We are providing Previous Statistical & Quantitative Methods SQM Question Papers of Pune University for new as well as old Pattern. We hope this will help you to get idea about University Question Papers Pattern and Question types. 

    MAY 2009 OLD

    Instructions to the candidates:

     1) Question NO.1 is compulsory.

     2) Attempt any two from the remaining.

     3) Use of electronic calculators and statistical tables are allowed.

     4) Figures to the right indicate full marks.

    Ql) a) The following data relate to ages of a group of workers in a factory.

              Draw the ogive curves. Compute the Median. Check the computed Median

              with the Median value obtained from the graph. Also find from the graph,

              the number of workers between ages 28 – 48. [10]

    Ages

    20 -25

    25 – 30

    30-35

    35 -40

    40-45

    45 – 50

    50 – 55

    55 -60

    No. of

    35

    45

    70

    105

    90

    74

    51

    30

    Workers

    b)  Consider a small plant which makes two types of automobile parts, say A and B. It buys castings that are machined, bored and polished. The capacity of machining is 25 per hour for A and 24 per hour for B, capacity for boring is 28 per hour for A and 35 per hour for B and the capacity of polishing is 35 per hour for A and 25 per hour for B. Castings for part A cost Rs. 2 and sell for Rs. 5 each and those for part B cost Rs. 3 and sell for Rs. 6 each. The three have running costs of Rs. 20, Rs. 14 and Rs. 17.50 per hour. Assuming that any combination of parts A and B can be sold. Formulate and solve this problem as an LPP to determine the optimum product mix and maximum profit. [10]

    Die A has four red faces and two white faces. Die B has two red faces and four white faces. A coin is flipped once. If it falls heads the game continues by throwing die A, if it falls tails die B is to be used.

              i)  Show that the probability of getting a red face at any throw is V2.

              ii) If the first two throws resulted in red faces, what is the probability of

              red face at the ;3fd throw?        [10] .

     

     

     

     

    Q2) a) Explain the concept of association of attributes and illustrate the coefficient

          of association. [5]

      b) A collar manufacturer is considering the production of a new style collar to

         attract   young men. The following statistics of neck circumference are available

          based on the measurement of a typical group of students.

    Mid value (in inches)

    12.5

    13

    13.5

    14

    14.5

    15

    15.5

    16

    No. of Students

    4

    19

    30 –

    63

    66

    29

    18

    1

    Compute the Mean and Standard Deviation for the size of collars. [5]

     

    c) From a city population, the probability of selecting (i) a male or a smoker is 7/10, (ii) a male smoker is 2/5 and (iii) a male, if a smoker is already selected is 2/3. Find the probability of selecting: [5]

    A non-smoker

    A male

    A smoker, if a male is first seleCted.

    Given rl2 = 0.6, r23 = 0.5, r21 = 0.7. Calculate rl23 and RU3"    [5]

    Q3) a) The marks secured by recruits in the selection test (X) and in the proficiency

              test (Y) are given below: Calculate Karl Pearson's coefficient of correlation. [8]

     

    X

    10

    15

    12

    17

    13

    16

    24

    14

    22

    20

    Y

    30

    42

    45

    46

    33

    34

    40

    35

    39

    38

     

    b) Hundred children took three examinations A, Band C; 40 passed the first 39, passed the second and 48 passed the third, 10 passed all three, 21 failed all three, 9 passed the fJrst two and failed the third, 19 failed the first two and passed the third. Find how many children passed at least t~o examinations. Show that for the question asked certain-of the given frequencies are not necessary. Which are they? [6]

    Assume the mean heights of soldiers to be 68.22 inches with a variance of 10.8 inch2• How many soldiers in a regiment of 1000 would you expect to be over 6 feet tail? (Given that the area under the standard normal curve between x = 0 and X = 0.35 is 0.1368 and between X = 0 and X = 1.15 is 0.3746). [6]

    Q4) a) Consider the game with the following pay off table: [6]

    MAY 2009 NEW

     

    Instructions to the candidates:

       1) Question No.1 is compulsory.

       2) Attempt any three questions from the remaining.

       3)  Use of electronic calculator and statistical table are allowed.

       4) Graph paper will be supplied on request.

     

    Q l) a) A firm uses lathes, milling machines and grinding machines to produce

             two machine parts. Table given below represents the machining times

            required for each part, machining times available on different machines

            and profits on each machine part.                  [8]

    ______________________________________________________________________________

             

    Type of         Machining time            Maximum time

    machine      required for                  available in

                       machine part in               minutes

    minutes

    ______________________________________________________________________________

     

                        I         II

                       ___________

     

    Lathes         12      6        3000

    Milling         4        10      2000

    Grinding      2        3        900

    ______________________________________________________________________________

    Profit per                        

    unit Rs.       40      100

    _____________________________________________________________________________

    Find the number of parts I and II to be manufactured per week to maximise the profit.

    b) The following data refer to the dividend (%) paid by the two companies

        A and B over the last seven years. [7]

    _______________________________________________________________________

               Company A           4        8        4        15      10      11      9       

              Company B 12      8        3        15      6        4        10

    ______________________________________________________________________

             

              Calculate coefficient of variation and comment.              

    c)       Explain the concept of multiple correlation.                      [5]

    d)       Explain the term pure strategy regarding a 2 x 2 game.    [5]

     

     

     

     

    Q2) a)          Obtain the equation of the regression line for husband's age (Y) on wife's

              age (X) at marriage from the following data:                     [8]

           _______________________________________________________________________

              Y        36      23      27      28      28      29      30      31      33      35

              X       29      18      20      22      27      21      29      27      29      28

            _______________________________________________________________________

     

    b) A company has 4 machines on which to do 3 jobs. Each job can be assigned

        to one and only one machine. The cost of each job on each machine is given

        in the following table. What are the job assignments for minimum cost? [7]

     

    Job

    Machine

    W       X       Y        Z

     

     

    A

    B

    C

     

     

     17     23      27      31

      7      12      16      18

      9      14      18      21

     

     

    Q3) a) Find the initial basic feasible solution of the following transportation

              problem using                 [8]

              i) North west corner method.   ii) Matrix minimum method.

             iii) Vogel's approximation method. Also find corresponding costs.

     

    Factory

    Warehouse

    A   B   C     D

    Capacity

    units

     

    X

    Y

    Z

     

    11 31   51    11

    71 31   41    61

    41  9   71    21

    7

    9

            18

    Requirement

    units

     

    5   8      7     15

                                                

    b) The following table gives the distribution of items of production and also the

        relatively defective items among them, according to size groups. Find the

         correlation coefficient between size and defect in quality. [7]

              Size   No. of items No. of defectives

          __________________________________________

              15                400              300

              16                540              324

              17                680              340

              18                720              36O

              19                800              360

              20                600              240

          _________________________________________

     

      Q4)   a) If 5% of electric bulbs manufactured by a company are defective, use

           poisson distribution to find the probability that in a sample of 100 bulbs

            none is defective. [4]

     

           [Given e-5 = 0.007, e-Q•5 = 0.6065]

     

       b) An investment firm purchases 3 stocks for one week trading purposes. It

           assesses the probability that the stocks will increase in value over the week

            is 0.8, 0.7 and 0.6 respectively. What is the chance that [4]

              i)  All three stocks will increase.

              ii) At least two stocks will increase?

    c) A trader deals in a perishable commodity. The daily demand is a random

        variable. Records of past 200 days show the following distribution.

            ___________________________________________

              Demand in units   10 20 30 40 50

              Number of days      20 40 60 60 20

            _____________________________________________

    A trader buys the commodity at Rs. 10 per unit and sells at Rs. 15 per unit. Calculate the profit in 10 days by simulating the system. [7]

     

    Use following random numbers.

         69, 01, 08, 74, 82, 20, 72, 14, 75, 12

    Q5) a) For the following 2 x 2 Game check whether & addle point exists or not?

              If not find the probabilities for each action of player A and each action for

               player B. Also find value of the Game. [8]

     

    Player A

    Player B

    B1      B2

     A1

     A2

    11      7

    9        10

                        

                        

                        

    b) In a certain interview there were 126 candidates, of which 70 were boys,

              36 candidates were successful, among them 20 were boys.        [7]

              i) Prepare a 2 x 2 contingency table.

             ii) Check whether the data is consistent or not?

            iii) Find coefficient of association.

            iv) Interpret the result.

     

    Q6) a)          Following table gives profits matrix for different events and actions.   

              Calculate E. Y.P.I.                                                                              [8]

                    Events                                                   Actions                          

              (States of Nature) Probability             AI       A2                  A3

                ____________________________________________________________________________________________________________

              E1                         0.20                      40      52                 45              

              E2                         0.35                      70      28                 40              

              E3                         0.35                      30      70                -50    

              E4                         0.10                      30      -50               -70

             ____________________________________________________________________      

    b)       Calculate mean, median and mode.                                           [7]

              _________________________________________________________________

              Marks (below)        5        10      15      20      25      30      35      40     

              No. of Students      2        7        14      27      48      64      72      80

             ____________________________________________________________________

    DECEMBER 2008

     

    Instructions to the candidates:

     

    1) Question No.1 and question No.4 are compulsory.

    2) Solve anyone question from question No.2 and question No.3 and anyone from

       Question No.5 and question No.6.

    3) Use of electronic calculator and statistical table are allowed.

    4) Figures to the right indicate full marks.

    5) Graph paper will be· supplied on request.

     

    SECTION – I

     

    Ql) Attempt any four:.

     

      a) Describe different measures of central tendency.                                          [5]

     

      b) The incidence of a certain disease is such that on the average 20% of  workers

         suffer from it. If 10 workers are selected at random, find the probability that

         more than 2 workers suffer from the disease.                                         [5]

     

      c) In a marketing organisation out of 200 sales representatives 150 were_ males,

         120 of them achieved their sales target during a financial year and 10 female

         representatives faile? to achieve their targets. Is there any association between the

         sex and sales performance?                                                                           [5]

     

     d) Draw the less than ogive for the data given below and answer the following from

         graph :

     

    Marks

    No.of students

    0-10

    10-20

     20-30

    30-40

    40-50

    50-60

    60-70

    70-80

    80-90

    90- 100

    5

    20

    40

    70

    85

    65

    50

    35

    20

    10

     

    i) If 25% students get the distinction, determine minimum marks for distinction.

    ii) If pass marks are 35, what percentage of students pass the examination?                                                                                                                                 [5]

     

    e) Probabilities of x, y, z becoming managers are 4, 2, and 1, respectively.

                                                                                       9 9         3

       

        Probability that bonus scheme will be introduced if x, y, z become

             

        manager are 2,  1, and "4 respectively. What is the probability that bonus

                           10   2         5

        Scheme will be introduced? What is the probability that manager was x if

        bonus    scheme is introduced?                                                               [5]

     

    Q. 2) a) The following data gives the experience of machine operator and their

                performance rating, calculate regression lines of performance rating

                on experience and estimate the performance if operator has

                7 years experience.                                                                         [8]

     

    Operator

     

    Experience

    Performance rating

    1

    2

    3

    4

    5

    6

    7

    8

    16

    12

    18

    4

    3

    10

    5

    12

    87

    88

    89

    68

    78

    80

    75

    83

     

    b) The scores of 2 batsmen A and B in 10 innings during a certain season   

     

    A: 32

    28

    47

    63

    71

    39

    10

    60

    96

    14

    B: 19

    31

    48

    53

    67

    90

    10

    62

    40

    80

     

       Find which of the two batsman, A or B, is more consistent in scoring.

     

    OR

     

    b) Explain partial and multiple correlation with illustrations and find multiple

        correlation RI23 if rl2 = 0.6, r23 = 0.7 and r l3 = 0.6.

     

    Q. 3) a) 44% of candidates get marks 55 or less and 6% of candidates get marks 80 or

             more assume that the distribution is normal, find average marks and standard

             deviation of marks.                                                                                    [8]

           

    Given

    Z

    0.15

    1.55

    Area from

    0.06

    0.44

    z = 0 to

     

    b)   Calculate rank correlation coefficient between marks assigned to 10

          students by judges x and y in a certain competitive test:                                 [7]

     

     

    S.No:

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    Marks by

    52

    53

    42

    60

    45

    41

    37

    38

    25

    27

    Judge X

     

     

     

     

     

     

     

     

     

     

    Marks by

    65

    68

    43

    38

    77

    48

    35

    30

    25

    50

    Judge y

     

     

     

     

     

    OR

     

    b) Calculate mode and Quartile Deviation for the following data:

     

    Marks

    10

    20

    30

    40

    50

    60

    70

    80

    below

     

     

     

     

    No.of.

    5

    13

    25

    40

    60

    74

    86

    92

    Students

     

    i-

     

     

     

     

     

     

     

    SECTION – II

           

    Q4) Attempt any three:                                                                [20]

    a)     A company is considering three magazines for releasing its advertisements.

         The relevant data are given here:

     

     

    Magazines

    A

    B

    C

    Readers

    1,00,00,000

    60,00,000

    40,000

    Prospectives buyers

    of the product

    10%

    15%

    70/0

    Cost per

    advertisement (Rs.)

    10,000

    5,000

    6,000

     

    The company has allocated Rs.l,OO,OOO for magazine advertising. It has also decided that there should be at least 2 insertions each in magazines A and C, and that the number or insertions in magazine B should not exceed 5. Formulate the LPP to maximize prospective buyers.

    b) In a bank every 15 minutes one customer arrives for cashing the cheque. The staff in the only payment counter takes 10 minutes for serving a customer on an average. It is assumed that the arrival time follows poisson distribution and service time follows exponential distribution. Customers are served on first come first basis. Find out,

     

    i) Average number of customers in a bank.

    ii) Average time required by a customer to cash the cheque.

    iii) Second counter is justified if waiting time of a customer in a queue is 15 minutes or more, find out what should be new arrival rate to justify second counter.

     

    c) A small industry finds from past data that the cost of making an item is Rs.25/-. The selling price of an item is Rs.30/-. If it is not sold within a week, it could be disposed off at Rs.20/-. At the end of a week. Data for sales is given below Find optimum number of items per week the industry should purchase,

    Find E.VP.I

     

    Weekly sales

    4

    5

    6

    7

    No. of weeks

    10

    20

    40

    30

     

    d) Describe markov process and steady state condition.

     

    Q. 5) a) A company has three plants and four warehouses. The supply and demand in

             units and the corresponding transportation costs are given. The table below

             has been taken from the solution procedure of a transportation problem. [9]

     

     

    unit transportation cost

     

    Plants

     

    Warehouses

     

    Supply

     

    I

    II

    III

    IV

     

    A

    5

    10

    4

    5

    10

    B

    6

    8

    7

    2

    25

    C

    4

    2

    5

    7.

    20

    Demand

    25

    10

    15

    5

    55

     

    The present distribution pattern is as follows:

     A to III – 10 units

     B to I – 20 units

     B to IV – 5 units

     C to I – 5 units

     C to II – 10 units

     C to III – 5 units.

     

    Answer the following questions:

    i) Is this solution feasible? Why?

    ii) Is this solution degenerate? Why?

    iii) Is this solution optimum? Why?

    iv) Does this problem have more than one optimum solution? If so, show all of them.

     

      b) Describe characteristics of queuing theory.

    OR  

       b) Describe different criteria of decision making.

     

    Q6) Attempt any three:

       a) Solve the following game Payoff Matrix

     

    B

    A

    I

    II

    III

    IV

    V

    VI

    I

    4

    3

    1

    3

    2

    2

    II

    4

    3

    7

    -5

    1

    2

    III

    4

    3

    4

    -1

    2

    2

    IV

    4

    3

    3

    -2

    2

    2

     

    [6] [5]

     

     

     

    b) A company has a team of four salesmen and there are four districts where the company wants to start its business. The following is the profit per day in rupees for each salesman in each district. Find the assignment of salesmen to various districts which will yield maximum profit.                                                               [5]

     

    District

    Salesman

    A

    B

    C

    D

    DJ

    16

    10

    14

    11

    D2

    14

    11

    15

    15

    DJ

    15

    ]5

    13

    12

    D4

    13

    12

    14

    15

     

     

    Max Z = 300 XI + 400 X2 [5]

    Subject to the constraints

    5X + 4X2 ≤ 200

             

    3X1 +5X2 ≤150

    5X1 + 4X2 ≤ 100

    8X1 + 4X2 ≥ 80

    X1 > 0, X2 ≥ o.

     

    d) A confectioner sells confectionery items. Past data of demand per week

            (in hundred kgs) with frequency is given below:                    [5]

     

    Demand

    Per week

    0

    5

    10

    15

    20

    25

    Frequency

    2

    11

    8

    21

    5

    3

     

    Using the following, sequence of random numbers, generate the demand for the next 1 () weeks. Also find the average demand per week.

    Random No's: 35, 52, 90, 13, 23, 73, 34, 57, 35, 83.

    MAY 2008

    Instructions to tile candidates:

    1) Question No.1 is compulsory.

    2) Attempt any TWO from tile remaining.

    3) Use of electronic calculators and statistical tables are allowed

           4) Figures to tile right indicate full marks.

     

    Q1)  a) Find the modal age of the workers using                                          [10]

            i)      graphical method and    

            ii)     by calculation.        

                   Age                   No. of workers      

                   More than   15 115 

                   " "             25   111 

                                    35   96   

                                    45   70   

                                    55   40   

                                    65   18   

                                    75   03   

    b) A firm makes two types of furnitures chairs and tables. The contribution to profit by each product as calculated by accounting department is Rs. 20 per chair and Rs. 30 per table. Both the products are to be processed on three machines M1, M2, M3' The time required in hours by each product and total time available in hours per week on each machine are as follows.

    Machine    Chair         Table             Available Time

                                                       (in hrs)

    M1            3             3                     36

    M2             5             2                     50

    M3            2             6                     60

    How should the manufacturer schedule the production in order to

    maximise the profit.                                                                      [10]

     

    Competitors

    Judges:

    A

    B

    C

    1

    6

    5

    4

    2

    5

    8

    9

    3

    3

    4

    8

    4

    10

    7

    1

    5

    2

    10

    2

    6

    4

    2

    3

    7

    9

    1

    10

    8

    7

    6

    5

    9

    8

    9

    7

    10

    1

    3

    6

    Discuss which pair of judges has the nearest approach to common tastes

    Of beauty.                                                                                                   [10]

     

    Q2) a) Find the association between proficiency in English and in Hindi among

    candidates at a certain test if 245 of them passed in Hindi, 285 failed in Hindi,

    190 failed in Hindi but passed in English and 147 passed in both.

    [5]

    b) The mean yield for one-acre plot is 662 Kg, with a s.d. of32 Kg. Assuming normal distribution how many one-acre plots in a batch of 1000 plots would you expect to have

    i) over 700 Kg.

    ii) below 650 Kg.

                   iii)    What is the lowest yield of the best 100 plots.                           [7]

    c) The products of 3 plants X, Y, Z are to be transported to 4 warehouses I, II, III, IV. The cost of transportation of each unit from the plant to the warehouses along with normal capacities of plants and warehouses are indicated below.

    Warehouse

    Plants

    I

    II

    ill

    IV

    Availability

    X

    25

    17

    25

    14

    300

    Y

    15

    10

    18

    24

    500

    Z

    16

    20

    08

    13

    600

    Requirement

    300

    300

    500

    500

     

     

    i) Solve the problem for minimum cost of transport.

    ii) Does there exist any alternative solution.                                        [8]

    Q3) a) The mean of binomial distribution is 4 and variance is 4/3. Find probability of getting

     

    i)                   no success ii) at least 5 success.                                                [5]

     

    b) Analysis of monthly wages paid to the workers of 2 firms A and B

          belonging to the same industry gives the following result.

    Firm A             Firm B

    No. of workers                   500                 600

     Average daily wage            186                175

    Variance of wages               81                100

    Find

    i) Which firm is more consistent in wages

          ii)     Variance of all  the worker taken together.                              [5]

    c) Find the value of the following game. Also determine the optimal strategies

    A            B1         B2  

    Bl           -5            2

    B2          -7            -4          [5]

    d) Customers arrive at a box office window, being manned by a single individual according to a poisson input process with a mean rate of 30 per hour. The time required to serve a customer has an exponential distribution with a mean of90 seconds. Find the average waiting time of a customer. Also determine the average number of customers in the system

          and average queue length.                                                               [5]

     

    Q4) a) The two regression lines are given as below 15X + 17Y = 395 and

            20X + 14Y = 440 Find

             i) Mean of X and Y

            ii) Both the regression coefficients

           iii) Correlation coefficient

           iv) 0y when 0x = 3.                                                                        [5]

    b) A company manufactures 200 motor cycles per day which changes according to availability of raw material.

          Production    196     197    198    199    200    201     202    203   204

    (Per day)

    Probability 0.05 0.09 0.12 0.14 0.20 0.15 0.11 0.08 0.06

    Using the following random numbers, stimulate the procedure for 12 days and find average production of sample drawn.

    R. Nos.  82,  89,  78,  24,  52,  61,  18,  45,  04,  23,  50,  77.          

                                  [5]

     

     

    .c) A project work consists of four major jobs for which an equal number of contractors have submitted tenders. The tender amount quoted (in lakhs of rupees) is given in the matrix.                                                 

                                                  Job

                                  A      B     C     D

    Contractor              1      10    24    30    15

                           2      16    22    28    12

                           3      12    20    32    10

                           4      09    26    34    16

    Find the assignment which minimizes the total cost of the project. [5]

    d) In a study of random sample of 120 students the following results were

    obtained.          

     Obtain the least square regression equation of X3 on Xl & X2 & hence estimate the percentage of marks of students in fmal examination if he gets 60% & 67% marks in test I & II respectively.        [5]

    MAY 2007

     

    Instructions:

              1)       Question No.1 &4 are Compulsory.

    2)       Solve any one question from Question no.2 and 3 and any one question from Question No. 5 And 6

              3)       Figures to the right indicate full marks.

              4)       Use of electronic calculators is allowed.

              5)       Graph Papers will be supplied on demand.

     

    SECTION-I

     

    Q1) a) The following table gives the distribution of out lay of the budget of a state under major head of development expenditure:

     

    Heads

    Expenditure

    (in Rs.Cr.)

    a)  Agriculture and community development

    b)  Irrigation and Power

    c)  Industry and mining

    d)  Transport and communication

    e)  Miscellaneous

    10,000

    4,000

    8000

    6000

    2000

     

     

    Represent the information by a suitable diagram or chart                             [5]

     

    b)                  You are given variance of Y=16.

    The regression equation are 4x-5y+33=0 and 20x-9y=107.

    Find i)     Average value of X and Y,

    ii)                 Correlation coefficient between x and Y, and

    iii)              Standard deviation of X                                                  [5].

     

    c)       There are 3 men aged 60, 65 and 70 years. The probability that to live 5 more years is 0.8 for a 60 years, 0.6 for 65 years old and 0.3 for a 70 years old person. Find the probability that at least two of the 3 person will live 5 years more.                                                                                                     [5]

     

    d)       The mean and standard deviation of 100 items are calculated as 60 and 7 respectively. Two of the items were found to be incorrect at the time of checking. 35 and 47 were wrongly copied as 53 and 74. Calculate correct mean and S.D.[5]

     

    Q2) a) Explain the meaning of partial and Multiple Correlation and regression giving illustration.                                                                                                [5]

     

    b)       Assuming that the probability of a fatal accident in a factory is 1/1200 in a year.         Calculate the probability that in a factory employing 300 workers, there will be at least two fatal accidents in a year. (Given e-0.25=0.7788)                         [5]

     

    c)                  Calculate Rank correlation coefficient between marks in statistics and Economics as given below:                                                               [5]

    Marks in Statistics

    45

    70

    65

    40

    80

    40

    50

    70

    85

    60

    Marks in  Economics

    35

    80

    70

    40

    90

    45

    60

    80

    80

    50

                                                                                         

     

    Q3) a) State what are the different measures of dispersion and state their merits and demerits.                                                                                                [5]

     

    b)                  Given the following:

    Total Population of a locality – 240000

    Total literate in the Locality -40000

    Total illiterate in the locality         -40000

    Total literate criminal in the Locality       -5000

     

    Determine the coefficient of the association between literacy and criminal behavior.

     

    c)                  Given                                                                                            [5]

     

    X

    Y

    Average

    S.D.

    20

    5

    25

    4

     

    Correlation coefficient between X and Y = 0.6. Find the two regression equation and estimate X for Y = 20.

     

    Q4) a) A refinery makes 3 grades of petrol A,B and C from crude oil D,E and F. Crude oil F can be used in any grade but the other must satisfy the following specifications.     [6]

    Grade

    Selling Price per liter (Rs.)

    Specifications

    A

     

    B

     

    C

    Rs.48

     

    Rs.50

     

    Rs.49

    Not less than 50% crude D.

    Not less than 25% crude E.

    Not less than 25% crude D.

    Not more than 50% crude E.

    No specifications.

     

    There are capacity limitation on the amount of 3crude elements can be used

    Crude

    Capacity (KL)

    Price per litre

    D

    E

    F

    500

    500

    360

    49.5

    47.5

    48.5

    Formulate LPP to maximize Profit.                                  

    b)      Find the optimal strategies for A and B in the following game. Also obtain the value of the game.

    A’s

    Strategies

    B’s Strategies

                 B1                                 B2                                   B3

    A1

     

    A2

     

    A3

                9                                    8                                      -7

     

               3                                    -6                                      4

     

              6                                      7                                      -7

     

     

    c)      Solve the following T.P. unit transportation cost (Rs.)                      [8]

    Factory

    Ware Houses

    Capacity

        D                E               F

    A

     

    B

     

    C

     

        5                1               7

     

        6                4               6

     

        3                2               5

    10

     

    80

     

    15

    Requirement

        75              20             50

     

     

     

    Q5)    a)       Solve by graphical method.

                       Minimize z = 6×1 + 14 x2

                       Subject to 5×1 + 4×2 >= 60,

                                      3×1 + 7 x2 >= 84,

                                     x1 + 2×2 >= 18,

                                          x1, x2 >=0.

     

    b)      Market survey is made on two brands of breakfast foods A and B. Everyday a customer purchases; he may buy the same brand or switch to another brand. The transition matrix is given below.                                         [8]

                                         

    From

    To

             A                               B

    A

     

    B

    0.8                                                                0.2

     

    0.6                             0.4

     

    At 60% of people buy brand A and 40% people buy brand B. determine Market share of Brand A and B in Steady State.

     

     

    Q6)    a)       Explain Queuing theory and its characteristics.                                  [8]

     

    b)       Solve the following assignment problem                                              [7]

                       Sales in thousands

    Salesmen

    Districts

            D1                        D2                      D3                          D4

    S1

     

    S2

     

    S3

     

    S4

            20                       25                        22                          18

     

           25                        24                        19                           21

     

            18                       20                         22                          20

     

           25                        20                         17                          22

     

     MAY 2006

     

    Instructions:          

    1) Question No. 1 and 4 are compulsory.

    2) Solve any one question from question No. 2 and 3 and any one question from

         question No.5 and 6.

    3) Figures to the right indicate full marks.

    4) Use of electronic calculator is allowed.

    5) Graph paper will be supplied on demand.

    SECTION-I

    Q-1) a) Draw Histogram and Ogive curve for the following distribution and read

                 mode and median for the distribution.

                                                    [5]

    c.i.

    10-20

    20-30

    30-40

    40-50

    50-60

    Frequency

    16

    19

    28

    20

    17

     

             b) Calculate coefficient of association between literacy and unemployment from

                  the following data:                                                                                    [5]

                  Total adults                                           -10,000

                  Total literates                                       –    1290

                  Total unemployed                               –    1390

                  Literate unemployed                           –      500

       

     

              c) The equations of the two lines of regression are

                  4X-5Y+ 33=0 and 20X-9Y-107=0

                

     

     

      Find  i) mean values of X and Y  

               ii) coefficients of correlation between X and Y.

              iii) coefficients of regression.

           

             d) One percent of articles produced by a company are defective. A sample of 100

                  item is selected by random process, find the probability that there are 3

                  defective items in the sample. [Given e-1=0.3679]                                      [5]

     

    Q-2) a) Given r12 =0.6, r13=0.7,r32=0.65

                 σ1 =1, σ2=8, σ3=9

                 Find     i) regression coefficients b12.3 and b13.2

                             ii) R2.13 and r32.1

          b) Determine which company’s share prices are more variable from data for last

                10 days share prices of company A and B given below:

                

    Co.A

    55

    54

    52

    56

    58

    52

    50

    51

    49

    53

    Co.B

    108

    107

    105

    106

    107

    104

    103

    104

    101

    105

     

       Q-3) a) In computing correlation following results were obtaions

                    ΣX=120, ΣY=90, ΣXY=356, ΣX2=600, ΣY2=250, N=30.

                    the time of checking it was found pairs ( X=8, Y=12 )

                    (X=10, Y=8) were wrongly copied as (X=8,Y=10) 

                    (X=10, Y=7), calculate correct coefficient of correlation.

               

                b) Ten competitors in a beauty contest were ranked by 3 judges as following:

                    

    Candidate

    A

    B

    C

    D

    E

    F

    G

    H

    I

    J

    Judge 1

    3

    5

    4

    10

    8

    8

    1

    6

    8

    2

    Judge 2

    5.5

    5.5

    1

    8.5

    4

    10

    2

    7

    8.5

    3

    Judge 3

    9

    9

    7

    5

    2

    2

    2

    5

    5

    9

                      Which pair of judges have nearest approach to beauty?

     

    SECTION-II

    Q-4) a) A manufacturing company makes 3 products, each of which require operations as part of manufacturing process. The company can sell a of the products it can manufacture but its production capacities are limited Additional related data are as below :

     

      

    Product

    Manufacturing    requirements

    Hours /unit

    Cost

    (Rs.)

    Selling Price

    (Rs.)

    Center 1

    Center 2

    Center 3

    A

    B

    C

    1

    3

    2

    3

    4

    2

    2

    1

    2

    11

    12

    10

    15

    20

    16

    Available

    160

    120

    80

     

     

    Formulate the L.P.P.

     

    b) Write a note on Game Theory.                                                                                 [5]

     

    c) Customers arrive at a service counter being manned by one individual at a rate of

        25 per hour. The server takes on an average 120 seconds per customer.

     Find i) average waiting time of customer

             ii) average number of customers in queue

     State your assumptions.

     

    d) The demand for bread is recorded for last 50 days with following results.

    Demand

    (No. of breads)

    0

    5

    10

    15

    20

    25

    Number of days

    2

    11

    8

    21

    5

    3

    Simulate demand for next 10 days, using Random Numbers?

    78, 99, 43, 62, 44, 02, 67, 32, 54, 75.

    Determine average demand.                                                                                [5]

     

    Q-5) a) Solve the following L.P.P. using graphical methods:                           [8]

                 Maximize Z=4X1+2X2

                 Subject to constraints

                 2X1 + 3X2 ≥30,

                 X1+X2 ≤ 14,

                 X1+2X2  ≤ 18,

                 X1+X2 ≥ 0,

     b) Solve the assignment problem:                                                              [7]

    Cost matrix       

    Person

    Job

    J1                            J2                               J3                              J4                               J5                  

    P1

    P2

    P3

    P4

    27                   18                       X                      20                      21      

    31                   24                       21                     12                      17 

    20                   17                       20                     X                       16   

    22                   28                       20                     16                      27

    Job J3 cannot be assigned to P1 and Job-J4 cannot be assigned to P3.

     

    Q-6) a) Solve the following transportation problem for maximum profit:

    Per Unit Profit (Rs.)

    safadfasdasfdds

    Warehouse

    A                              B                       C                     D

    Capacity

    X

    Y

    Z

    12

    8

    14

    18

    7

    3

    6

    10

    11

    25

    18

    20

    200

    500

    300

    Demand

    180

    320

    100

    400

    1000

     

     

             b) Market share of Brand A,B,C are 50%, 30% and 20% Customers their brands. Brand switching matrix every quarter is given below:

    From

    To

    A                             B                       C

    A

    B

    C

                   50%                        30%                   20%

                   20%                        70%                   10%

                   20%                          20%                 60%              

       Find market shares at the end of quarter.

    MAY 2006 OLD

     

    Instructions:          

    1) Solve any Two questions from Section-I and Two questions from

        Sections – II.

    2) Figures to the right indicate full marks.

    3) Use of electronic calculator is allowed.

    4) Graph paper will be supplied on demand.

    SECTION-I

    Q-1) a) Draw Histogram, Frequency Polygon and Ogive curve for the following

                 distribution and read mode and median from it .                           [5]

     

     

    Marks less than

    10

    20

    30

    40

    50

    60

    70

    80

    90

    Number of students

    4

    6

    24

    46

    67

    86

    96

    99

    100

     

     

            b) Calculate R2.13 and r 23.1,

                 Given r12 = 0.6, r13=0.7, r23 = 0.65                                                         [5]

     

            c) The defective production of a company is 1% A sample of 100 units produced

                by company is selected by random process. Find the probability that it

                contains 3 defective items [ Given e-1 = 0.3679]                                     [5]

     

    Q-2) a)  Calculate mean and quartiles for the following distribution .           [5]

    c.i.

    20-40

    40-60

    60-80

    80-100

    100-120

    Frequency

    6

    11

    18

    32

    27

     

     

           b) Find coefficient of correlation between X and Y from the following

                information:                                                                                                [5]

                n= 12, ΣX =523 ΣY =13208, ΣX2 =2347 and

                  ΣY2 = 8504.

     

           c) Find co-efficient of association between Literacy and Crime

               Total number tested                            2,40,000

               Total Literate                                         40,000

               Total Literate Criminals                        5,000

               Total Illiterate Criminals                     40,000

     

    Q-3) a) The incidence of occupational disease is 80%. If 6 workers are select at

                 random, find the probability that

                 i) no worker contracts the disease

                 ii) at least 2 workers contract the disease.

     

            b) The runs scored by two batsmen A and B are given below :

    A

    0

    93

    48

    58

    23

    36

    74

    67

    47

    84

    B

    81

    13

    47

    59

    25

    46

    34

    35

    22

    98

    Who is better run getter?

    Who is more consistent?

     

    Q-4) a) Find both lines of regression and estimate Y when X = 22 estimate X when

                 Y =20, from the following data:

    X

    35

    25

    29

    31

    27

    24

    33

    36

    Y

    23

    27

    26

    21

    24

    20

    29

    30

    The wages of workers in a factory are normally distributed, with average wages Rs. 4000 and S.D. of wages Rs. 400. Find the highest wage lower paid 10% workers. Also find lowest wage paid to highest paid 15% of workers.

     

    Given

    Z

    1.04

    1.28

    Area under SNV from Z=O

    0.40

    0.40

     

    SECTION-II

    Q-5) A manufacturing company makes 3 products, each of which require 3

            operations as part of manufacturing process. The company can sell all of the

            products it can manufacture but its production capacities are limited, related

           data are as below.                                                                               [5]

    Product

    Manufacturing requirements

    Hours/unit

    Cost (Rs.)

    Selling Price (Rs.)

    Center I        | Center II        |  Center III

    A

    B

    C

    1

    2

    3

    3

    4

    2

    2

    1

    2

    11

    12

    10

    15

    20

    16

    Available

    Hours

    160

    120

    80

     

     

    Formulate the L.P.P.

     

          b) Solve the following assignment problem                                            [5]

     

    Cost matrix

    Operator

    Job

    A                        B                                C                         D

    P

    Q

    R

    S

    120                   100                            80                            90             

    80                     90                              110                          70

    110                   110                            140                          100     

    90                       90                              80                           90

     

    c) Write a note on Markov Process.                                                              [5]

     

    Q-6) a) Determine the decisions based on                                                     [7]

                 i) maximax,       ii) Maximin,        iii) Laplace,      iv)regret criterion

                for the following pay off matrix :

    Pay off matrix

    Decision alternatives

    Demand

    High

    Moderate

    Low

    Nil

    Large expansion

    Small expansion

    No change

    50

    70

    30

    25

    30

    15

    -25

    -40

    -1

    -45

    -80

    -10

            

     

     

          b) Solve the L.L.P. using graphical method.

               Maximize Z = 4X1 + 2X2

               Subject to 2X1 + 3X2  ≥ 30,

                                   X1 + X2 ≤ 14,

                                   X1 + 2X2 ≤ 18,

                                   X1 + X2 ≥ 0.

     

    Q-7) a)  Solve the following Transportation problem:

    Profit Per unit (Rs.)

    Warehouse

    Market

    Capacity

    A                B                 C                 D  

    X

    Y

    Z

    12             18                  6                 25  

    8                7                   10               18

    14              3                   11               20

    200

    500

    300

    Demand

    180        320                 100            400

     

     

           b) Solve the following game:

    Player

    A

    Player B

              B1                                                          B2                                                   B3

    A1

    A2

    A3

              30                                         40                                    -80

               0                                          15                                    -20

             90                                          20                                      50

     

    Q-8) a) Customers arrive at a service counter being manned by one attendance at a

                 rate of 25 per hour. The service time per customer is on an average 120

                 seconds.                                                                                                                 [5]

                 Find i) average time customer spends in system

                        ii) average number of customers in the queue.

     

           b) Write short notes on:

               i) Decision theory

              ii) Simulation model.

    OCTOBER 2006

     

    Instructions:

    N.B.:

    1) Question No. 1 and 4 are compulsory.

    2) Solve any one question from question No. 2 and 3 and any one question from

         question No.5 and 6.

    3) Figures to the right indicate full marks.

    4) Use of electronic calculator is permitted.

    5) Graph paper will be supplied on demand.

    SECTION-I

    Q-1) a) Prepare frequency distribution table . Present the information graphically to

                 Determine median and mode of the distribution.                                          [5]

            26, 35, 61, 29, 36, 48, 57, 67, 69, 50, 48, 40, 47, 42, 41, 37, 62,

            51, 63, 33, 31, 32, 35, 40, 38, 37, 60, 51, 54, 56, 37, 46, 42, 38,

            61, 59, 58, 44, 39, 57, 18, 44, 45, 45, 47, 38, 44, 47, 47, 64 .

     

    b) To test whether there are any defects in the production inspection is

        conducted, cost of inspection (X) is Rs. ‘000, the number of defects (Y) found

        are in hundreds. Following and data are generated.  

        n = 10, ∑X = 424, ∑Y=363, ∑XY = 12815, ∑X2 =21926,

        ∑Y2 = 15123.

       Fit regression equation of X on Y. Estimate expenditure on inspection if

       number of defects found is 40.                                                                          [5]

     

    c) Calculate coefficient of association between smoking and coffee drinking habits

         from the following data ;

     

     

     

     

    Habits

    Coffee drinkers

    Non-coffee drinkers 

    Smokers

    Non-smokers

    90

    260

    65

    110

              Interprete the the                                                                                                 [5]

     

     d) A factory has two machines M1 and M2 producing the same product. M1

         manufactures 60% and M2 manufactures 40% of the total production. The past  

         experience shows 7% produced by M1 and 5% produced by M2 is defective

         production . At the end of days production one unit is selected at random from it

         and is found to be defective. What is the chance that it is produced by M1?   [5]

     

    Q-2) a) The average wage and S.D. of  workers from 3 departments of a factory are

                 given below.

            n1 = 150, n2 = 200,  n3 = 250. __

           __             __              __

           X1= 5000, X2 = 4500, X3= 4000

           σ1 = 150, σ2  = 200, σ3 = 100

          Calculate average wage of all workers of the factory and S.D. of wage for all 

          workers.                                                                                                             [5]

     

      b) Given r12  = 0.5, r13 = 0.3, r23, = 45

          Calculate R2.13 and r23.1.                                                                                [5]

     

     c) Of a large group of men 30% are under 165 cms and 60% are between 165 cms

         and 185 cms in height. Find mean and S.D. of the group of men, assuming

         heights are normally distributed.                                                                    [5]

         Given

    Z

    0.525

    1.28

    Area under SNV from Z=0

    0.2000

    0.4000

     

     

    Q-3) a) State merits and demerits of Mean, Median Mode.                                    [5]

             b) Explain how correlation coefficient is calculated for qualitative type of data.

                                                                                                                                           [5]

             c) Explain giving illustrations what is understood by the terms:

                 Partial, multiple correlation, regression, coefficient of regression of

                 regression, using 3 variables X1, X2, X3.                                                    [5]

     

    SECTION-II

    Q-4) a) The orient manufacturing company produces three types of writes: Tik-tok,

                 Mik-Mik and Pik-Pik. All the 3 types are required to first to be machines and

                 then to be assembled. The time required for the various types are as follows:

    Type

    Machine Time

    (hrs)

    Assembly Time

    (hrs)

    Tik-tok

    Mik-Mik

    Pik-Pik

    15

    13

    12

    4.4

    3.5

    4.0

         

        The total available machine time and assembly times are 4000 and 1240

        hours/month. The sale price and costs and contribution margin for the three

        are

     

    Tik-tok

    Mik-Mik

    Pik-Pik

    Selling price

    Labour Material & other

    Variable expenses

    Contribution margin

    Rs.11,000

     

    8,000

    3,000

    Rs.5,000

     

    2,400

    2,600

    Rs.3,000

     

    1,500

    1,500

     Company sells all 3 on month credit basis. The Labour Material, and other variable expense, must be paid in cash. Available cash Rs.1,30,000. Formulate LPP.                                                                                                                   [7]

    b) Explain the concept of Markov process, giving illustration.                           [6]

     

    c) Shruti Ltd. and Purnima Ltd. are two competitors in the market. Shruti has devices 4 strategies S1, S2, S3, S4, and Purnima 3 strategies P1, P2, P3, The pay- offs corresponding to all 12 combinations of strategies are given below. Considering the information state which strategy is better for Shruti? Which is better for Purnima What is value of the game? Is the game fair?

     

    Pay-offs

    Shruti’s

    strategies

    Purnima’ strategies

    P1                      P2                        p3

    S1

    S2

    S3

    S4

    30,000            -21,000                 1,000

    18,000             14,000                12,000

    -6,000              28,000                  4,000

    18,000               6,000                  2,000

                                                                                                                              [7]

     

    Q-5) a) Solve the following LPP using graphical method .

     Maximize Z = 5X1 + 2X2

    4X1 + 2X2 < 16

    3X1 + X2 < 9

    3X1 – X2 < 9

      X1, X2 > O                                                                                                 [8]

     

    b) Solve the assignment problem

    Expected sales data

    Salesman

    Districts

    D1                  D2                D3            D4

    S1

    S2

    S3

    S4

      20                   25                 22              18

      25                   24                 19              21

      18                   20                 22              20

      25                   20                 17              22        

     

    Q-6) a) A tailor specialises in ladies dresses. The no. of customers approaching the tailor appear to be poisson with a mean of 6 customers per hour. The tailor attends the customers on first come first serve basis and customers wait if the need be. The tailor can attend the customers at an average rate of 10 customers per hour with the service time exponentially distribution

                Calculate :                                                                                                                                                                                                                          

           1) Utilization parameter.

           2) Prob. That queue-system is idle. 

           3) Average time tailor is free on a 10 hour working day.

           4) Average waiting time of customer in queue before service.             [6]  

       b) A company has 4 factories situated at different places in the country and four

           sales agencies also situated at different places. The cost of prod . Of a unit, at a

           factory varies from factory to factory and sale prices also vary from agency.

           The table below gives units transportation cost from factories to sales agencies,

           prod. Cost and sale price .

     

     

     

     

     

     

     

     

     

     

    Units transportation cost Rs.

    Factory

    Sales agencies

    S1      S2      S3     S4

    Monthly capacity units

    Cost of prod Rs./unit

    F1

    F2

    F3

    F4

    7         5        6       4

    3         5         4       2

    4         6         4       5

    12       7         6       5   

    10

    15

    20

    15

    10

    15

    16

    15

    Monthly requirement units

    8         12       18    22 

    Sale price Rs.

    20     22         25   18

    Find the production and distribution schedule which will maximize profit.       [9]

    MAY 2005

     

    Istructions:

    1)     Solve any two question from Section I and any two question from Section II.

    2)     All question carry equal marks.

    3)     Write answers to both sections in the same answer book.

    4)     Use of electronic calculator and statistical in the same answer book.

    5)     Mobile calculators are not allowed.

    SECTION-I

    Q-1) a) Given the following distribution for overtime draw Histogram and estimate mode from it. 

                                                                                                                                                                [15]

     

            Over time in ( hours)

            No. of Workers

              4-8          8-12           12-16          16-20        20-24          24-28

               4              8                 16               18              20               18      

     

             b) Find arithmetic mean from the following data.

    Classes

    Frequency

      1-10           11-20            21-30            31-40                   41-50

        8                15                  25                  10                        7       

     

              

           c) The management of hotel has employed 2 managers, 5 cooks and 8 waiters. The monthly

               salaries of the mangers, the cook and the waiters are Rs. 3,000 Rs. 1,200 and Rs. 1,000

               respectively. Find the mean salary of the employees.

     

    Q-2) a) Particulars regarding the income of two villages are given below .                                   [15]

     

    Village X

    Village Y

    No. of People

    Average (Rs)

    Variance (Rs)

    600

    175

    100

    500

    186

    81

     What is combined standard deviation if the village X and Y are put together?

     

           b) Given

              __       __

    N= 8, X= 10, Y=8,

                  __      __                         __2                           __2  

          ∑(X-X) (Y-Y) = 43,        ∑X-X) = 32,              (Y-Y) =72.

     

    Find coefficient of correlation.

     

    c) Given r=0.8

     

     

    Price Rs.

    Amount demanded ‘000units

                                              Mean

                                              S.D.

    10

    2

    35

    2

        

     

    Q-3 a) If r3 =0.4, r13=0.2, and R1.23,=0.75  obtain possible value of r12.  [15]

     

      b) In an examination at which 600candiates appeared , boys out numbered

    girl by 16% of all candidates . Number of passed candidates exceeded the number of failed candidates by 310. boys failing in the examination umbered 88. Find coefficient of association between sex and success.

     

           c) In a certain locality there were 320 patients. 10 were suffering from typhoid, 12 were suffering from malaria and 2 were suffering from both. If one is selected at random. What is the probability that he is suffering either from ryphoid or malaria?

     

    Q-4) a) An insurance company insured 1500 scooter drivers, 3500 car drivers and 5000 truck drivers. The probability of an accident is 0.050, 0.02 and 0.10 respectively in case of scooter, car and truck drivers. One of the insured person meets an accident. What is the  probability that he is a car driver?

     

    b) Ina certain factory it was found that average absentee rate is 3 orkers per shift. Find the probability that on a given shift.

             i) Exactly two workers will be absent.

      ii) More than four workers will be absent.                  

      [Given e3=0.04979, e0.3=0.7408]

     

    c) It is observed that 80% of T.V. viewers watch Aap ki Adalat  program . What is the probability that at least 80% of viewers in a random sample of 5 watch this program?                       [15]

     

    SECTION-II

     

    Q-5) A company has two grades of inspectors I& II, who are to be assigned for a quality control inspection. It is required that at last 2000 pieces be inspected per 8 hour day. Grade I inspectors can check pieces at the rate of 50 per hour with an accuracy of 97% Grade II inspectors can check pieces at the rate of 40 per hour with an accuracy of 95% . The wage rate of Grade I inspector is Rs. 4.50 per hour and that of grade II is Rs. 2.50 per hour. Each time an error is

    made by an inspector the cost to the cost to the company is one rupee. The  company has available for the inspection job 10 grade I and 5 grade II  inspectors. Formulate  the problem to find how many grade I and II inspectors. To be engaged to minimize the total cost.  [8]

     

    b) Solve the game with pay off matrix as below                                                                            [7]

     

    Player A

                                     PlayerB

    B1                             B2                                 B3

     

    A1

    A2

    A3

    1                                   7                                2

    6                                 2                                  7

    5                               1                                   6

         

    Q-6) Solve the following transportation problem given problem given the unit transportation costs,

            demand and supply as below.                                                                                                         [8]

     

     

    Sources

    Write houses

    A                    B              C

                          Supply

    1

    2

    3

    5                   1                   7

    6                   4                   6

    3                   2                  5 

    10

    80

    15

    Demand

    75                20                 50

     

     

    b) A company has 5 jobs to be done. The following matrix shows the return in Rs. of assigning 1th job. Assign the five jobs to the five machines so as to maximize the total return.                                                                                                        [7]

     

     

                                                     Mchine

    Job

    A        B             C             D          E

                                                        1

                                                        2

                                                        3

                                                        4

                                                        5

    5          11             10           12         4   

    2          4               6             3          5               

    3          12             5            14         6

    6          14             4             11        7

    7           9              8             12        5     

                                

      

    Q-7) a) At a bus terminus every bus should leave with driver. At the terminus they keep 2 drivers as reserved, if any one on, scheduled duty is sick and could not come. Following is the probability distribution that driver become sick.[8]                     

     

    Number of sick drivers

    0              1              2              3           4             5

    Probability

    0.30       0.20           0.15        0.10        0.13      0.12   

     

    Simulate for 10 days and find utilization of reserved drivers. Also find how many days and how many buses cannot run because of non availability of drivers.Use following random numbers.

     30,54,34,72,20,02,76,74,48,22.

     

    b) A.T.V. repairman finds that the time spent on his jobs has an exponential distribution with mean 30 minutes.If he repairs sets in the order in which they come in, and if the arrival of sets is approximately poisson with an average of 10per 8 hour day. What is the repairmans expected idle time each day? How many jobs are ahead of the average set just brought in?                             [7]

     

    Q-8) a) Write short notes on any two                                                                                             [8]

                 i) Multiple channel queueing system.

                 ii) Markov chain.

                iii) Uses of transportation models.

          

          b) Find regret table from the following pay off table.                                                           [7]

     

     

     

     

     

                           Events

     

     

                       Actions

    ___________________________________

    A1            A2                A3                A4

    E1

    E2

    E3

    E4

    80           430                -20               30            

    330         30                  230              330

    120         130                  30              330   

    80           30                  130               30

     

    Also find Expected Regret for each action if

    P(E1)= 0.15,P(E2)=0.45,P(E3)=0.25, P(E4)=0.15.

    OCTOBER 2005

     

    Instructions:

    N.B.:

    1) Question No. 1 compulsory any one question out of Q.No.2 and Q.3.

    2) Question No. 4 is compulsory solve any one out of Q.No.5 and Q.No.6.

    3) Figures to the right indicate full marks.

    4) Use of electronic calculator is permitted.

    5) Graph paper will be supplied on demand.

    SECTION-I

    Q-1) a)  Draw histogram, frequency polygon Ogive curve for the following distribution .       [6]

    Marks Less than

    10

    20

    30

    40

    50

    60

    70

    80

    90

    Number of Students

    4

    6

    24

    46

    67

    86

    96

    99

    100

     

    Q-2 b) The distribution of wages of workers in two factors A and B is given below. Determine in which factory total wages paid to all the workers is more and in which factory the wages of the workers is more and in which factory the wages of the workers are more variable.                 [7]

     

    Wages in Rs.

    No. of Workers

    A

    B

    50-100

    100-150

    150-200

    200-250

    250-300

    300-350

    2

    9

    29

    54

    11

    5

     

    6

    11

    18

    32

    27

    11

     

           c) Calculate the equation of regression of X1 on X2 and X3 and estimate X1 when X2 =165 and

               X3 =175.                                                                                                                                                           [7]

               

     

              Given__                     __                       __  

                             X1 =170,           X2 =160,               X3 = 168,

                         σ1 =2.4,          σ2 = 2.7,             σ3 = 2.7,

                         r12 = 0.28,       r13 =0.49,         r23 = 0.51

     

    Q-2) The expenditure of 1000 families is given below :                                                                [6]

      

    Expenditure in Rs.

    40-59

    60-79

    80-99

    100-119

    120-139

    No. of families 

    50

    500

    50

    The median of the distribution is Rs. 87. Calculate missing frequencies and for the completed distribution table calculate mode.

     

             b) The advertisement cost and effected sales are given in the following table. Calculate the line

                 of regression of sales on advertisement expenses. ( cost ) and estimate the sales when

                 advertisement cost is Rs. 1,00,000.

     

    Advt.Cost (Rs.)

    39

    65

    62

    90

    82

    75

    25

    98

    36

    78

    Sales (Rs. Lakhs)

    47

    53

    58

    86

    62

    68

    60

    91

    51

    84

    Calculate co-efficient of correlation between advertisement cost and sales.                                 [9]

     

    Q-3) a) It is observed that a person going to petrol pump for filling the petrol checks the air

                 pressure of the tyres of the vehicle 12% of times and checks the level of engine

                 oil 29% of    the times . It is also abserved that 7% persons check both – air pressure and the level of oil.                                                                                                                              [5]

    i)                   Calculate the probability that the persons going to the petrol pump neither checks the air pressure nor the level of oil.

    ii)                 Calculate the probability that the person checks the air pressure but not the level of oil.

     

         b) The following table gives the analysis of the examination results.                                   [5]

                     

     

    Boys

    Girals

    No. Candidates appeared

    Married Candidates

    Married and Successful

    Unmarried and Successful

    800

    150

    70

    550

    200

    50

    20

    110

    Determine whether there is an association between marital status and success.

     

         c) The life of battery cells supplied by company A was tested and it was found that the average life is 50 hours with a standard deviation 3 hours. It the company has supplied 1000 battery cells.                                                                                                                                        [5]

              i) how many of these will have life than 55 hours.

             ii) how many of these will have life than 44 hours.

                 Given area under normal curve

                 for (1) Z =1.67 is  0.4525

                 and (2) Z = 2    is 0.4772

     

    SECTION-II

     

    Q-4) a)  A firm produces 3 products A,B and C. It uses 2 raw materials I and II of which 5,000 and  7,500 units can be used for production of A, B and C. Product A requires 3 units of raw material I and 5 units of raw  material II per units corresponding requirements per units of B are 4 and 3 units of raw material I and to respectively and per units of C5 units of raw material I and 5 units of raw material II.                   

                   The labour time to produce 1 unit of A is twice required to produce 1 unit of B and is three

                   Times required to produce 1 unit of C. The entire labour force of the firm can produce

                   equivalent of 3,000 units of product A.

                   The minimum demand for 3 products is 600, 650 and 500 units respectively. Assuming

                   profit per units of A, B and C are Rs. 50 , Rs.60 and Rs. 80 respectively formulate the L.P.P. to maximize profit satisfying constraints.                                                             [5]

     

              b) A T.V. repairman finds that the time spent on his job has an exponential distribution with mean 30 minutes. If he repairs sets in the order the sets arrive and arrival of sets in poisson distribution pattern with an average rate of 10 sets per 8 hour day. Find the expected idle time of server repairman each day. Find the average number of sets a head of a new arrival of set.                                                                                                                                       [5] 

     

             c) IN a cricket season for a one –day match a bowler bowled 50 balls. The frequency distribution of runs scored per ball is as given below:                                                 [5]

    Runs/Ball

    0

    1

    2

    3

    4

    5

    6

    Number of balls

    15

    10

    10

    4

    8

    1

    2

                Simulate the system for 2 overs and average runs given in 2 overs by him. Use following 

                random number.

                88,03,05,29,28,48,65,19,55,17,37,82.

     

             d) Write a note on decision theory.                                                                                           [5]

     

    Q-5) a) Finance faculty in a management school decided to hold seminars on 4 topics-leasing,

                 portfolio, management, private mutual funds and swaps and options.

                 The seminars are to be held once a week, so that number of students unable to attend is to be 

                  kept minimum. The past experience indicates certain number of students can not attend the

                  seminar on particular day of week as shown the table below:

     

    Leasing

    Protfolio Management

    Private Mutual Fund

    Swaps and Options

    Monday

    Tuesday

    Wednesday

    Thursday

    Friday

    50

    40

    60

    30

    10

    40

    30

    20

    30

    20

    60

    40

    30

    20

    10

    20

    30

    20

    30

    30

              Find the optimal schedule of seminars, so that minimum number of students will miss the

              seminar. Find the total number of students who will be missing at least one seminar.           [7]

     

       

     

      b) Write short notes on:

           i) Markov chin.                                                ii) Multiple channel queueing system.

     

    Q-6) a) A manufacturer of jeans in interested in developing an advertising campaign that will reach

                 4 different age groups. Advertising campaign can be conducted through T.V., radio and

                 magazines. The following table gives estimated cost per exposure for each group in

                appropriate units of money, according to medium employed. The maximum exposure levels

                possible in each of the media T.V., radio and magazine are 40, 30 and 20 million respectively.

                Also desired exposures in each age groups 12-18,19-25,26-35 and 36 and above are 30, 25, 

               and 10 million.                                                                                                                        [9]

               The objective is to minimize the cost of obtaining the minimum exposure level in each age

               group.

             

    Media

    Age Group

    12-18

    19-25

    26-35

    36 and above

    T.V.

    Radio

    Magazine

    12

    10

    14

    7

    9

    12

    10

    12

    9

    10

    10

    12

     

               Formulate above problem as Transportation problem and find the optimal solution.

     

           b) Solve the following game                                                                                                       [6] 

     

    Pay –off

     

    Player B

    B1                          B2                    B3                 B4 

     A1

    Player A                         A2

    A3

                                            A4

         3                        2                    4                0

          3                        4                     2               4

          4                        2                     4               0

          0                        4                     0               8