information of the market, what would be the expected profit? [47.80; 50.70

Also compute EVPI [2.90]

Chapter 12 Linear Programming Problem

1. Food X contains 6 units of vitamin A per gm, 7 units of vitamin B per gm and

cost Rs 5 per gm. Food Y contains 8 units of vitamin A per gm and 12 units of

vitamin B per gm and cost ra 18 per gm. The daily minimum requirement of

vitamin A and B are 100 units and 120 units respectively. Formulate the given

problem in a LPP model with the objective of minimizing the cost.

[5][2075][Minimize Z =Rs (5x+ 18 y) subject to constraints 6x+8y≥100; 7x

+12≥ 120; x ≥0, y≥0]

2. Solve the following Linear Programming Problem by Graphic Method.

Objective Function : Zmax = 600 x + 300y Subject to constraints

X+2y ≤8 x+y ≥2 5x+3y ≤ 15 x≥0, y ≥ 0 [10][2074][ Zmax = 1800 at x=3 and

y = 0]

3. Solve the following linear programming problem using graphical method.

Maximize Z = 30x+50y subject to constraints x+y ≤ 30; x+2y ≤40; x,y ≥ 0.

[10][2074 old][ Zmax =1100 at x=20 and y = 10]

4. Solve the following Linear Programming Problem graphically:

Objective function : Min Z = 3x + 2y subject to constraints 5x + y ≥ 10; 2x+4y

≥12; x+y ≤6; x≥0, y≥0. [10][2073][Min Value = 9.11 at x = 14/9 and y = 20/9]

5. One unit of food X contains 2 grams of vitamin A and 2 grams of vitamin B

and one unit of food Y contains 1 unit of vitamin A and 3 units of vitamin B. A

man wants to produce as many foods as possible when 20 grams of vitamin

A and 24 grams of vitamin B are available. It the profit of selling food X and

food Y are Rs 50 and Rs 30 per unit respectively. Formulate the problem in a

mathematical form and solve it graphically in order to maximize the profit.

[10][2073 old][Max profit = Rs 510 at food at food X =9 units and Food Y = 2

units]

6. Using graphical method, solve the following linear programming problems.

[10][2072]

Maximize Z = 2x+3y Subject to 3x +5y ≤45, 6x+4y ≤48; x,y ≥0 [Zmax = 83/3 at

x = 10/3 and y =7]

7. A dealer wishes to purchase a number of fans and sewing machines. He has

only 5,760 to invest and has spaces for at most 20 items. A fan costs him Rs

360 and a sewing machine Rs 240. His expectation is that he can sale a fan

at a profit of Rs 22 and sewing machine at a profit of Rs 18. How should he

invest his money in order to maximize his profit by using graphic method?

[10][2072 old][Rs 392, 8 fans and 12 sewing machines]

8. Find graphically the minimum value of Z= 40 x + 20 y subject to Constraints:

x+2y ≥ 2; 3x+y≥ 3; 4x+3y≥ 6; x, y≥ 0. [10][2071][ Zmax = 48 at x = 3/5 and y =

6/5

9. Solve the following Linear Programming problem by using graphic method.

Maximize the profit = Rs 300 A + Rs 500 B Subject to the limitations 9A + 6B

≤ 54; 3A ≤ 18; B≥ 6 Where A , B ≥ 0 non-negative conditions. [10][2070][Max

prfit = Rs 4500 at A =0, and B=9]

Chapter 13 Determinant

1. Prove the following determinant by using properties: [5][2076]

|

(𝑥 + 𝑦 + 𝑧) 𝑥 𝑦𝑧 (𝑦 + 𝑧 + 2𝑥) 𝑦𝑧 𝑥 (𝑧 + 𝑥 + 2𝑦)

| = 2(x+y+z)3

2. Hari sells 7 share of P and buys 9 shares of Q thus increasing his cash by Rs

70. Gopal sells 9 share of P and buys 14 shares of Q, decreasing his cash by

Rs 80. Find the price of P and Q by using Cramer’s rule. [5][2075][Ans Rs 100

and Rs 70]

3. Prove the following by using the properties of determinant.[10][2074]

|

𝑥2 + 1 𝑥𝑦 𝑥𝑧

𝑥𝑦 𝑦2 + 1 𝑦𝑧

𝑥𝑧 𝑦𝑧 𝑧2 + 1

|= 1+x2+ y2+z2

4. Prove the following determinant: [10][2074 old]

|𝑎 𝑏 𝑐

𝑎2 𝑏2 𝑐2

𝑎3 𝑏3 𝑐3|= abc(a-b) (b-c) (c-a)

5. Solve the following by using the properties of determinant [10][2073][x=0 or

21]

|7 + 𝑥 7 − 𝑥 7 − 𝑥7 − 𝑥 7 + 𝑥 7 − 𝑥7 − 𝑥 7 − 𝑥 7 + 𝑥

| = 0

6. Prove the following using properties of determinants[10][2073 old]

|𝑥 𝑥 + 𝑦 𝑥 + 2𝑦

𝑥 + 2𝑦 𝑥 𝑥 + 𝑦𝑥 + 𝑦 𝑥 + 2𝑦 𝑥

| = 9y2(x+y)

7. By using properties of determinant, prove that [10][2072]

|𝑎 + 𝑏 + 𝑐 −𝑐 −𝑏

−𝑐 𝑎 + 𝑏 + 𝑐 −𝑎−𝑏 −𝑎 𝑎 + 𝑏 + 𝑐

| = 2. (a+b). (b+c). (c+a)

8. Show that [10][2072]

|𝑎 𝑎2 𝑎3

𝑏 𝑏2 𝑏3

𝑐 𝑐2 𝑐3

|= abc (b-c) (c-a)(a-b)

9. The price of 3 commodities X,Y,Z are rupees x, y, z per unit respectively. A

purchases 4 units of Z and sells 3 units of X and 5 units of Y. B purchases 3

units of Y and sells 2 units of X and 1 unit of Z. C purchases 1 unit of X and

sells 4 units of Y and 6 units of Z. In this process A, B and C earn Rs

6,000;Rs 5,000; RS 13,000 RESPECTIVELY. Find the price per unit of 3

commodities by using Cramer’s rule. [10][2072 old][x=3,000; y = 1,000 z =

2000]

10. Prove by using the properties of determinant that

|

(𝑥 + 𝑦 + 2𝑧) 𝑥 𝑦𝑧 (𝑦 + 𝑧 + 2𝑥) 𝑦𝑧 𝑥 (𝑧 + 𝑥 + 2𝑦)

| = 2(x+y+z)3

11. Find the numerical value of the following determinant by using

properties.[10][2071][(x+y+z)(x-z)2

|𝑦 + 𝑧 𝑥 𝑦𝑧 + 𝑥 𝑧 𝑥𝑥 + 𝑦 𝑦 𝑧

|

12. Prove the following by using properties of determinants.[10][2071 old][2070]

|1 1 1𝑎 𝑏 𝑐

𝑎3 𝑏3 𝑐3| = (a-b) (b-c) (c-a) (a+b+c)

Chapter 14. Matrix

1. Find the product (A.B) of the following matrices[2][2076]

A= [1 3 42 1 3

]and B = [2 4 61 3 52 1 3

] Ans[13 17 3311 14 23

]

2. Given A = [4 2

−1 8]and B =[

1 35 0

] Show that AB ≠BA. [2][2074]

3. If A = [3 25 7

]then find A -1 [2][2073] Ans [

7

11

−2

11−5

11

3

11

]

4. Find the adjoint matrix of the matrix given below: [2][2072]

A = [1 23 −5

] Ans [−5 −2−3 1

]

5. Find A -1 where A = [2 11 6

] Ans [

6

11

−1

11−1

11

2

11

]

6. Prove that the two matrices A and B are inverse of each other, where A =

[3 21 1

] and B = [1 −2

−1 3] [5][2076]

7. Evaluate A2 – 4A – 5I, where A = [1 32 4

] Ans [

6

11

−1

11−1

11

2

11

]

8. An electronic company has obtained orders for 10 laptops of type A, 9 laptops of type B and 6 laptops of type C . The following matrix shows the amount of raw materials needed for each type of laptops. [10][2074]

Laptops Raw Materials

R1 R2 R3

A 40 20 10

B 50 15 15

C 30 12 12

If the cost of each unit of R1, R2 and R3 are Rs 5000, Rs 6000 and Rs 4000 respectively, find the using matrix algebra,

a. The cost of materials for laptop of each type. [Rs360,000 Rs 400,000 and Rs 270,000]

b. The cost of materials for all laptops. [Rs 8,820,000]

9. Solve the following equations by using matrix methos: [10][2074 old][x=2, y =3 and z = 4]

X+2y+3z = 20: 3x+6y+2z = 32: x+y+z = 9

10. From the given matrix: A = [5 2 −12 4 11 2 3

]. Prove that AA-1 = A-1A=1.

[10][2073 old]

11. Solve the following equations by using elementary row operation method.[10][2073][x=1, y=2 and z =-1]

2x+5y+3z =9; 3x+y+2z=3; x+2y-z =6

12. A manufacturing company Ltd. Produces two types of drugs D1and D2 with the help of three chemicals C1, C2 and C3. The quantity of chemical requirements per kg of each D1 and D2 are given below in suitable units.

D1 D2

C1 10 12

C2 16 15

C3 8 16

The price of chemicals in three different markets M1, M2 and M3 are as follows:

C1 C2 C3

M1 20 19 16

M2 8 9 7

M3 6 7 8

Find the price per kg of each drug in each market by using matrix algebra. [10][2072][the price per kg of drug D1 in markets M1, M2, M3 are 632,280, 236 respectively. The price per kg of drug D2 in markets M1, M2, M3 are 781, 343, 305]

13. Solve by using matrix method [5][2072]

6x+8y = 30; 9x+5y =42; x=31/7 and y = 3/7

14. If A = [1 23 1

] then show that A2 – 2A – 5I = 0 where 0 is the 2*2 null matrix

and I be the identity matrix of order 2. [5][2072 old]

15. If A =[2 11 3

] show that AA-1= I.[5][2072 OLD]

16. A biscuit manufacturing company has two plants located at two different places. Each plant produces three different types of biscuits Tiger, Glucose and Salty. Matrix A shows the number of packets of each type of biscuit produced per day and matrix B shows their cost and selling price.[10][2072 old]

Tiger Glu. Salty C.P. S.P.

A = 𝑃𝑙𝑎𝑛𝑡 𝐼𝑃𝑙𝑎𝑛𝑡 𝐼𝐼

[300 200 100200 300 200

] B = [10 1311 1212 14

]

If plant I and Plant II operate for 20 and 25 days respectively in a month. Using matrix algebra, find the total.

i. Production of biscuits from each plant per day. [Plant I:600 and Plant II: 700]

ii. Production of each type of biscuits in the month, [11,000;11,500;7,000]

iii. Profit in the month. [Rs 58,500]

17. The three commodities C1. C2, C3 are purchased and Ssold by the three persons A, B and C. Mr. A purchases 4 units of C3 and sells 3 units of C1 and 5 units of C2. Mr B purchases 3 units of C1 and sells 2 units of C2 and 1 units of C3. Mr C purchases 1 units of C2 and sells 4 units of C1 and 6 units of C3. In the purchase – sell process Mr. A suffered a loss of Rs 1,000; Mr B earns no profit and Mr C earns Rs 40,000. Find the prices per unit of these commodities by using matrix method. [2071][10][Rs 3,000,

Rs 2,000 and Rs5,000]

18. Diary Corporation is planning to produce 1200 kgs of baby food mixing 3 nutrients A, B and C. The 3 nutrient components per kg cost Rs 12, Rs 16 and Rs 14 respectively. The final output of 1 kg of baby food pack must contain nutrient C to be twice that of nutrient A . The budget available to buy component nutrients are to the corporation is Rs 18,480. Find amount of each nutrient which should be mixed in the final output by using matrix method. [2071][10][A =90 kg, B = 930 kg and C = 180 kg]

19. Three different chemiclas are mixed in a specific proportion to produce 100 units of a product. The company has to meet a constraint in the production of the product that chemical X should be one-third of the difference between chemicals Y and Z used. The cost of chemicals X, Y and Z are respecvtively Rs 50 Rs 20 and Rs 40 and the total cost of all the chemicals used is Rs 2900. Determine each unit of chemical X, Y and Z used by matrix method. [10][2070][X=10, Y = 60 and Z = 30]

20. Objective Questions. [2075][2]

a. If the distribution of data is negatively skewed then mean is ___than mode.

b. The product of two regression coefficients can not exceed____.