B.Sc Corporate Statistics Question Paper 2008

Loyola College B.Sc. Statistics Nov 2008 Statistical Mathematics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 08

 

THIRD SEMESTER – November 2008

ST 3501 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

 

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

 

  1. Define lower sum of a function for a given partition of a closed interval [a, b].
  2. Give an example of a non-monotonic function that is integrable.
  3. Define probability density function (p.d.f.) of a random variable (r.v).
  4. Give one example each for improper integrals of I and II kinds.
  5. Show that the improper integral  converges absolutely.
  6. Find the value of
  7. If L[f(t)] = F(s), express L[ tn F(t)] in terms of F.
  8. State the general solution for the linear differential equation  + Py = Q
  9. Define linearly dependent vectors.
  10. Define a positive definite quadratic form.

 

SECTION – B

 

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

 

  1. For any partition P of [a, b], show (under usual notations) that

m (b – a ) ≤ L(P , f)  ≤ U(P , f) ≤ M (b – a )

 

  1. State (without proof) a necessary and sufficient condition for R-integrability of a function in a closed interval [a, b]. Using this condition, show that if g R[a, b] and if g is bounded away from 0, then 1/g  R[a, b]
  2. If a r.v. X has p.d.f. f(x) = ,  – ∞ < x < ∞, find the moment generating function of X. Hence find its mean and variance.
  3. Show that: (a)  is divergent (b)   is convergent
  4. Find L[ cos at] and hence find L[ sin at] using the result for Laplace transform of the derivative of a function.

 

  1. Solve: (x + y)2 d x = 2 x2 d y.

 

  1. Find any non-trivial solution which exists for the following system of equations:

2 x1 +3 x2 – x3 + x4 = 0

3 x1 + 2 x2 – 2 x3 + 2 x4 = 0

5 x1 – 4 x3 + 4 x4 = 0

 

  1. Show that if λ is a characteristic root of A, then λn is a characteristic vector of An with the same associated characteristic vector. Establish a similar result for A– 1.

 

SECTION – C

 

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

 

  1. (a) State and prove the First and Second Fundamental Theorems of Integral Calculus.

 

(b) If f(x) = c x2, 0 < x < 2, is the p.d.f. of a r.v X, find ‘c’ and P( 1 < X < 2)

(16+4)

  1. (a) Show that  is divergent

(b) Discuss the convergence of gamma integral.                                                         (8+12)

 

  1. (a) Evaluateover the region between the parabola y = x2 and the line x+y =2.

(b) Evaluate  over the half circle x2 + y2 ≤ a2 with y ≥ 0.

(10+10)

 

  1. (a) State and prove Cayley-Hamilton Theorem.

 

(b) Show that a polynomial of degree ‘n’ has (n +1) distinct real roots if and  only

if all its coefficients are zero.                                                                                        (12+8)

 

 

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Loyola College B.Sc. Statistics Nov 2008 Probability And Random Variables Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 02

 

   B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

Answer ALL the Questions                                                               (10×2=20 Marks)

 

  1. What do you mean by a random experiment?
  2. Using Axioms of Probability, show that P(Ac∩B) = P(B)-P(A∩B)
  3. A bag contains 3 white and 6 green balls. Another bag contains 6 white and 5 green balls. A ball is chosen from each bag. What is the probability that both will be green?
  4. If the letters of the word ‘REGULATION’ are arranged at random, what is the chance that there will be exactly 4 letters between R and E?
  5. A speaks truth in 60 percent cases and B in 70 percent cases. In what percentage of cases are they likely to contradict each other in stating the same fact?
  6. Define mutually independent events and pairwise independent events.
  7. The odds against Manager X settling the wage dispute with the workers are 8:6 and odds in favour of Manager Y settling the same dispute are 14:16. What is the chance that neither settles the dispute if they both try independently of each other?
  8. A problem in statistics is given to three students A, B and C, whose chances of solving it are 0.5, 0.75 and 0.25 respectively. What is the probability that the problem will be solved, if all of them try independently?
  9. Define Random Variable.

10.For the following probability distribution

  1. X : 1          2          3
  2. p(x): 1/2       1/3       1/6

Find variance of X.

 

SECTION – B

Answer any FIVE Questions                                                             (5×8 = 40 Marks)

 

  1. A committee of 4 people is to be appointed from 3 officers of production department, 4 officers of purchase department, 2 officers of sales department and 1 chartered accountant. Find the probability of forming the committee in the following manner

(a). There must be one from each department.

(b). It should have atleast one from the purchase department.

(c). A chartered accountant must be there in the committee.                        (2+4+2)

  1. (i). An MBA applies for a job in two firms X and Y. The probability of his being selected in the firm X is 0.7 and being rejected at Y is 0.5. The probability of atleast one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the two firms?

(ii). If two dice are thrown, what is the probability that the sum is neither 7 nor 11?                                                                                                                                                                    (4+4)

  1. For any three events A, B and C, Prove that

P(AUB/C) = PA/C)+P(B/C)-P(A∩B/C)

  1. Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys and 1 girl and 3 boys. One child is selected at random from each group. What is the chance that three selected consists of one girl and two boys?
  2. A bag contains 17 tickets marked with the numbers 1 to 17. A ticket is drawn and replaced, a second drawing is made. What is the probability that (a) the first number drawn is even and the second is odd. (b) the first number is odd and the second is even? Find the corresponding probabilities under without replacement. (2+2+4)
  3. State and prove addition theorem on expectation when the random variable are continuous.
  4. State and prove Chebchev’s inequality.
  5. An urn contains four tickets marked with numbers 112, 121, 211, 222 and one ticket is drawn at random. Let Ai (i=1, 2, 3) be the event that ith digit of the number of the ticket drawn is 1. Discuss the independence of the events A1, A2 and A3.

 

SECTION – C

Answer any TWO Questions                                                             (2×20 = 40 Marks)

 

19.(i). State and prove Multiplication Theorem of Probability for n events.

(ii) The odds that a book on statistics will be favourably reviewed by three independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that of the three reviews: (a) all will be favourable (b) exactly two reviews will be favourable (c) atleast one of the reviews will be favourable.                  (10+10)

  1. (i). A and B play for a prize of Rs. 1000. A is to throw a die first and is to win if he throws 6. If he fails, B is to throw and is to win if he throws 6 or 5. If he fails, A is to throw again to win if he throws 6, 5 or 4 and so on. Find their respective expectations.

(ii). Two persons X and Y appear in an interview for two vacancies in the same post. The probability of X’s selection is 1/7 and that of Y’s selection is 1/5. What is the probability that (a) both of them will be selected (b) only one of them will be selected (c) none of them will be selected (d) atleast one of them will be selected?     (10+10)

  1. (i). State and prove Baye’s Theorem.

(ii). A factory produces a certain type of output by three types of machines. The respective daily production values are

Machine I : 3000 units      Machine II : 2500 units           Machine III : 4500 units

Past experience shows that 1% of the output produced by machine I is defective, the corresponding fraction of defectives for other two machines are 1.2% and 2% respectively. An item is drawn at random from the day’s production run and is found to be defective. What is the probability that it comes from machine II.    (10+10)

  1. An experiment consist of three independent tosses of a fair coin. Let X = The number of heads, Y = The number of head runs and Z = length of head runs. A head run being defined as consecutive occurrence of atleast two heads, its length being the number of heads occurring together in three tosses of the coin. Find the probability functions of (a) X (b) Y (c) Z. Also compute the mean and variance of the number of heads.

 

 

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Loyola College B.Sc. Statistics Nov 2008 Estimation Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 10

 

FIFTH SEMESTER – November 2008

ST 5500 – ESTIMATION THEORY

 

 

 

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions.                                                                              10 X 2 = 20

  1. Define Consistent estimator. Give an example.
  2. Give two examples for unbiased estimator.
  3. Define UMVUE.
  4. Describe the concept of Bounded completeness.
  5. Describe the Method of Minimum Chi-square estimation.
  6. State an MLE ofλ based on a random sample of size n form a Poisson Distribution

with parameter λ.

  1. Describe the concept of Baye’s estimation.
  2. Define Loss function.
  3. Describe the Method of Least Squares.
  4. Define BLUE.

SECTION – B

Answer Any FIVE questions.                                                                                       5 X 8 = 40

  1. Derive an unbiased estimator of , based on a random sample of size n form B (1,).
  2. Let { Tn = 1, 2,3, ….. } be a sequence of estimators such that

and  .Then show that Tn is

consistent for .

  1. If is a random sample from P (λ),, then show that

is a sufficient statistic for .

  1. Show that the family of Binomial distributions {B (1,).0 < θ < 1} is complete.
  2. Describe estimation of parameters by “Method of Maximum Likelihood”
  3. Describe any two properties of MLE, with examples.
  4. Explain prior and posterior distributions.
  5. Derive the least square estimator of β1 under the model Y = β0 + β1X+

 

SECTION – C

Answer any TWO questions.                                                                          2 X 20 = 40

  1. a. State and prove Chapman-Robbin’s inequality. [12]

b Using Factorization theorem derive a sufficient statistic for μ based on a random

sample of size n from N (μ, 1), MϵR                                                                      [8]

 

  1. a. State and prove a necessary and sufficient condition for an unbiased estimator to be a

UMVUE.                                                                                                                      [15]

  1. If T1 and T2 are UMVUES of y1(q) and  y2(q) respectively, then show that T1+T2 is the UMVUE of  y1(q) and  y2(q).                                                                                                                [5]

 

 

21 a.  Explain the concept of estimation by the method of modified minimum chi-square.  [8]

  1. Let be a random sample from a distribution with density function

f (x, θ) =                                                            [12]

Find the maximum likelihood estimator of  and examine whether it is consistent.

 

  1. Explain: i) Risk function.              ii) Method of Moments

 

iii) Completeness     iv). Gauss –Markov model                           [ 4 x 5 ]

 

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Loyola College B.Sc. Statistics Nov 2008 Econometric Methods Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 18

 

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – November 2008

ST 5405 – ECONOMETRIC METHODS

 

 

 

Date : 14-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

            PART A

                                                                                                                                                                                                                                                                                                                                              

Answer ALL questions                                                                                                         (10 x 2 = 20)

 

  1. Define the term ‘Econometrics’.
  2. What is meant by Population regression line?
  3. Is the model Y = β1 + β2eX + u linear with respect to X? Why or why not?
  4. Interpret the following confidence interval.

                   Pr (-1.5 < β2 < 2.2) = 0.95, where β2 is the regression coefficient in a

      two variable linear model.

  1. Mention any two properties of OLS estimators.
  2. For a two variable linear model, the sample regression function is found to be Y = 1.7 + 4.5X + e. Interpret the regression equation.
  3. The observed and predicted values of the dependent variable Y in a linear model is as follows:

                  Observed Y:  14         12        20        14        11

                  Predicted Y:  11         10        18        13        14

      Find the value of R2.

  1. What is meant by multicollinearity?
  2. When are two regression equations said to be parallel and dissimilar?
  3. How is the ‘bench mark category’ interpreted in a regression model involving

      dummy independent variables?      

 

 

PART B

 

Answer any FIVE questions                                                                                                            (5 x 8 = 40)

 

  1. Explain the concept of ‘Population regression function’ through an example.
  2. Mention the assumptions underlying the classical linear regression model.
  3. Suppose that a researcher is studying the relationship between grade points on

     exam(Y) and hours studied for the exam(X) for a group of 20 students.   

     Analysis of the data reveals the following:

                 Sum of Y = 1600:      Sum of X = 400

                 Sum of xy = 1800;     sum of x2 = 600

Residual sum of squares = 43,200 where x and y are the deviations of X and Y from their respective means.

     Find the following:

  • Mean of X and Y
  • Least squares intercept and slope
  • Standard error of regression
  • Standard error of slope coefficient
  1. For a two variable linear model, derive the variance of the OLS estimators.
  2. Describe the method of testing the overall significance of a regression

             model.

  1. Find the residual sum of squares for the following data by assuming a linear

            model of Y on X.

.                       Y: 10          12                         15        17        13

                        X: 1.3         1.5            1.7       2.0       1.6.

  1. Explain the various remedial measures available to overcome the problem of multicollinearity. 

 

 

  1. Consider the following OLS regression results with standard errors in

parenthesis:

           S = 12,000 – 3000X1 + 8000(X1 + X2)          

                                  (1000)    (3000)                        n = 25

where S = annual salary of economists with B.A. or higher degree

           X1 = 1 if M.A. is highest degree; 0 otherwise

           X2 = 1 if Ph.D is highest degree; 0 otherwise

  • What is S for economists with a M.A. degree?
  • What is S for economists with a Ph.D degree?
  • What is the difference in S between M.A.’s and Ph.D’s?
  • At 5% level of significance, would you conclude that Ph.D’s earn more per year than M.A.’s?

 

PART C

 

Answer any TWO questions                                                                                                (2 x 20 = 40)

 

  1. Consider the following data on Y and X:

                 Y: 2.57    2.50    2.35    2.30    2.25    2.20    2.11    1.94

                 X: 0.77    0.74    0.72    0.73    0.76    0.75    1.08    1.81

      Fit a linear model of Y on X and test the hypothesis that the intercept and 

      slope coefficients are statistically significant at 5 % level.

 

  1. ) Explain the method of constructing a 100(1-α) % confidence interval for
    the slope parameter in a two variable regression model.

      b.) What is a ‘k’ variable linear model? Give an example for the same.

      c.) Explain the method of computing r2 based on a two variable linear model.

                                                                                                                                                           (10+4+6)

  1. a.)  Consider the following ANOVA table based on a linear regression:

                                           Source                 df        Sum of squares

                                           Regression           4          ?

                                           Residual              ?          128

                                           Total                    19        500

                     1.) Find the missing values.

                     2.) Compute the F-ratio and test the hypothesis that R2 is significantly

                          different from zero at 5 % level.

                     3.) Write the form of the PRF.

                     4.) Obtain an estimate for the variance of the disturbance term.

 

               b.) Consider the following results based on a linear regression:

                    The estimated regression line is

                                            = 53.428 + 0.895 X1 – 0.926 X2         R2 = 0.167

                                                   (11.462)  (0.607)       (0.607)           n = 16

                    where the numbers in the parenthesis denote the standard errors.

                    Does the above information reveal the presence of multicollinearity in the

                    sample data? Justify your answer.                                                                              (10+10)

 

  1. a.) Consider the following data on annual income (in 000’s $) categorized by gender and age.

                                      Income: 12         10       14     15        6       11       17

                                      Gender:  0            1          1      0         0         1         1

                                                 Age:  1            1          0      1         0         0         1

                   where Gender = 1 if male; 0 if female

                                   Age = 1 if less than or equal to 35; 0 if greater than 35

                      Perform a regression of Income on Gender and age. Interpret the results.

 

               b.) Define Heteroscedasticity. Explain the Spearman’s rank correlation test to detect the      

                     presence of   heteroscdasticity.                                                                                    (10 + 10)

 

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Loyola College B.Sc. Statistics Nov 2008 Computational Statistics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

BA 13

 

FIFTH SEMESTER – November 2008

ST 5503 – COMPUTATIONAL STATISTICS

 

 

 

Date : 10-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL the Questions. Each question carries 20 marks.

 

  1. The table given below presents the data for complete census of 2010 farms in a region.  The farms were stratified according to farm-size in acres into seven strata. The number of farms in the different strata, the strata standard deviation are given below. Find the sample sizes under each stratum under (i) Proportional allocation  and (ii) Neyman Optimum allocation.
Stratum No. No. of farms Stratum standard deviation
1 394 8.3
2 461 13.3
3 391 15.1
4 334 19.8
5 169 24.5
6 113 26
7 148 35.2

 

 

 

 

 

 

 

 

 

(OR)

In a population size N = 6, the values of Yi are 24, 56, 12, 45, 25, 55. Calculate the sample mean for all possible simple random samples (without replacement) of size n = 2, also calculate s 2 for all samples and verify E (s 2 ) = S 2.

 

  1. Calculate fixed base index numbers and chain base index numbers for the

following    data.

 

Commodity 2002 2003 2004 2005 2006
I 2 3 5 7 8
II 8 10 12 4 18
III 4 5 7 9 12

 

 

 

 

 

 

 

(OR)

 

Calculate seasonal variations given the average quarterly price of a commodity for

5 years by ratio to trend method.

 

Year I Quarter II Quarter III Quarter IV Quarter
2001 28 22 22 28
2002 35 28 25 36
2003 33 34 30 35
2004 31 31 27 35
2005 37 36 31 36

 

 

 

  1. (a) Glaucoma is an eye disease that is manifested by high intraocular pressure. The

distribution of intraocular pressure in the general population is known to be normal

with mean 16 mm Hg and standard deviation 3 mm Hg. Pressures in the range of 12

mm Hg to 20 mm Hg are considered safe. What percentage of the population is

unsafe?

&

(b)  The scores in a certain test from 12 men and 10 women candidates are reported below:

Men:      56, 67, 45, 78, 86, 64, 78, 88, 91, 46, 45, 84

Women:  67, 48, 91, 75, 58, 90, 46, 69, 70, 82

Test whether there is significant difference in the average scores of the two groups at 5% level of significance. (Variances are considered to be equal but are not known)

(6 + 14)

(OR)

(c) In a pediatric clinic a study is carried out to test the effectiveness of aspirin in

reducing temperatures. The temperatures of twelve five-year old children suffering

from influenza were observed before and after one hour of administering aspirin and

the paired observations are reported below:

 

Patient Temp ( 0F) before

taking  aspirin

 Temp (0F) 1 hr

after taking aspirin
1

2

3

4

5

6

7

8

9

10

11

12

 102.4

103.2

101.9

103.0

101.2

100.7

102.5

103.1

102.8

102.3

101.9

101.4

 99.6

100.1

100.2

101.1

99.8

100.2

101.0

100.1

100.7

101.1

101.3

100.2

 

Test whether aspirin is effective in reducing the temperature at 5% level of

significance.

                                                                   &                                                       

(d) A textile mill attempts to control the yarn defects that appear on manufactured

cloth. The occurrence of defects has been found to follow Poisson law and the

historical average number of defects per 100 m of cloth is 1.25. Recently, due to

changes implemented by the HR department, the occurrence of defects is expected

to reduce significantly. The quality control department wishes to test whether this

improvement has happened. The following numbers of defects were observed from

12 bales of cloth (each of length 100 m): 1, 2, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1. Carry out the

relevant test at 5% level of significance and report your findings to the quality

control department.                                                                                           (8 +12)

 

  1. (a) The daily iron intake among children in the age group 9 – 11 has been

established to be normally distributed with average14.4 mg and standard deviation

4.75 mg. A social service organization working among ‘Below-Poverty-Line’

(BPL) families, took a sample of 50 children and reported that the average intake

among them was 12.5 mg. It is believed that the standard deviation is not different

(from the known 4.75 mg) for this group. Does this information indicate that the

iron intake is significantly lower in the BPL families?

&

 

 

(b) In a study on infarct size among people, the following data were obtained from 6

treated children and 8 untreated children:

Treated Children:       20.3, 21.0, 21.4, 18.9, 19.5, 19.7

Untreated Children:   19.6, 18.7, 19.9, 20.6, 20.1, 19.6, 19.0, 21.0

Test whether the variances differ significantly at 2% level of significance

(6 + 14)

(OR)

(c) A small factory A produces fasteners for use in machine tools and supplies it to a machine tool manufacturing company B. Factory A supplies fasteners in large lots every day and maintains a low percentage of 5% defectives. Company B carries out sampling inspection every day by taking a sample of 20 fasteners. It rejects the entire lot and sends it back to Factory A if more than one defective is observed in the sample. Find the proportion of days in which lots are sent back to factory A.

&

(d) The dispersion in the quality characteristic is an important indicator of the quality conformance of a production process. It has been historically found that a production process was operating with a variance of 18 for a normally distributed quality characteristic, But, due to certain changes made in the processes, it is believed that there could be a change in the process dispersion. The following data on the quality characteristic are available after the changes were implemented:

65, 60, 67, 70, 67, 62, 68, 63, 59, 69, 70, 58, 75, 75, 78

Test whether there is a significant change in the process variance at 1% level of

significance.                                                                                                      ( 8 +12)

 

  1. The following are the weight gains (pounds) of two random samples of young Indians fed on two different diets but otherwise kept under identical conditions:

 

Diet I:  16.3     10.1     10.7     13.5     14.9     11.8     14.3     10.2     12.0     14.7     23.6     15.1     14.5     18.4     13.2            14.0

Diet II:            21.3     23.8     15.4     19.6     12.0     13.9     18.8     19.2     15.3     20.1     14.8     18.9     20.7     21.1            15.8     16.2

Use U test at 0.01 level of significance to test the null hypothesis that the two population samples are identical against the alternative hypothesis that on the average the second diet produces a  greater gain in weight.

(OR)

 

The same mathematics papers were marked by three teachers A, B and C. The final marks were recorded as follows:

Teacher A 73 89 82 43 80 73 66 60 45 93 36 77
Teacher B 88 78 48 91 51 85 74 77 31 78 62 76
Teacher C 68 79 56 91 71 71 87 41 59 68 53 79

Use Kruskal-wallis test, at the 0.05 level of significance to determine if the marks distributions given by the three teachers differ significantly.               (10 + 10)

 

 

 

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Loyola College B.Sc. Statistics Nov 2008 C And C ++ Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 15

 

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – November 2008

ST 5401 – C AND C ++

 

 

 

Date : 14-11-08                       Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

PART A

 

Answer ALL questions.                                                                                                           (10×2=20)

 

  1. Mention the difference between signed and unsigned integer.
  2. Give the syntax of scanf() function.
  3. If x = 12 and y = 10, what is the value of x&y?
  4. What are recursive functions?
  5. Write a C program to find the sum of squares of the first 10 integers.
  6. Mention the difference between local and global variable.
  7. What is meant by data hiding?
  8. Give the syntax of declaring a structure.
  9. Write a C++ program to calculate the simple interest.
  10. What are header files?

PART B

 

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

 

  1. Explain the various steps to be followed in writing a C program.
  2. Write a C program to accept three integers and display the maximum among them.
  3. Explain the syntax of switch – case statement and give an example.
  4. Write a C program to accept ‘n’ integers and display their mean and variance.
  5. What are public, private and protected access specifiers? Give an example for each.
  6. Write a C++ program that makes use of the concept of polymorphism.
  7. Write short notes on inline functions and operator overloading.
  8. Write a C++ program to print the multiplication table form 1 to 12.

 

PART C

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

 

  1. a.) Explain the different data types available in C.

b.) Write a C program to accept two matrices each of order m x n and display

     their sum and difference.                                                                                                (10+10)

 

  1. a.) Write a C program to arrange ‘n’ integers in ascending order.

b.) Create a structure by name ‘student’ that contains the following

     information: student id_no, marks in test 1 and test 2.  Write a C program to

     accept the above information for 10 students and display their average marks.       (10+10)

 

  1. a.) Explain the different components of object oriented programming.

b.) What are friend functions? Give an example for the same.                                        (10+10)

 

  1. a.) Write a C++ program to overload the operator ‘++’.

b.) Write a C++ program that makes use of the concept of hierarchical

     inheritance.                                                                                                                       (10+10)

 

 

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Loyola College B.Sc. Statistics Nov 2008 Basic Sampling Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

BA 09

 

THIRD SEMESTER – November 2008

ST 3502 / 4500 – BASIC SAMPLING THEORY

 

 

 

Date : 08-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART A

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

 

  1. Define the term “statistic”  in finite population sampling.
  2. What do you mean by probability sampling?
  3. What will happen to the variance of sample mean under simple random sampling without replacement, if the sample size is increased?
  4. Mention an unbiased estimator of in simple random sampling without replacement where is the sample mean.
  5. Under what condition PPSWR reduces to SRSWR?
  6. Give an unbiased estimator of the population total in PPSWR, when the sample size is one.
  7. What is meant by Neyman allocation?
  8. Under what condition Neyman allocation reduces to Proportional allocation?
  9. Write all possible Circular systematic samples of size 3 when the population     size is 7.
  10. Write the down the model used for representing populations exhibiting linear trend in Sampling Theory

 

PART B

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

 

  1. Compare Sampling with Census.
  2. Illustrate with an example thatUnbiasedness is a sampling design dependent property.
  3. Derive under simple random sampling without replacement.
  4. Define Hansen – Hurwitz estimator and obtain its variance under PPSWR.
  5. Show that Lahiri’s method of sample selection is a PPS selection method.
  6. Derive the formula for Optimum allocation for fixed variance in stratified random sampling.
  7. Explain Balanced systematic sampling and derive the under BSS assuming the population values possess linear trend
  8. Derive the condition under which LSS is more efficient than SRS in terms of

 

PART C

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

 

  1. (a) Bring out the relationship between Mean Square Error, Bias and Variance of an estimator

(b) Show that is unbiased for in SRSWOR

  1. Assume that there are two strata with and relative weights and . Show that for the given cost , . Assume the strata sizes are large.
  2. Derive the Yates corrected estimator under Linear Systematic Sampling for populations possessing linear trend.
  3. Write short notes on the following
  • Cumulative Total Method
  • Modified Systematic Sampling
  • Proportional allocation
  • Limitations of Systematic Sampling

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***********

Loyola College B.Sc. Statistics Nov 2008 Applied Stochastic Processes Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 

BA 14

 

FIFTH SEMESTER – November 2008

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

Answer ALL the questions:                                                                (10 x 2 = 20)

                  

  1. What is meant by a stochastic process?
  1. What is a state space of a stochastic process?
  2. Explain ‘Independent Increments’.
  3. Define ‘Markov process’.
  4. Define Transition Probability Matrix.
  1. Define accessibility of a state from another state.
  1. Ifis a stochastic matrix, fill up the missing entries in the

matrix.

  1. Define aperiodic Markov chain.
  2. Define absorbing state.
  3. Define irreducible Markov chain.

                                                                                      

Section – B

Answer any FIVE of the following:                                                                       (5 x 8 = 40)

                                               

  1. State the classification of stochastic processes based on time and state space.

Give an example for each type.

  1. Prove that a Markov chain is completely determined by the one step transition  

       probability matrix and the initial distribution.

  1. Let { Xn, n ³ 0} be a Markov chain with three states 0,1 and 2. If the transition

       probability matrix is

                                  

                                   

 

and the initial distribution is Pr{ X0 = i}= 1/3, i = 0,1,2,

            find

            i).    Pr{X1 = 1 ½ X0 = 2}

            ii).   Pr{X2 = 2 ½ X1 = 1}

            iii).  Pr{X2 = 2, X1 = 1 ½ X0 = 2} and

            iv).  Pr{X2 = 2, X1 = 1, X0 = 2}

 

 

 

 

  1. Obtain the equivalence classes corresponding to the transition probability matrix

 

 

  1. Form the differential – difference equation corresponding to Pure birth process.
  2. Describe the one dimensional random walk and write down its tpm.
  3. Describe second order process, covariance function and its properties.
  4. Derive any one property of Poisson process.

      

                                                          Section – c

 

Answer any TWO of the following:                                                             (2 X 20 =40)

 

 19.a) Let { Zi, i = 1,2…} be a sequence of random variable with mean 0. Show that

       

           Xn =   is a Martingale.

  1. b) Consider a Markov chain with state space {0, 1, 2, 3} and tpm 

 

 

P  = 

            

            

 

 

   Find the equivalence classes and compute the periodicities of all the 4 states

      

  1. Sociologists often assume that the social classes of successive generations in

              family can be regarded as a Markov chain. Thus the occupation of son is

              assumed to depend only on his father’s occupation and not his grandfather’s.

              suppose that such a model is appropriate and that the transition probability

              matrix is given by

                                               

 

               

 

               For such a population what fraction of people are middle class in the long run?

  1. Define the Poisson process and find the expression for Pn(t).
  2. Write short notes on the following:

(a). Stationary distribution

(b). Communicative sets and their equivalence property

(c). Periodic states

 

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Loyola College B.Sc. Statistics Nov 2008 Applied Statistics Question Paper PDF Download

by

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

BA 12

 

FIFTH SEMESTER – November 2008

ST 5502 – APPLIED STATISTICS

 

 

 

Date : 07-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A      ( 10 x 2 = 20 Marks)

Answer ALL questions.

 

  1. Define an index number.  Mention its uses.
  2. Discuss the concept of deflating an index number.
  3. Define a Time series and give examples.
  4. List down different methods of measuring trend.
  5. Define total and specific fertility rates.
  6. Specify Pearle’s Vital index in the measurement of population growth.
  7. Define multiple correlation coefficient.
  8. Write down the formula for partial correlation coefficient r12.3.
  9. What is meant by financial statistics?
  10. Write a note on live stock statistics.

 

SECTION – B     (5 x 8 = 40 Marks)

Answer any FIVE questions.

 

  1. What are the steps to be followed in the construction of consumer price index number?
  2. The following table gives the quantities and prices of five commodities for two periods.  Compute the quantity and the price indices using Fisher’s formula.

 

Commodity Base Year Current Year
Quantity Price Quantity Price
A 6 5 8 6
B 8 3 10 2
C 12 2 10 4
D 2 8 2 7
E 5 9 6 9
  1. What are the components of a time series? Explain them.
  2. The following data gives the yearly sales of a product.  Compute the linear trend by the method of least squares.  Estimate the trend for the year 2008.

 

Year 2001 2002 2003 2004 2005
Sales(Lakhs) 26 30 38 50 56

 

  1. What are the components of a Life Table.  Explain them.
  2. Explain the Gross and Net Reproduction Rates.
  3. Discuss the functions of Central Statistical Organisation (CSO).
  4. Given the values of  r12=0.8, r13=0.7 and r23=0.9, compute the values of R1.23 and 2.31.

 

 

 

 

SECTION-C     (2 x 20 = 40 Marks)

Answer any TWO questions.

 

  1. (a). Discuss the unit test, time reversal test, factor reversal test and the circular test associated with index numbers.

(b). Construct the wholesale price index number for the years 2007 and 2008 given the following data.

 

Commodity Wholesale price in rupees per quintal
2006 2007 2008
I 140 160 190
II 120 130 140
III 100 105 108
IV 75 80 90
V 250 270 300
VI 400 420 450

 

  1. (a). Discuss various methods of measuring Seasonal indices in Time series.

(b). The following table gives the cost of an item for the period of five years.  Using the method of Link relatives, compute the Seasonal indices.

 

Quarter Year
2001 2002 2003 2004 2005
I 60 62 65 70 72
II 65 68 70 75 80
III 62 65 64 68 70
IV 69 68 62 67 78

 

  1. (a). Compute the crude and standardized death Rates of the two populations X and Y regarding X as the standard population.

 

Age group Under 10 10-20 20-40 40-60 Above 60
Population  in X 20000 12000 50000 30000 10000
Deaths in X 600 240 1250 1050 500
Population in Y 12000 30000 62000 15000 3000
Deaths in Y 372 660 1612 325 180

 

 

(b). Derive the multiple regression equation of X1 on X2 and X3 in the usual notation.

 

  1. Write short notes on the following
  • Partial and mulitiple correlations.

(ii)  Discuss the methods of National Income estimation.

(iii) Economic census and labour statistics.

 

 

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