## Loyola College B.Sc. Statistics April 2007 Statistical Methods Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 01

FIRST SEMESTER – APRIL 2007

# ST 1500 – STATISTICAL METHODS

Date & Time: 24/04/2007 / 1:00 – 4:00 Dept. No.                                                Max. : 100 Marks

# PART-A

Answer all the questions:                                       10×2=20

1. Define statistics.
2. Explain ordinal data,nominal data.
3. State any two merits of median.
4. Find the values of Q1 and Q3 for the following data.

20,28,40,12,30,15,50

1. Write the formulas for β1 and β2 in terms of the moments.
2. Mention the properties of the correlation coefficient.
3. Given the two regression equations

8X – 10Y= – 66

40X – 18Y = 214

find the mean values of X and Y.

1. What are the normal equations for fitting

Y= abx?

1. Find whether the given data

(A)=100, (B)=150, (AB)=60,N=500 is consistent.

1. Explain scatter diagram.

PART-B

Answer any 5 questions:                                                                                   5×8=40

1. Draw the Box-whisker plots for the following data and compare.

Scores of jayanth   58  59  60  54  65  66   52  75   69  62

Scores of vasanth   87  89  78  71  73  84  65  66  56   46

1. Obtain the mean deviation about median for the marks give below;

Marks                                           Frequency

• 7
• 12

20-30                                                    18

30-40                                                    25

40-50                                                    16

50-60                                                    14

60-70                                                     8

1. Calculate the rank correlation coefficient for the variables X and Y  from the  following data:

X   75    88    95    70    60     80    81     50

Y  120  134  150  115  110   140  142   100

1. Fit a parabolic curve to the following time series:

Year    1997    1998    1999    2000    2001    2002    2003

Production     42         49        62        75        92      122      158

1. In a group of 800 students , the number of married is 320.But of 240 students who

failed,96 belonged to the married group.Find out whether the attributes marriage

and failure are independent.

1. Find the geometric mean for the following data given below.

Marks                        Frequency

• 6
• 10
• 18
• 30
• 15
• 12
• 10
• 6
• 2
1. An algebra test was given to 400 school children of whom 150 were boys and 250

girls.The results were as follows.

Boys                   Girls

Mean              72                       73

SD                7                       6.4

Sample size      150                     250

Find the combined mean and combined standard deviation.

1. Explain the various types of diagrams used in statistical applications.

### PART-C

Answer any two questions:                                                                            10×2=20

1. The following table gives the aptitude test scores and productivity indices of 10

workers selected at random.

Aptitude scores(X)     60   62   65    70   72   48   53   73   65   82

Productivity

index(Y)     68   60   62    80   85   40   52   62   60   81

i)Obtain the regression equation of Y on X

ii)Obtain the regression equation of X on Y

iii)Obtain productivity index of a worker with test score=92

1. iv) Obtain the test score of a worker whose productivity index is 75

v)obtain the correlation coefficient between X and Y through regression

equations.

1. Two brands of tyres are tested with the following results.

No    of    tyres

Life(000 miles)                                        X        Y

20-25                                                 1          0

25-30                                                22        24

30-35                                                64        76

35-40                                                10         0

40-45                                                 3          0

i)Which brand of tyres has greater average life?                                    (5)

ii)Calculate coefficient of variations and state which one is consistent.            (15)

1. Find β1 and β2 for the following data and interpret the results.

Age                                    Frequency

• 2
• 8
• 18
• 27
• 25
• 16
• 7
• 2
1. a)Explain the contingency tables and the method of calculating chi-square for a

contingency table.

b)Coefficient of correlation between X and Y for 20 items is 3,mean of X is 15 and

that of Y 20,standard deviations are 4 and 5 respectively.At the time of calculation

one item 27 has wrongly been taken as 17 in case of X values and 35 instead of 30

in  case of Y series.Find the correct coefficient of correlation.

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## Loyola College B.Sc. Statistics April 2007 Statistical Process Control Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.  DEGREE EXAMINATION –STATISTICS

 AC 24

SIXTH SEMESTER – APRIL 2007

ST 6602STATISTICAL PROCESS CONTROL

Date & Time: 20/04/2007 / 9:00 – 12:00       Dept. No.                                                                 Max. : 100 Marks

PART-A

Answer all the questions:                                                     10×2=20

1. Mention any 4 advantages of a control chart.
2. Write the 3-σ control limits for a c-chart with the process average equal to 4 defects.
3. The control limits for a p-chart are given below

UCL=.161,CL=.08,LCL=0,n=100

Find the equivalent control limits for an np chart.

1. Write the need for an EWMA control chart.
2. Define consumers risk,producers risk.
3. Mention any 4 advantages of acceptance sampling.
4. Write the expression for AOQ of a double sampling plan.
5. Write the control limits for an s-chart when σ is given.
6. Explain the term process capability.
7. What are the uses of a stem and leaf plot?

PART-B

Answer any 5 questions:                                            5×8=40

1. The number of defective switches in samples of size 150 are shown  below. Construct a fraction   defective chart for these data.

Sample number                       number of defective

switches

1                                                   8

2                                                   1

3                                                   3

4                                                   0

5                                                   2

6                                                   4

7                                                   0

8                                                   1

9                                                  10

10                                                   6

11                                                   6

12                                                   0

1. Explain the oc curve of a control chart in detail.

1. Explain the theory behind the construction of control limits for  and S charts.
2. Explain the double sampling plan in detail.
3. Draw box-whisker plots for the following data on two variables and compare.

Sample no:    1      2      3      4       5       6      7      8       9     10      11     12

x1       :    6     10     7      8       9      12    16     7       9     15       8       6

x2       :   15    11     5     13     13     10     9      4      12    13      16     11

1. Explain the CUSUM chart in detail.
2. Explain the methods of measuring process capability in detail.
3. A p-chart is used with UCl=.19,CL=.1,LCL=.01 to control a process
• If 3-σ limits are used , find n .
• Obtain the α-risk.
• Obtain the β-risk if p has shifted to p=.2

PART-C

Answer any 2 questions:                                                      2 x 20=40

19.a) A paper mill uses a control chart to monitor the defects in finished rolls of a

paper. Use these data to set up chart for defects per roll of paper. Does the

process  appear to be in control?

Day                  Number of Rolls                Total number of defects

1                                18                                         12

2                                18                                         14

3                                24                                         20

4                                22                                         18

5                                22                                         15

6                                22                                         12

7                                20                                         11

8                                20                                         15

9                                20                                         12

10                               20                                         10

1. b) Explain the need for

i)Pareto diagram

ii)Cause and effect diagram

iii)Defect concentration diagram in statistical process control .

1. a)The data shown below represents from nominal diameter value for holes drilled

in aerospace  manufacturing.

Sample no                       x1               x2              x3                x4              x5

1                            -30               50             -20               10              30

2                               0                50            -60              -20              30

3                            -50               10              20               30               20

4                            -10              -10              30              -20              50

5                              20              -40              50               20             10

6                               0                  0               40             -40             20

7                               0                  0               20             -20             10

8                              70               -30             30              -10              0

9                               0                  0               20              -20            10

10                             10                 20             30               10             50

Set up  & R charts. Is the process  in control?

b)A normally distributed quality characteristic is controlled through use of an

and R  charts.

char                                                      R-chart

UCL=626                                              UCL=18.795

CL=620                                                 CL=8.236

LCL=614                                              LCL=0

1. i) If specifications are 610±15 what percentage of defective items is produced?
2. ii) What is the probability of detecting a shift in the process mean to be 610 on

the first sample? (σ remains constant).

iii) What is the probability of type I error?

1. iv) If S chart were to be used for the R chart what would be the approximate

parameters of the S chart?           (4x 2.5)

1. a) Draw EWMA control chart for the following data on the sample mean with λ = .2 ,σ = 2.

Sample number                                   x

• 45
• 55
• 37
• 64
• 95
• 08
• 5
• 87
• 25
• 46
• 39
• 69

b)Also obtain the tabular CUSUM values with Δ=.5σ ,α=.005

1. a) Draw the oc curve for a single sampling plan with n=100,c=2,N=1000

1. b) Obtain the single sampling plan for which

P= .01, α = .05, P= .06, β= .10

1. c) Obtain the expressions for AOQ and ATI. (10+10)

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## Loyola College B.Sc. Statistics April 2007 Testing Of Hypothesis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 17

FIFTH SEMESTER – APRIL 2007

# ST 5501 – TESTING OF HYPOTHESIS

Date & Time: 28/04/2007 / 1:00 – 4:00 Dept. No.                                              Max. : 100 Marks

# PART – A

Answer ALL the questions.                                                        10 X 2=20 marks

1. Define a simple and composite statistical hypothesis and give an example each.
2. What are Type-I and Type-II errors in testing of hypothesis.  Also define the power function.
3. Define a best critical region (BCR) of size for testing the simple hypothesis against an alternative simple hypothesis.
4. When do you say that a BCR is uniform?
5. When do you say that a distribution belongs to an exponential family?
6. Under what situation likelihood ratio test is used?
7. Define Sequential Probability Ratio Test (SPRT).
8. State any two differences between SPRT and other test procedures.
9. Write the statistic for testing the equality of means when the sample is small.
10. What is a non-parametric test?

# PART – B

## Answer any FIVE questions.                                                        5 X 8 =40 marks

1. Let Y have a binomial distribution with parameters n and p. We reject Ho: p = ½ and accept     H1 = p> ½ if y  Find n and c to give a power function K(p)  which is such that K  = 0.10 and K = 0.95 approximately.
2. State and prove Neyman Pearson theorem.

1. If X1, X2,…Xn is a random sample from normal distribution with mean q and variance 1, find a BCR of size a for testing Ho: q = 0 against H1 : q =1.

1. Let X1, X2, … Xn denote a random sample from a distribution that is N(q, 1). Show that  there is no uniformly most powerful test of the simple hypothesis Ho: q = q¢ where q¢ is a fixed number , against  the alternative composite hypothesis H1 : qq¢.
2. Let X1, X2, ….Xn be a random sample from a Poisson distribution with parameter q where q>0. Show that the distribution has a monotone likelihood ratio in the statistic Y= .
3. Let Y1,< Y2 < …. < Y5 be the order statistics of a random sample of size n=5 from a distribution with pdf f(x;q) =   e|x – q | , -<x<, for all real q.  Find the likelihood ratio test for resting Ho: q = q0 against H1 : q q0.
4. Ten individuals were chosen at random from a population and their heights     were   found to be in inches 63, 63, 66, 67, 68, 69, 70, 70, 71, 72.  Test the hypothesis   that the height in the population is 66 inches. Use 5%  significance level.

1. Explain the wilcoxon

# PART – C

## Answer any TWO questions.                                                              2 X 20 =40 marks

1. (a) Let X1, X2,…, X10 denote a random sample of size 10 from a Poisson

distribution with mean q.  Show that the critical region C defined by

is a best critical region for testing H0 : q = 0.1 against

H1 : q = 0.5.  Determine the significance level a and the power at q = 0.5

for this test.

• Let X have a pdf of the form f (x;q) = , 0 < x < q  zero elsewhere.  Let

Y1< Y2 < Y3 ,< Y4 denote the order statistics of a random sample of size 4

from this distribution. we reject H0 : q = 1 and accept H1 :q1 if either

y4 or YFind the power function K (q),  0 < q, of the test.   (10+10)

1. Let the random variable X be N(q1,q2). Derive a likelihood radio test for testing                         H0 : q1 = 0, q2 >0 against H1 : q1 0  , q2 >0.

1. (a) Let X be N(0,q) and let q¢ = 4 , q” = 9, a0 = 0.05, and o = 0.10.  Show

that  the sequential probability ratio test can be based upon the statistic

. Determine c0(n) and c1(n) .

• In a survey of 200 boys, of which 75 were intelligent, 40 has skilled

fathers while 85 of the intelligent boys has unskilled fathers.  Do these

figures support the hypothesis that skilled fathers have intelligent boy?  Use

5% significance level.                                                                       (10+10)

1. (a) An IQ test was administered to 5 persons before and after they were trained.

The results are given below :

 Candidates I II III IV V IQ before training 110 120 123 132 125 IQ after training 120 118 125 136 121

Test whether there is any improvement in IQ after the training programme. Use 1%

significance level.

• Let m be the median lung capacity in litres for a male freshman. Use sign test to test at the a = 0.0768  significance level, the null hypothesis  Ho : m = 4.7 against the two sided alternative hypothesis H1 :.m > 4.7.  The observations are : 7.6   4   4.3   5.0   5.7   6.2   4.8   4.7   5.6   5.2   3.7   4.0   5.6   6.8   4.9   3.8   5.6                                                   (10+10)

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## Loyola College B.Sc. Statistics April 2007 Statistical Mathematics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 06

DEGREE EXAMINATION –STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3500STATISTICAL MATHEMATICS – II

Date & Time: 21/04/2007 / 1:00 – 4:00            Dept. No.                                                     Max. : 100 Marks

SECTION A

Answer ALL questions. Each carries 2  marks                                 [10×2=20]

1. Define Skew-Hermitian matrix and give an example of 3×3 skew Hermitian

matrix.

1. State Cayley-Hamilton theorem.
2. Evaluate the primitive integral : .
3. Define order and degree of differential equations and give an example
4. What is meant by double limit? Give an example
5. Define improper integral of second kind
6. Define Integrability and Integral of a function
7. Solve (2- 4x2)dy = (6x-xy) dx
8. If f(x) =     is a p.d.f. , find the value of K
9. Evaluate

SECTION B

Answer any FIVE questions                                                                    (5×8 =40)

1. Find the inverse of the matrix A =by using 2×2 partitioning
2. State and prove a necessary and sufficient condition for integrability of a

function

1. Evaluate :
2. Define Beta distribution of 1st kind and hence find its mean and variance by stating the conditions for their existence.

1. Show that double limit at the origin may not exist but repeated limits exist for

the following function :

f(x,y) =

1. Investigate the extreme values of f(x,y) = (y-x)4 + (x-2)2, x, y Î R.

1. Prove that

1. Compute mean and variance for the following p.d.f

## SECTION C

Answer any TWO questions                                                             (2 x 20 =40)

19.a] Find the rank of the matrix A=

[b] Find the characteristic roots of the following matrix. Also find the

inverse using  Cayley-Hamilton theorem:

A=

1. a] Test if    converges absolutely

b] Solve the differential equation

1. a] If f(x,y) =

is the joint p.d.f of (X,Y), find the joint d.f. F(x,y).

[b] Let  , x, y Î R. Find  fxx , fyy , fxy fyx

1. [a] Let f(x,y) = be the joint p.d.f of (x,y). Find the

co-efficient of  correlation    between X and Y.

[b]  Change the order integration and evaluate

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## Loyola College B.Sc. Statistics April 2007 Statistical Mathematics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 04

SECOND SEMESTER – APRIL 2007

# ST 2500 – STATISTICAL MATHEMATICS – I

Date & Time: 20/04/2007 / 1:00 – 4:00 Dept. No.                                                Max. : 100 Marks

SECTION A

Answer ALL questions. Each carries 2  marks                                         [10×2=20]

1. Define a bounded function and give an example.
2. Define a monotonic sequence and give an example.
3. What is ‘permutation of indistinguishable objects’? State its value in factorial notation.
4. Define the limit of a sequence and give an example.
5. If A and B are independent events, show that A and Bc are independent.
6. What are the supremum and infinum of the function f(x) = x – [x], x €R ?
7. Find noÎN, such that |(n/n+2) -1| < 1/3 .
8. Define probability mass function (p.m.f) of a discrete random variable and state its properties.
9. Define Moment Generating Function and state its uses.
10. Is the following a P.G.F:  F(s) = s/ (3-s) ?

# SECTION B

Answer any FIVE   questions.  Each carries 8  marks                          [5×8=40]

1. State and prove the Addition Theorem of Probability for two events. Extend it for three events.
2. Discuss the convergence of the series
3. A and B play a game in which their chances of winning are in the ratio 3 :2    Find A’s chance of winning at least 2 games out of 5 games.
4. Find the sum of the series .
5. State and prove Baye’s theorem.
6. Investigate the extreme values of the function f(x) = 2x5 -10X4 +10x3 + 8
7. Identify the  distribution for which  is the P.G.F
8. Let X be a random variable with the probability mass function

x:    1           2              3

P(x): 1/2        1/4          1/4         Find P.G.F and hence Mean and Variance

SECTION C

Answer any TWO  questions.  Each carries 20 marks                         [2×20=40]

19 [a] Let Sn = 1 + , n=1,2,3,…prove that Lim Sn exists  and lies between

2 and 3.                                                                                                           [12]

[b] Let Sn =  and   tn =    , verify that the limit of the difference   of

two convergent sequences is the difference  of their limits.                              [8]

20 Discuss the convergence of the Geometric series

for  possible variations in x                                                                             [12]

[b] Let X be a random variable with the following distribution:

x  :    -3      6      9

P(x):    1/6    ½    1/3

Find [i] E(X)     [ii] E(X2)        [iii] E(2X+1)2              [iv] V(X)                     [8]

21 [a] What is the expectation of the number of failures preceding  the first success

in an infinite series of independent trials with constant probability ‘p’ of

success in each trial.

[b] Let X be a random variable with the probability mass function

x:   0    1          2          3          ….

P(x): ½   (½)2   (½)3     (½)4           …

Find the  M.G.F and hence Mean and Variance

22 [a] Let X be a random variable which denotes the product  of the  numbers on

the upturned faces, when two dice are rolled:

[i] Construct the probability distribution

[ii] Find the probability for the following :

1) the product of the faces is more than 30

(2) the product of the faces is maximum 10

(3) the product of the faces is at least 9 and  at most 24

[iii] Find E(X)

[b]Define Poisson distribution. Find the moment generating function (m.g.f.).Hence find the

mean and the variance.

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## Loyola College B.Sc. Statistics April 2007 Statistical Mathematics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 05

SECOND SEMESTER – APRIL 2007

# ST 2501 – STATISTICAL MATHEMATICS – I

Date & Time: 20/04/2007 / 1:00 – 4:00 Dept. No.                                                Max. : 100 Marks

 SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

1. Define a bounded function and give an example.

1. State the values of and .
2. Write down the distribution function of the number of heads in two tosses of a fair coin.
3. Investigate the nature (convergence / divergence / oscillatory) of the series

1 – 2 + 3 – 4 + 5 – ∙∙∙∙∙∙∙∙

1. State the Leibnitz test for convergence of alternating series.

1. Apply first principles to find f ‘(a) for the function f(x) = xn.

1. Show the validity of Rolle’s Theorem for f(x) = , x [– 1, 1].

1. Define a vector space.

1. If M(t1,t2) is the joint m.g.f. of  (X,Y), express E(X) and E(Y) in terms of M(t1,t2).

1. Define an Idempotent Matrix.
 SECTION – B

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

1. Show that Inf f + Inf g  ≤  Inf (f + g)  ≤  Sup (f + g) ≤ Sup f + Sup g.
2. Show by using first principles that  = 0.
3. Discuss the convergence of the following series (a) ,  (b)
4. A discrete r.v. X has p.m.f. p(x) = , x = 0, 1, 2, …… Obtain the m.g.f. and hence mean and variance of X.

1. Show that differentiability implies continuity. Demonstrate clearly with an example that continuity does not imply differentiability.

1. Obtain the coefficients in the Taylor’s series expansion of a function about ‘c’.

(P.T.O)

1. State any two properties of a bivariate distribution function. If F(x , y)  is the bivariate d.f. of (X, Y), show that

P( a < X ≤ b , c < Y ≤ d) = F( b, d) – F ( b , c) – F( a, d) + F( a, c)

1. Establish the ‘Reversal Laws’ for the transpose and inverse of product of two matrices.

 SECTION – C

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

1. (a) Establish the uniqueness of limit of a function as x → a (where ‘a’ is any real number). Also, show that if the limit is finite, then ‘f’ is bounded in a deleted neighbourhood of ‘a’.

(b) Identify the type of the r.v. X whose distribution function is

F ( x) =

Also, find P( X ≥ 4 / 3 ) and P(X ≤ 1).                                                          (12 + 8)

1. (a) Investigate the extreme values of the function f(x) = ( x + 5)2(x3 – 10)

(b) State the Generalized Mean Value Theorem. Examine its validity for the functions f(x) = x2, g(x) = x4 for x[1, 2].                                                   (12 + 8)

1. (a) For the following function, show that the double limit at the origin does not exist but the repeated limits exist:

f (x ¸y) =

(b) Show that the mixed partial derivatives of the following function at the origin

are unequal:

f (x , y) =              (8 + 12)

1. (a) Show that if two (non-zero) vectors are orthogonal to each other, they are

linearly independent.

(b) Find the inverse of the following matrix by ‘partitioning’ method:

(5 + 15)

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## Loyola College B.Sc. Statistics April 2007 Resource Management Techniques Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 LO 19

DEGREE EXAMINATION –STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3100 – RESOURCE MANAGEMENT TECHNIQUES

Date & Time: 02/05/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

# PART – A

Answer all the questions.                                                           (10 x 2 = 20 Marks)

1. What is the need for an artificial variable in a linear programming problem?
2. How many basic solutions can be obtained for a system of 3 equations with 5 variables?
3. Explain the need for a transportation problem.
4. Express assignment problem as a linear programming problem.
5. What is the objective of a sequencing problem?
6. When an activity is called critical in a project?
7. Distinguish between CPM and PERT.
8. Define holding cost and shortage cost in an inventory model.
9. Write the formula for EOQ in a single item static model explaining the notations used.
10. What are the assumptions in a single item static model?

# PART – B

Answer any five questions.                                                        (5 x 8 = 40 Marks)

1. Nerolac produces both interior and exterior paints from 2 raw materials R1 and R2. The following data provides the basic data of the problem:

Tons of raw material               Maximum availability

per ton

Interior                    Exterior

Raw material, R1                         6                  4                                  24

Raw material, R2                         1                  2                                    6

Profit per ton in 000’s                 5                  4

A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton.  Also, the maximum daily demand of interior paint is 2 tons.

Nerolac wants to find the optimum product mix of interior and exterior paints that maximizes the total daily profit.

Formulate the problem as a linear programming problem.

1. Solve the following linear programming problem graphically.

Max     Z  =  4 x1 + 3 x2

Subject to

2x1 + x2 £ 1000

x1+ x2 £  800

x1 £ 400

x£ 700

x1, x2  ³ 0

1. Obtain the initial basic feasible solution to the following transportation problem using least cost method.

Distribution Centre                                         Availability

W                    X                     Y                     Z

A    20                    25                    50                    12             450

Factory     B    45                    50                    15                    40             500

C     22                   10                    45                    45             550

Requirement    500                  400                  300                  300

1. Four operators are to be assigned to 4 jobs in a company. The time needed by the operators for the jobs are given below.  How should the jobs be assigned so that the time is minimised?

Operators

A         B         C         D

I           15        13        14        17

II         11        12        15        13

Jobs

III        18        12        10        11

IV        15        17        14        16

1. A book binder has one printing press, one binding machine and the manuscripts of a number of different books. The time required to perform the printing and binding operations for each book are known.  Determine the order in which the books should be processed in order to minmise the total time required to process all the books.  Find also the total time required.

Processing time

1         2         3         4         5

Printing time               40        90        80        60        50

Binding time               50        60        20        30        40

1. Draw the network for the data given below and compute the critical path.

Activity                     Predecessor             Time (weeks)

A                                 ¾                                3

B                                 ¾                                5

C                                 ¾                                4

D                                 A                                 2

E                                  B                                 3

F                                  C                                 9

G                                 D, E                             8

H                                 B                                 7

I                                   H, F                             9

1. Find the optimum order quantity for a product for which the price breaks are as

follows

Quantity                                  Unit cost (Rs.)

0 £ y <500                                    10

500 £ y                                         9.25

The monthly demand for the product is 200 units, the cost of storage is 2% of the unit cost and the cost of ordering is Rs.350.

1. Discuss in detail the factors affecting inventory control.

# PART – C

Answer any Two questions.                                                       (2 x 20 = 40 Marks)

1. Use simplex method to solve

Max Z  =  2x1  +  3x2  subject to

x1 + x£  4,   – x1  +  x2  £ 1,  x1  +  2x2  £  5

x1,  x2  ³  0

1. A company has 3 factors A, B, C and four distributors W, X, Y and Z. The monthly production capacity and demand for the distribution centers and the unit transportation costs are given below.

Distribution center                   Availability

W        X        Y        Z

Factory      A         20        25        50        10                    4500

B         45        50        15        40                    5000

C         22        10        45        35                    5500

Demand              5000    4000    3000    3000

1. A project consists of activities A, B, C, _ _ _ _ H, I. Construct the network diagram for the following constraints.

A < D;  A < E;  B < F;  C < G;  D < H:  E, F < I

The project has the following time estimates (in days).

Task                             A         B        C        D        E         F          G         H         I

Optimistic time           5         18        26        16        15        6         7          7          3

Pessimistic time           10        22        40        20        25        12        12        9          5

Most likely time          8         20        33        18        20        9         10        8         4

Obtain the expected times and their variances.  Also obtain the critical path and total float, free float for the activities.

1. a) Explain a single item static model with no shortages in detail.
2. b) An oil manufacturer purchases lubricants at the rate of Rs.42 per piece from a vendor. The requirement of these lubricants is 1800 per year. What should be the order quantity per order, if the cost per placement of an order is Rs.16 and inventory carrying charge per piece per year is only 20 paise?

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## Loyola College B.Sc. Statistics April 2007 Probability And Random Variables Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 02

DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2007

ST 1501PROBABILITY AND RANDOM VARIABLES

Date & Time: 26/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

 SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

1. Define Random Experiment and Sample Space.
2. Using the Axioms of Probability, show that P(Ac) = 1 – P(A).
3. Draw a Venn diagram to represent the occurrence of exactly two of three events A, B, C.
4. State the exhaustive number of ways of choosing 2 balls one-by-one from a collection of 5 balls (i) with replacement, (ii) without replacement.
5. A restaurant menu lists 3 soups, 10 rice varieties, 5 desserts and 3 beverages. In how many ways can a meal consisting of all the four be ordered?
6. How many ‘distinct words’ can be formed from the letters of the word MISSISSIPPI.
7. Find the probability that number 2 shows up in one of the two dice thrown given that the sum of the two numbers got is 7.
8. Two persons A and B can independently solve a problem in mathematics with probabilities 1/5 and 1/3 respectively. If a problem is posed to them, what is the probability that the problem will be solved?
9. If A and B are independent events show that Ac and B are independent.
10. Define a Random Variable.

 SECTION – B

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

1. Using the Axioms of Probability, prove the following:

(a) If AB, P(A) ≤ P(B)

(b) P(A – B) = P( A ) –  P(AB)                 (4 + 4)

1. State the Binomial Theorem. Using Pascal’s Triangle, write down the expansion of  (a + b )5.

(4 + 4)

1. An urn contains 3 red balls, 4 white balls and 5 blue balls. Another urn contains 5 red, 6 white and 7 blue balls. One ball is selected from each urn. Find the probability that (i) blue and red balls are selected; (ii) the two are of different colours.                                       (3 + 5)

1. A box contains 10 tickets numbered 0 to 9 and from it three are chosen one by one. By placing the numbers in a row, an integer between 0 and 999 is formed. What is the probability that the integer so formed is divisible by 39 (regarding 0 as divisible by 39). Solve this under ( i) with replacement, (ii) without replacement, sampling schemes.

1. A man tosses two fair dice. What are the conditional probabilities that the sum of the two dice is 7 given that (i) the sum is odd? (ii) the sum is greater than 5?

1. Two fair dice are thrown. Discuss the independence of the following three events:

A: First die shows odd number

B: Second die shows odd number

C: Sum of the two numbers is odd

1. Consider two events A and B with P(A) = ¼. P(B| A) = ½, P( A| B) = ¼. Verify whether the following statements are true:

(i) A is a sub-event of B; (ii) P(A | B) + P( A| Bc) = 1

1. State and prove the ‘Multiplication Theorem of Probability’ for many events.

 SECTION – C

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

19.(a) State and prove the Addition Law of Probability for two events. State the

extension for three events.

(b) Consider the following statements about the subscribers of a magazine with

respect to their gender, marital status and education:

P(Male) =0.312, P(married) = 0.470, P(Graduate) = 0.525, P(Male Graduate) = 0.042,                                                                   P(Married Graduate) = 0.147, P(Married Male) = 0.086, P(Married male graduate)= 0.025.

Show that the information is wrong.

• There are three urns with the following contents.

Urn I:    3 white, 2 red, 5 black balls

Urn II:   4 white, 1 red, 5 black balls

Urn III:  4 white, 4 red, 2 black balls.

One ball is chosen from each urn. Find the probability that in the sample drawn  (a) there are exactly 2 black balls, (b) Balls of any two colours are found.   (8 +12)

1. (a) State and prove the “Law of Total Probability’. Hence establish Baye’s Theorem.

(b) Three Companies X, Y, Z manufacture tube lights. The market shares of the companies are 50% for X, 30% for Y and 20% for Z. 5% of the tubes manufactured by X are defective, 1% from Y and 2% from Z are defective. A bulb is chosen at random and is found to be defective. What is the probability that it was manufactured by Z?                                                                           (10 +10)

1. A loaded coin with Heads twice as likely as Tails in any toss is tossed thrice. Write down the sample space of the experiment. Obtain the p.m.f. and hence the c.d.f. of the number of Heads. Also, compute the mean and variance of the number of Heads.

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## Loyola College B.Sc. Statistics April 2007 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 23

SIXTH SEMESTER – APRIL 2007

# ST 6601 – OPERATIONS RESEARCH

Date & Time: 18/04/2007 / 9:00 – 12:00 Dept. No.                                             Max. : 100 Marks

SECTION A

Answer all questions.                                                                                    (10×2=20)

1. Define: Basic Feasible Solution.
2. Mention any two limitations in solving a LPP by graphical method.
3. Express the following LPP in its standard form:

Maximize Z = 3x1 + 2x2

subject to x1 + x2 ≤ 2; 5x1 + 4x2 ≥4; x1,x2 ≥ 0;

1. What is the difference between Big-M and Two Phase method?
2. What is an unbalanced Transportation Problem?
3. Comment on the following statement:

“Assignment problem is a particular case of Transportation problem”.

1. Define: Activity and Node with reference to network analysis.
2. Draw the network, given the following precedence relationships:

Event           :   1          2,3       4          5          6          7

Preceded by:   —          1          2,3       3          4,5       5,6

1. Verify whether saddle point exists for the following game:

Player B

Player A          B1        B2        B3

A1        5          8          1

A2        10        16        13

A3        25        22        20

1. Give the Laplace criteria for making decisions under uncertainty.

SECTION B

Answer any FIVE questions.                                                        (5×8=40)

1. Show graphically that the maximum or minimum values of the objective function for the following problem are same:

Maximize (Minimize) Z = 5x1 + 3x2

Subject to:           x1 + x2 ≤ 6;

2x1 + 3x2 ≥ 3;

x1 ≥ 3;

x2 ≥ 3;

x1,x2 ≥ 0;

1. Use Big M- method to maximize Z = 2x1 + 3x2 subject to the constraints:

x1 + 2x2 ≤ 4;       x1 + x2 =3;     x1,x2 ≥ 0

1. Explain the MODI algorithm to obtain an optimum solution for a transportation problem.

1. Solve the following assignment problem.

 1 2 3 4 5 A 3 8 2 10 3 B 8 7 2 9 7 C 6 4 2 7 5 D 8 4 2 3 5 E 9 10 6 9 10

1. Explain the Floyd’s method of finding the shortest route between any two given nodes.
2. Draw the network based on the following information and determine the critical path.

Activity           Duration (in days)                   Activity           Duration (in days)

(1,2)                             3                                  (3,4)                             3

(1,3)                             1                                  (3,7)                             10

(1,4)                             15                                (4,5)                             10

(1,6)                             7                                  (4,7)                             22

(2,3)                             8                                  (5,6)                             5

(2,5)                             10                                (5,7)                             12

(6,7)                             7

1. A farmer wants to decide which of the three crops he should plant on his 100 acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high, medium and low. His estimated profit for each crop is shown in the table below:

Rainfall           Crop A                        Crop B                        Crop C

High                8000                3500                5000

Medium           4500                4500                5000

Low                 2000                5000                4000

If the farmer wishes to plant only one crop, decide which should be his ‘best

Crop’ using Maximin, Savage and Hurwicz (α = 0.5) rule

1. Use simplex method to find the inverse of the following matrix

A =

SECTION C

Answer any TWO questions.                                                                 (2×20=40)

1. a.) Explain the mathematical formulation of a Linear Programming Problem.

b.) Use two phase method to verify that there does not exist a feasible solution

to the following LPP:

Maximize z = 2x1 + 3x2 + 5x3 subject to

3x1 + 10x2 +5x3 ≤ 15;

33x1 – 10x2 + 9x3 ≤ 33;

x1 + 2x2 + x3 ≥ 4;

x1 , x2 , x3 ≥ 0.

1. Obtain an optimum solution to the following transportation problem where the entries denote the unit cost of transporting commodities from source to destination:
 From↓To→ I II III IV Supply A 2 3 11 7 6 B 1 0 6 1 1 C 5 8 15 9 10 Demand 7 5 3 2

Find initial solution by least cost method and Vogle’s method and use the best

among them as the starting solution.

1. a.) Solve the following traveling salesman problem so as to minimize the total

distance traveled:

To

From           A         B         C         D         E

A         —        20        4          10        —

B         20        —        5          —        10

C         4          5          —        6          6

D         10        —        6          —        20

E          —        10        6          20        —

• Explain the calculations in PERT to identify the critical path of a project.

1. ) Consider the following data that gives the distance between pairs of cities

1, 2…,6:

Route: (1,2)     (1,3)     (2,3)     (2,4)     (2,5)     (3,4)     (3,5)     (4,5)

Distance:   5          1           1          5          2          2          1          3                                           Route:   (3,6)        (4,6)     (5,6)

Distance:   4           4          3

Use Dijkstra’s algorithm to find the shortest route between the cities

i.) 1 and 4    ii.) 1 and 6.

b.) Solve graphically the 2 x 4 game whose pay-off matrix is given below

Player B

Player A          1          3          11        7

8          5          2          5

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## Loyola College B.Sc. Statistics April 2007 Numerical Methods Using C Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 12

FOURTH SEMESTER – APRIL 2007

# ST 4202 – NUMERICAL METHODS USING C

Date & Time: 26/04/2007 / 9:00 – 12:00 Dept. No.                                            Max. : 100 Marks

SECTION A

Answer all questions.                                                                        (2×10=20)

1. Mention the difference between float and double data type in C.
2. Write the syntax of scanf() and getc() function.
3. What is meant by nested loop?
4. What are recursive functions?
5. Mention any two library functions available in C with their use.
6. Find the characteristic root of the matrix
7. Mention any two iterative methods to find the root of a polynomial equation.
8. What is Simpson’s 3/8 rule?
9. Mention the use of Gauss Seidal iteration method.
10. Find the inverse of the matrix

SECTION B

Answer any five questions.                                                         (5×8=40)

1. What is a structured programming language? Explain the structure of a C program.
1. Explain the syntax of if – else and if – else – if statements with examples.
2. What are arrays? Explain the concept of single dimension and two dimension

arrays with examples.

1. Write a program in C using function to display all the prime numbers between 1 and 100.
2. Explain the regula – falsi method to find the real root of an equation f(x) = 0.
3. Solve the following equations by Gauss – Jordan method:

10x + y + z = 12; 2x + 10y + z = 13; x + y + 5z = 7.

1. Use Newton’s forward interpolation formula to find the value of f(142) based on the following data:.

x:         140      150      160      170      180

f(x):         3.685   4.854   6.302   8.076   10.225

1. Obtain the value of f ¢(90) using Stirling’s formula based on the following data:

x:         60        75        90        105      120

f(x):         28.2     38.2     43.2     40.9     37.7

SECTION C

Answer any TWO questions.                                                          (2×20=40)

1. a.) Mention the various input and output statements available in C.

b.) What are control statements? Explain the syntax of the ‘go to’ and

‘switch – case’ statements with examples.                       (8+12)

1. Explain the Lagrange’s Interpolation method and hence find the value of y when

x=10 based on the following data:

x:         5          6          9          11

y:         12        13        14        16

1. Find the inverse of the following the matrix using Gauss elimination method.

22.. a.) Write a program in C for finding  using Simpson’s 1/3 rule.

b.) Evaluate  by using Trapezoidal rule.                               (12+8)

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## Loyola College B.Sc. Statistics April 2007 Mathematical Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 11

DEGREE EXAMINATION –STATISTICS

FOURTH SEMESTER – APRIL 2007

ST 4201MATHEMATICAL STATISTICS

Date & Time: 19/04/2007 / 1:00 – 4:00            Dept. No.                                                     Max. : 100 Marks

Part A

Answer all the questions.                                                                                  10 X 2 = 20

1. Define Sample space and events.
2. Let the random variable X1 and X2 have the joint pdf f(x1, x2) = 2, 0<x1<x2<1, zero elsewhere. Find the marginal pdf of X1.
3. State any two properties of a distribution function.
4. State how the mean and variance are obtained from the m.g.f.
5. If X is a Poisson variable with P(X =1) = P( X = 2), find the variance of X.
6. State the p.d.f. of Exponential distribution and state its mean.
7. State the conditions under which Binomial distribution tends to Poisson distribution.
8. Define Student’s ‘t’ distribution.
9. Define a Statistic with an example.
10. What are Type I and Type II errors?

Part B

Answer any five questions.                                                                                 5 X 8 = 40

1. State and prove addition theorem of probability for two events.
2. The probabilities of X, Y and Z becoming managers are 4/9, 2/9 and 1/3 respectively. The probabilities that Bonus scheme will be introduced of X, Y and Z becomes managers are 3/10, 1/2 and 4/5 respectively.
1. a) What is the probability that Bonus scheme will be introduced, and
2. b) If the Bonus scheme has been introduced, what is the probability that the

manager appointed was x?

1. Obtain mgf of Binomial distribution.
2. Derive the mean and variance of Gamma distribution.
3. The joint probability distribution of two random variables X and Y is given by

P(x = 0, y = 1) = 1/3, P(x = 1, y = -1) = 1/3 and P(x = 1, y = 1) = 1/3.

Find i). Marginal distributions of X and of Y ii). The conditional probability

distribution of X given Y = 1.

1. Calculate mean and standard deviation for the following p.d.f.:

f(x) = (3 + 2x)/18, for 2 ≤ x ≤ 4;    0, otherwise.

1. Calculate the mean and variance of Beta distribution of second kind.
2. Derive t – distribution.

Part C

Answer any two questions.                                                                               2 X 20 = 40

1. a) State and prove law of total probability and hence Baye’s theorem.

b). Obtain the m.g..f of Normal distribution.                                       (12 +8)

1. a) Derive the recurrence relation for the moments of Poisson distribution. Obtain   beta one and beta two.
1. b) Find the mean and variance of the distribution whose p.d.f. is

f(x) = 1 / ( b – a ), a < x < b.                                                           ( 12 + 8 )

1. a) Variables X and Y have the joint probability density function is given by

f(x, y) = 1/3 (x +y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.

i). Find coefficient of correlation between X and Y

1. b) Let X and Y have joint pdf:

e-(x + y) x3y4

f(x, y) =                         ,  x > 0, y > 0

G4 G5

Find the p.d.f. of U = X / (X + Y).                                        (10 + 10)

1. a)  Derive chi – square distribution.

b). Derive the m.g.f of chi-square distribution and hence establish its additive

property.                                                                                         ( 12 + 8)

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## Loyola College B.Sc. Statistics April 2007 Financial Accounting & Fin. State. Analysis Question Paper PDF Download

 TH  04

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.  DEGREE EXAMINATION –STATISTICS

SECOND SEMESTER – APRIL 2007

CO 2101/CO 3101 – FINANCIAL ACCOUNTING & FIN. STATE. ANALYSIS

Date & Time: 23/04/2007 / 9:00 – 12:00          Dept. No.                                                     Max. : 100 Marks

SECTION A

ANSWER ALL THE QUESTIONS                                                             10 X 2 = 20

1. Explain (a) Cash discount (b) Trade discount?
2. Give any two advantages of subsidiary books?
3. What is deferred revenue expenditure?
4. What do you mean by Contra entries? Illustrate with example.
5. Define Ledger
6. What is overdraft?
7. Prepare a Bank Reconciliation Statement from the following:

(a)        Bank overdraft as per Pass book                                Rs.80,000

(b)        Cheque issued but not presented for payment           Rs.  3,000

(c)        Cheque deposited but not yet collected by the bank Rs.  2,000

(d)       Bank charges not yet recorded in cash book              Rs.     300

1. From the following information calculate income from subscriptions to be presented in Income and Expenditure account for the year ending 31 – 3 – 07

Subscriptions received during 2006- 2007                 Rs. 75,000

Subscriptions received during 2005 –2006

For  the year 2006 – 2007                 Rs.   8,000

Subscriptions outstanding on 31 – 3 – 07                   Rs. 15,000

Subscriptions for the year 2007 – 2008 included

In the collections for  2006 – 2007                 Rs. 10,000

1. Fill in the blanks:

Goodwill is an _______ asset.

A journal is  known as a book of _________ entry.

1. From the following particulars, write up a Single column cash book;

2003

Jan 1    Cash in hand                                       Rs. 1,600

Jan 2    Paid for postage                                  Rs.      50

Jan 5    Sold  goods to Babu for cash             Rs. 1,000

Jan 8    Purchased furniture                             Rs.    600

SECTION B

ANSWER ANY FIVE                                                                             5 X 8 = 40

1. What do you mean by Bank Reconciliation Statement ?
2. Distinguish between Receipts and Payments Account   and    Income and Expenditure Account?
3. What are accounting conventions? Name and explain them in detail?

1. Enter the following transactions in proper subsidiary books of Mr. Raja:

2006

Mar 1   Purchased 500 bags of wheat from Paul at Rs. 900 per bag, less

• Bought 300 bags of rice from Kamal at Rs. 1,000 per bag, less trade discount 5%
• Sold to Lalitha 120 bags of rice 1,100 per bag less trade

Discount 5%

7   Returned to Paul 15 bags of wheat which were purchased on  1 – 3 – 06

• Sold to Harris 200 bags of wheat  1,250 per bag less trade discount 10%

15 Harris returned wheat worth Rs. 4,500

• Returned 40 bags of rice to Kamal
• Bought of Shankar 300 bags of rice at Rs. 900 per bag
• Purchased from Dayalan 200 bags of wheat at Rs. 700 per bag

1. Record the following in Journal, the transactions for April 2006 is as follows:

2006

Apr 1   Commenced business with cash                     Rs. 1,00,000

2   Purchased goods from David                         Rs.      2,400

3   Sold goods to Bosco                                       Rs.         950

5   Cash paid to David                                         Rs.      2,400

7   Paid for postage stamps                                  Rs.           25

9   Received from Bosco in full settlement         Rs.         900

10  Withdrew cash for personal use                      Rs.         500

12  Cash deposited into bank                               Rs.    10,000

1. (a) During the year a machine costing Rs. 10,000 (accumulated depreciation Rs. 3,000) was sold for Rs. 5,000. Calculate profit or loss on sale of machinery.

(b)        Capital of a Sole trader on 1- 4 – 06  was Rs. 1,25,000 and on 31 – 3 – 07 was Rs. 1,53,000. Net profit earned during the year was Rs. 45,000. Compute Drawings.

1. From the following Balance Sheet of Arth Ltd., Calculate (a) Debt Equity Ratio (b) Fixed assets to Currents assets

BALANCE SHEET

LIABILITIES           Rs.                 ASSETS                Rs.

Equity share capital   2,00,000     Goodwill                    1,20,000

Reserve                         40,000     Fixed assets(at cost)   2,80,000

P & L Account             60,000     Stock                             60,000

Secured loan              1,60,000     Debtors                          60,000

Creditors                   1,00,000     Advances                       20,000

Provision for tax           40,000     Bank                              60,000

—————                                         ————-

6,00,000                                           6,00,000

—————                                        ————–

1. From the following balances you are required to calculate Cash from operating activities:

31 – 12 – 05      31 – 12 -06

Rs.                   Rs.

P & L Account                                          50,000             3,10,000

Debtors                                                     90,000                84,000

Creditors                                                   40,000                52,000

Bills Receivable                                        24,000                30,000

Prepaid expenses                                         3,200                  2,800

Bills payable                                              30,000                32,000

Outstanding expenses                                 2,400                  3,200

Outstanding income                                    1,600                  1,800

SECTION C

ANSWER ANY TWO                                                                             2 X 20 = 40

1. From the following Trial balance as on 31 – 12 – 2006, Prepare Trading , Profit and Loss Account and Balance sheet as on that date:

TRIAL BALANCE

Particulars                                                       Dr.                   Cr.

Balances           Balances

Stock 1-1-06                                                5,840

Cash on hand                                                  192

Drawings                                                      2,840

Rent                                                                480

Machinery                                                    3,800

Tax                                                                  600

General expenses                                         1,760

Purchases                                                  41,448

Debtors                                                      16,800

Sales returns                                                    840

Provision for bad debts                                                               420

Capital                                                                                     17,000

Interest                                                                                         320

Bank overdraft                                                                             960

Sales                                                                                        47,624

Creditors                                                                                   8,000

Purchase returns                                                                        1,164

————               ———-

75,488                 75,488

————               ———-

• Depreciation on machinery @ 10% p.a
• Rent outstanding Rs. 500
• Tax prepaid Rs. 100
• Provision for bad debts is to be increased to 5 % debtors
• Closing stock Rs. 3,500

1. The following is the Receipts and Payments of Delhi football association for the first year ending 31st December 2006:

Receipts and Payments Account

Receipts                          Rs.                Payments                         Rs.

To Donation                            50,000             By Pavillion office

(constructed)                40,000

To Reserve fund                                              By expenses in

(Life membership fees                         connection with

And entrance fee                                             matches                            900

Received)                                4,000              By furniture                   2,100

To receipts from football                                 By investments

Matches                                 8,000              at cost                         16,000

Revenue receipts                                    Revenue payments

To subscription                        5,200               By salaries                   1,800

To locker rents                  50               By wages                         600

To interest on securities              240               By insurance                    350

To sundries                     350               By telephone                   250

By electricity                   110

By sundry expenses         210

By balance on hand     5,520

———                                                 ———

67,840                                                 67,840

———                                                 ———

• Subscriptions outstanding for 2006 are Rs. 250
• Salaries unpaid for 2006 are Rs. 170
• Wages unpaid for 2006 are Rs. 90
• Outstanding bill the sundry expenses is Rs. 40
• Donations received have to be capitalized

Prepare from the details given above Income and Expenditure A/c for the year ended 31 – 12 – 2006 and the Balance Sheet of the association as on that date.

1. With the following ratios and further information given below prepare a Trading Account , Profit and Loss Account and a Balance Sheet of Shri Nataraj:
• Gross profit ratio 25 %
• Net profit/ sales 20%
• Stock turnover ratio 10
• Net profit / Capital 1/5
• Capital to Total liabilities ½
• Fixed assets / Capital 5/4
• Fixed assets / Total current assets 5/7
• Fixed assets 10,00,000
• Closing stock    1,00,000

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## Loyola College B.Sc. Statistics April 2007 Estimation Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 16

DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5500ESTIMATION THEORY

Date & Time: 27/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

# Answer all the questions.                                                                  10 X 2 = 20

1. State the problem of point estimation.
2. Define asymptotically unbiased estimator.
3. Define a consistent estimator and give an example.
4. Explain minimum variance bound estimator.
5. What is Fisher information?
6. Write a note on bounded completeness.
7. Examine if { N (0, σ2), σ2 > 0 } is complete.
8. Let X1, X2 denote a random sample of size 2 from P(q),q > 0. Show that X1 + 2X2 is not sufficient for q.
9. State Chapman – Robbins inequality.
10. Explain linear estimation.

# Answer any five  questions.                                                              5 X 8 = 40

1. Let X1, X2, … ,Xn denote a random sample of size n from B(1,q), 0<q<1. Show that  is an unbiased estimator of q2, where T = .
2. If Tn  is consistent for Y(q) and g is continuous, show that g(Tn ) is consistent for g{Y(q)}.
3. State and establish Cramer – Rao inequality.
4. Show that the family of binomial distributions { B (n, p), 0 < p < 1, n fixed } is complete.
5. State and establish Rao – Blackwell theorem.
6. Let X1, X2, … , Xn denote a random sample of size n from U (0, q), q > 0.

Obtain the UMVUE of q.

1. Give an example for each of the following
1. MLE, which is not unbiased.
2. MLE, which is not sufficient.
1. Describe the method of minimum chi-square and the method of modified minimum

chi-square.

##### Part C
###### Answer any two questions.                                                                     2 X 20 = 40

1. a). Show that the sample variance is a biased estimator of the population variance.

Suggest an unbiased estimator of s2.

b). If Tn is asymptotically unbiased with variance approaching zero as n

approaches infinity then show that Tn is consistent.                              (10 + 10)

1. a). Let X1, X2, … , Xn denote a random sample of size n from U (q-1, q +1).

Show that the mid – range U = is an unbiased estimator of q.

b). Obtain the estimator of p based on a random sample of size n from

B(1, p), 0 < p < 1by the method of

i). Minimum chi-square

ii). Modified minimum chi-square.                                                    (12 + 8)

1. a). Give an example to show that bounded completeness does not imply completeness.

b). Stat and prove invariance property of MLE.                                           (10 +10)

1. a). State and establish Bhattacharyya inequality.

b). Write short notes on Bayes estimation.                                                    (12 + 8)

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## Loyola College B.Sc. Statistics April 2007 Distribution Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.  DEGREE EXAMINATION – STATISTICS

 AC 15

FOURTH SEMESTER – APRIL 2007

ST 4501  – DISTRIBUTION THEORY

Date & Time: 24/04/2007 / 9:00 – 12:00         Dept. No.                                                     Max. : 100 Marks

Section A

Answer ALL the questions                                                                                  (10×2=20)

1. State the properties of a distribution function.
2. Define conditional  and marginal distributions.
3. What is meant by ‘Pair-wise Independence’ for a set of ‘n’ random variables.
4. Define Conditional Variance of a random variable  X given a  r.v. Y=y.
5. Define Moment Generating Function(M.G.F.) of a  random variable X.
6. Examine the validity of the statement “ X is a Binomial variate with mean 10 and standard deviation 4”.
7. If X is binomially distributed with parameters n and p, what is the distribution of

Y=(n-x)?

1. Define ‘Order Statistics’ and give an example.
2. State any two properties of Bivariate Normal distribution.
3.  State central limit theorem.

Section B

Answer any FIVE questions                                                                                                     (5×8 =40)

1. Let (X,Y) have the joint p.d.f. described as follows:

(X,Y) : (1,1)   (1,2)   (1,3)   (2,1)   (2,2)   (2,3)

f(X,Y) :  2/15   4/15    3/15    1/15   1/15   4/15. Examine if X and Y are independent.

1. The joint p.d.f. of  X1 and X2 is : f(x1, x2) =.

[a] Find the conditional p.d.f. of X1 given X2=x2.

[b] conditional mean and variance of X1 given X2=x2

1. Derive the recurrence relation for the probabilities of Poisson distribution
2. Obtain Mode of Binomial distribution
3. Obtain Mean deviation about mean of  Laplace distribution
4. The random variable X follows Uniform distribution over the interval(0,1). Find the

distribution of Y = -2 log X.

1. Obtain raw moments of Student’s ‘t’ distribution, Hence fine the Mean and the Variance.
2. Let Y1, Y2, Y3, and Y4 denote the order statistics of a random sample of size 4 from a

distribution having p.d.f.   f(x) =.       Find P (Y3 > ½ ).

Section C

Answer any TWO questions only                                                                                 (2×20 =40)

19.[a]  The random variables X and Y have the joint p.d.f. f(x, y) =.

Compute Correlation co-efficient between X and Y.                                                       [15]

[b] Let X and Y are two r.v.s with the p.d.f. f(x, y) =.

Examine whether X and Y are stochastically independent.                                            [5]

20 [a] Derive M.G.F. of Binomial distribution and hence find its mean and variance.

[b] Show that  E(Y| X=x) = (n-x) p2/ (1-p1),if (X,Y) has a Trinomial distribution with

parameters n, p1 and p2.

1. [a] Prove that Poisson distribution is a limiting case of Binomial distribution, stating the

assumptions involved.

[b] Let X and Y follow Bivariate Normal distribution with;m1=3  m2=1  s12 =16  s22 =25 and

r = 3/5.

Determine the following probabilities: (i) P[(3<Y<8)|X=7]   (ii) P[ (-3<X<3) | Y=4]

1. [a[ If X and Y are two independent Gamma variates with parameters m and n respectively.

Let U=X+Y and V= X / (X+Y)

[i] Find the joint p.d.f. of  U and V

[ii] Find the Marginal p.d.f.s’ of U and V

[iii] Show that the variables U and V are independent.                                                [12]

[b] Derive the p.d.f of F-variate with (n1, n2) d.f.                                                            [8]

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## Loyola College B.Sc. Statistics April 2007 Design And Analysis Of Experiments Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AC 22

B.Sc. DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – APRIL 2007

ST 6600 DESIGN AND ANALYSIS OF EXPERIMENTS

Date & Time : 16.04.2007/9.00-12.00      Dept. No.                                                      Max. : 100 Marks

SECTION A

Answer ALL questions. Each carries TWO marks.                                                 10 X 2 = 20

1. Give an example of a Contrast.
2. Write the number of error degrees of freedom in a Latin Square Desgin of order 4.
3. When do you recommend RBD instead of CRD ?
4. Give the model representing one way classified data.
5. Write all possible treatment combinations in a design.
6. What is confounding ?
7. Give a consistent set of values for the parameters involved in a BIBD.
8. Write the contrast defining highest order interaction in design.
9. What is a generalized effect ?
10. What is an Incidence matrix ?

SECTION B

Answer any FIVE. Each carries EIGHT marks.                                                        5 X 8 = 40

1. Estimate the block effects in two way statistical model.
2. Show that the mean of available values can be taken as the missing value when a single observation is missing in CRD.
3. Explain the preparation of a Randomised Block design with 4 blocks and 3 treatments.
4. Explain how various sums of squares are computed in design.
5. Explain Yates method of computing various sums of squares in design.
6. Show that (under usual notations).
7. Illustrate with an example of your choice how partial confounding is executed in  designs.
8. Obtain the factorial effect of the highest order interaction in three different ways known to you in the case 23 design.

SECTION C

Answer any TWO. Each carries TWENTY MARKS marks.                                2 X 20 = 40

1. Explain the analysis of RBD in detail.
1. With the help of a suitable statistical model illustrate how block differences are

made to  contain a confounded treatment effect.

1. (a) Explain how confounding is done in designs

(b) Show that is always non-singular where is the incidence matrix of a                 BIBD.

1. Explain the analysis of BIBD .

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## Loyola College B.Sc. Statistics April 2007 C And C ++ Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 21

FIFTH SEMESTER – APRIL 2007

# ST 5401 – C AND C ++

Date & Time: 04/05/2007 / 1:00 – 4:00 Dept. No.                                              Max. : 100 Marks

## Answer all the questions.                                                                        10 X 2 = 20

1. Explain briefly “ scanf” and “printf” with an illustration.
2. Write a program to output the following multiplication table.

5 X 1 = 5

5 X 2 = 10

……….

……….

5 X 10 = 50

1. Explain decrement operator with an example.
2. Write a program to convert the Fahrenheit to Celsius the following conversion formula: C = (F-32)/ 1.8.
3. Identify syntax errors in the following program. After correcting, what output would you expect when you execute it?

Main ()

{

int r,c;

float perimeter: area,

c=3.1415;

r=5;

perimeter = 2.0 c*r;

area= c*r*r;

printf(“%f”,”%d”,&perimeter, &area)

}

1. Mention any one salient feature of C++ when compared to C.
2. What is the output of the following program?

main ()

{

int a = 2, n=100;

while(a<n)

{

cout<<a<<endl;

a=a*a;

}

}

1. Write a program to find the largest number of the two numbers in C++.
2. What is function in C++? And give an example for function.

1. What is the output, corresponding to the following segment?

a=10;

b=15;

x=(a>b)?a:b;

cout<<x;

# Answer any five questions.                                                               5 X 8 = 40

1. Explain different types of constants in c programming.
2. The traveling time by train from one city to another city is given in terms of minutes (eg. 280 minutes). Write a c program that reads the given value, converts it into hours: minutes format and display the same.
3. When the principal, rate of interest and period of deposit are given, write a c program to compute the simple interest and compound interest using function.
4. There are three categories (A, B, C) of employees in a company. They are charged 30%, 20% and 15% respectively of their salaries as income tax. Write a c program to find the income tax to be paid by him.
5. The commission on a representative’s sales is calculated as follows:

a). if sales<Rs. 500, then there is no commission.

b). if Rs. 500 < sales < Rs. 5000, then commission = 10% of sales.

c). if sales > Rs. 5000, then commission = 5000 + 8% of sales .

Write a C++ program which reads in total sales and calculates commission.

1. Write a C++ program to calculate perimeter with help of function overloading using the following conditions.

a). when two sides are equal then perimeter = 2(a+(a2 – h2)) units.

b). when three sides are known but not equal perimeter = a + b +c

1. Write a C++ program to calculate addition of two matrices with the order of 4X4.
2. Develop a function in C++  and calculate the factorial value of an integer n. Using this in the main () function to compute  ncr.

# Answer any two questions.                                                               2 X 20 = 40

1. a). Explain switch statement in C and give an example with the same.(5+5)

b). Write a C program to calculate mean and variance for N values with help of

array.                                                                                                         (10)

1. a). Explain nested if statement with suitable example.                                   (5+5)
2. b). Explain one and two-dimensional arrays with examples.                          (5+5)

b). Write a C++ program with help of function overloading to find out the largest number among two and three numbers.                                                             (15)

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## Loyola College B.Sc. Statistics April 2007 Basic Sampling Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 AC 14

FOURTH SEMESTER – APRIL 2007

# ST 4500 – BASIC SAMPLING THEORY

Date & Time: 21/04/2007 / 9:00 – 12:00 Dept. No.                                              Max. : 100 Marks

SECTION – A

——————-

Answer ALL questions                                                 ( 10 x 2 = 20 marks)

1. What is meant by Census ? What are the constraints for carrying out a census?
2. If T1 and T2  are unbiased for θ, then show that one can construct uncountable number of unbiased estimators for θ.
3. Explain how a sample of size `n’ is drawn in SRSWOR using Lottery method.
4. In SRSWOR, let yi denote the y-value of the ith drawn unit. Find the discrete probability distribution of yi .
5. In PPS sampling, find the probability of selecting   ith   population unit in a given draw.
6. Show that under SRSWOR is more efficient than  under SRSWR.
7. Write all possible linear systematic samples , when N = 12 and  n = 4.
8. Describe Centered Systematic Sampling Scheme.
9. Compute the number of units to be sampled from each stratum when there are 4 strata of sizes 40, 30, 60 and 70. The total sample size is 40.
10. State V (st ) under proportional allocation for a given sample size.

SECTION – B

——————-

Answer any FIVE questions                                            ( 5 x 8 = 40 marks)

1. Show that an estimator can become biased under one sampling scheme even though it is unbiased under another sampling scheme.
2. Under usual notations, derive cov (yi , yj ) ; i ≠ j , in SRSWOR.
3. Using the probability of selecting a subset of the population as a sample, prove that sample mean is unbiased for population mean.
4. Show that Lahiri’s method of selection is a PPS selection.
5. A population contains 5 units. It is known that

Yi / Pi   –  Y)2 P i  =  100 .  Under PPSWR, compare

‘   =        and    ”   =   .

1. Deduct the formula for ,  V()   and  v() in SRSWR using the formula  for  ,  V()  and   v() available in PPSWR.
2. Describe circular systematic sampling with an example.
3. Derive values of nh  such that co  +  is minimum for a given

value of  V (st).

SECTION – C

——————-

Answer any TWO questions                                            ( 2 x 20 = 40 marks)

1. ( a ) In SRSWOR, derive V () by considering all possible samples and their

corresponding probabilities.                                            ( 14 )

( b ) Let ν denote the number of distinct units in a simple random with replacement

sample of size 3 drawn from a population containing 4 units.  Find   P(ν =1) ,

P(ν =2)   and   P(ν = 3).                                                        ( 6 )

1. ( a ) In SRSWOR, derive V () using probabilities of inclusion. ( 10 )

( b ) In CSS, assuming linear trend, prove the following :

( i )  The sample mean coincides with population mean when

k is odd.

( ii ) The sample mean is unbiased for population mean when

k is even.                                                                 ( 10 )

1. ( a ) Compare the mean based on distinct units with the sample mean under

( i )  SRSWR ,

( ii ) SRSWOR ,  taking   N  =  4  and  n  =  3.                           ( 8 )

( b ) Show that  s2 / n  is an unbiased estimator of   V()  under SRSWR.    ( 12 )

1. With 2 strata, a sampler would like to have n1 = n 2 for administrative convenience instead of using the values given by the Neyman allocation. If V and Vopt denote the variances given by n1 = n2 and the Neyman allocation respectively, show that  2  , where

r  =  as given by Neyman allocation.

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## Loyola College B.Sc. Statistics April 2007 Applied Stochastic Processes Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AC 20

B.Sc.  DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5400APPLIED STOCHASTIC PROCESSES

Date & Time: 03/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

SECTION A

ANSWER ALL QUESTIONS.                                                                               (10 X 2 =20)

1. Give an example for a discrete time and continuous state space stochastic process
2. Define : Covariance Stationary.
3. Give an example of an irreducible Markov chain.
4. Give an example for a  stochastic process having “Stationary Independent Increments” ?
5. Mention the usefulness of classifying a set of states as closed or not in a Markov chain.
6.  Define the term : Mean recurrence time.
7. When do you say a given state is “positive recurrent” ?.
8. Give an example for symmetric random walk…
9. Under what condition pure birth process reduces to Poisson process ?
10. For what values of p and q the following transition probability matrix becomes a doubly stochastic matrix ?

SECTION B

Answer any FIVE questions                                                                      (5 X8 =40)

1. Consider the process where and are uncorrelated random variables with mean 0 and variance 1. Find mean and varaince functions and examine whether the process is covariance stationary

1. Show that every stochastic process with independent increments is a Markov process

1. Show that a Markov chain is completely determined if its transition probability matrix and the distribution of is known.

1. A player chooses a number from the set of all non negative integers. He is paid an amount equivalent to the number he gets. Write the TPM corresponding to his earnings, given that probability of getting the number is
2. Consider the following Transition Probability Matrix explaining seasonal changes on successive days (S- Sunny, C-Cloudy)

Today

(S,S)  (S,C)      (C,S) (C,C)r

(S,S)    0.8       0.8       0          0

Yesterday        (S,C)    0          0          0.4       0.6

(C,S)    0.6       0.4       0          0

(C,C)   0          0          0.1       0.9

Compute the stationary probabilities and interpret your results

1. A radioactive source emits particles at a rate of 5 per minute in accordance with a poisson process. Each particle emitted has a probability of 0.6 of being recorded, Find in a 4 minute interval the probability that the number of particles recorded is 10..

1. Given the transition probability matrix corresponding to the Markov chain with states {1,2,3,4}Find the probability distribution of n which stands for the number of steps needed to reach state 2 starting from the same and also find its mean. Offer your comments regarding the state 2

1. Obtain the differential equation corresponding to Poisson process

SECTION C

Answer TWO questions.                                                                            (2 X 20 =40)

1. (a) Let be a sequence of random variables with mean 0 and is a Martingale.    (10)

(b) Show that every stochastic process with independent increments is a Markov process.

1. (a)  Show that for a process with independent increments is linear in t                                                                                                 (12)

(b) State and prove the reproductive property of Poisson process     (8)

1. Explain the postulates of Yule-Furry process and find an expression for
2. Write short notes on the following

(a) Stationary Distribution      (5)

(b) Communicative sets and their equivalence property (5)

(c) Interarrival time in Poisson process (5)

(d) Periodic States (5)

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## Loyola College B.Sc. Statistics April 2007 Applied Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 AC 18

DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5502APPLIED STATISTICS

Date & Time: 30/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

SECTION– A

Answer ALL questions.                                                              (10 x 2 = 20 marks )

1. Define a Time Series and give two examples.
2. What are the merits and drawbacks of Graphic method of studying Trend ?
3. Describe the Simple Aggregate Method of calculating Price Index Number and write the drawbacks of this method.
4. State the mathematical tests which are used for measuring formula error in the construction of index numbers.
5. What is meant by Base shifting of Index Numbers?
6. Explain Census Method of obtaining Vital Statistics. What is the main drawback of this method?
7. Explain Crude Death Rate used in measuring mortality.
8. Describe partial correlation with an example.
9. Write a note on economic census.
10. Briefly explain financial statistics.

SECTION –B

Answer any FIVE questions.                                                                    (5 x 8 = 40 marks)

1. Briefly explain the Trend Component in a Time Series.
2. Describe Fitting of Straight Line and Exponential Curve by Least  Square
3. The Seasonal indices of the sale of readymade garments of a particular type in a certain store are

given below :

Quarter                                    Seasonal Index

Jan. – March                                       98

Apr – June                                          89

July – Sep.                                          82

Oct. – Dec.                                        130

If the total sales in the first quarter of the year be worth Rs. 10,000, determine how much worth

of garments of this type should be kept in  stock by the store to meet the demand in each of the

remaining quarters.

1. Given the data :

——————————————————————

Commodity             p0                q0            p1        q1

——————————————————————

A                   1               10             2             5

B                   1                 5             x             2

——————————————————————

where p and q respectively stand for price and quantity and the subscripts stand for time period.

Find  x , if the ratio between Laspeyre’s  (L)  and Paasche’s  (P) index numbers is L : P : : 28 : 27

1. Discuss the uses of cost of living index number.

1. What is the purpose of standardizing death rates? Describe the direct method of standardization.

1. Given the following table for lx ,  the number of rabbits living atage x ,

complete the life table for rabbits.

x …        0         1        2         3        4         5           6

lx . . .       100     90      80       75      60       30         0

1. Discuss in detail about livestock and poultry statistics.

PART – C

Answer any TWO questions.                                                     (2 x 20 = 40 marks)

1. Given the three selected points U1 , U2, and U3 corresponding to t1 = 2, t2 = 30 and t3  = 58 as follows:

t1   = 2,    U1 = 55.8

t2   = 30,  U2 = 138.6

t3   = 58,  U3 = 251.8

Fit the Logistic curve by the method of selected points. Also obtain the

trend values  for   t  =  5, 18, 25, 35, 46, 50, 54, 60, 66, 70

1. (a) A price index number series was started with 1972 as base. By 1976 it rose by 25%. The index number for 1977 was 118.75. In this year a new series was started. This new series rose by 15 points in the next year. But during the following four years the rise was not rapid. During 1982 the price level was only 5% higher than 1980 and in 1980 these were 8% higher than 1978. Splice the two series and calculate the index numbers for the various years by shifting the base to 1978. (10 marks)

• You are given the inventory position of a company for six years. Find out the index number of physical volume of inventory . Comment upon the nature of the inventory position.

Year                  1977        1978        1979       1980        1981        1982

Inventory

(in ‘000 Rs)      425.6        447.8     472.4        492.6      524.7       540.8

Wholesale

Price Index

(1971=100)       108.2        121.5     158.0        173.9      162.6     181.5            (10 marks)

21 ( a ) Describe the Registration Method of obtaining Vital Statistics.

Discuss the shortcomings of this method.              ( 12 marks )

( b )  Estimate the standardized death rates for the following two  countries :

Age Group               Death Rate per 1000                    Standardized

( in years )               Country A      Country B           Population ( in lakhs)

——————————————————————————————–

0  –  4                   20.00                 5.00                         100

5  –  14                   1.00                 0.50                         200

15  –  24                   1.40                 1.00                         190

25  –  34                   2.00                 1.00                         180

35  –  44                   3.30                 2.00                         120

45  –  54                   7.00                 5.00                         100

55  –  64                 15.00               12.00                           70

65  –  74                 40.00               35.00                           30

75 & above          120.00             110.00                           10                               (8 marks )

1. (a)  On the basis of observations made on 39 cotton plants ,  the total

correlation of yield of cotton ( X1  ), number of seed vessels ( X2 ) and

height ( X3  ) are found to be :

r12  =  0.8 ,   r13  =  0.65  and  r23  =  0.7

Compute the partial correlation between yield of cotton and the number

of seed vessels eliminating the effect of height.                                                    (5 marks)

• The following are the zero-order correlation coefficients :

r12   =  0.98 ,  r13  =  0.44  and  r23  =  0.54

Calculate multiple correlation coefficient treating first variable as dependent variable , second and third variables as independent variables.                                                                   (5 marks)

( c ) Describe the main functions of National Sample Survey Organisation.  (10 marks)

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## Loyola College B.Sc. Statistics Nov 2007 General Economics Question Paper PDF Download

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