LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

B.Sc. DEGREE  EXAMINATION  – STATISTICS

FOURTH SEMESTER  – APRIL 2003

ST   4500/ sta  504   BASIC SAMPLING   THEORY

11.04.2003

9.00 – 12.00                                                                                           Max: 100 Marks

SECTION – A                                  (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries TWO marks.

1. Define a population and mention the assumptions made regarding population size in sampling Theory.
2. Explain ‘Sampling frame’ of a population.
3. What are the constraints for carrying out a census?
4. Distinguish between a statistic and a parameter.
5. Derive the expression for the mean square error of an estimator T in estimating q in terms of the variance and bias of the estimator T.
6. In SRSWOR, find the probability of selecting r^th population unit in i^th
7. Explain the cumulative total method of selecting a PPS sample.
8. Write all possible linear systematic samples of size n = 6 given N = 24.
9. Explain stratified random sampling.
10. Name any three allocation schemes used in stratified sampling.

 

 

SECTION –B                                                (5 ´ 8 = 40 Marks)

Answer any FIVE questions.  Each carries EIGHT marks.

 

11. Consider a population containing 4 units with values 7,4,10,5. Two units are drawn one by one without replacement with equal probabilities.  Verify whether or not the statistic T =(3y₁+ 4y₂) /5 is unbiased for the population mean.
12. Show that, under usual notations, in SRSWOR

 

13. Show that Lahiri’s method of selection is a PPS selection.
14. Examine whether the estimator is unbiased for the population total under PPSWR, where y_i is the value of i^th drawn unit (i = 1,2,….,n).
15. Deduce is SRSWR using expressions for available in PPSWR.
16. Letdenote the sample mean of only distinct units under SRSWR.

Find

17. Show that in Balanced systematic sampling, the sample mean coincides with

population mean in the presence of linear trend.

18. In stratified sampling, under proportional allocation, derive Vand find

an unbiased estimator of V.

 

 

 

 

 

 

SECTION – C                                (2 ´ 20 = 40 Marks)

Answer any TWO questions.  Each carries TWENTY marks.

 

19. (a) Illustrate that an estimator T can become biased under a sampling scheme

even though T is unbiased under another sampling scheme..                         (8)

(b) A population contains 5 units and it is known that   (12)

Under the usual notations, (i) compare  with

(ii) Find the values of a for which  is less efficient than.(10)

20. (a) consider the following data:

 

Plot number	No. of plants	No. of roses
1	20	56
2	32	65
3	14	30
4	19	47
5	22	37
6	7	28

 

Assume that a PPSWR sample of size 2 is drawn.  Compute Vfor the

data given.  Compare it with V.                                                       (10)

• Derive an unbiased estimator of Vin PPSWR scheme.                  (10)

 

21. (a) Describe centered systematic sampling.                                                 (5)

• Explain in detail the principal steps involved in the planning and

execution of  a sample survey.                                                               (15)

 

• (a) If  C_o is the overhead cost and  C_h is the cost of collecting  information per

unit in stratum h, then the total  cost of the survey is given by C_o+

In optimum allocation, determine the values of n₁, n₂,……,n_L  for which

V  is minimum subject to a fixed total cost.                                          (12)

• Show that if Y_i = a + b_i, i =1,2,..,N, then ,

where a stratified sample of size n is drawn as follows:

Divide the population into n groups of k units each.  And draw 1 unit from each group where the first stratum contains the first k units, the second

stratum contains the next k units and so on.                                             (8)

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2003

ST  4500 / STA 504 – BASIC SAMPLING THEORY

01.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. Explain sampling frame and give two examples.
2. If there are two unbiased estimators for a parameter then show that one can construct, uncountable number of unbiased estimators.
3. If T is an estimator for , then show that MSE (T)  =  V(T)  +  [B(T)]² .
4. Explain Lottery method for drawing random numbers.
5. Show that probability of including the i^th population unit (i =  1, 2, …, N) when a SRSWOR of size n is drawn from a population containing N units is .
6. Find the probability of selecting i^th population unit in cumulative total method.
7. Examine whether the estimator is unbiased for the population total under PPSWR.
8. Show that the sample mean under SRSWOR is more efficient than under SRSWR.
9. Explain Linear Systematic Sampling Scheme.
10. When do we use Neyman allocation?

 

SECTION-B

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

 

11. Examine the validity of the following statement using a proper illustration :

“property of unbiasedness depends on the sampling scheme under use”.

12. Prove that, under usual notations, in SRSWOR, P[y_i =
13. What is PPS sampling? Describe cumulative total method?
14. Deduce expressions for , V() and v () under SRSWR using the expressions for , V() and v () available under PPSWR.
15. Prove that is an unbiased estimator for population mean under stratified random sampling.  Derive .
16. Derive the formula for Neyman allocation.
17. Prove that the sample mean coincide with the population mean in Centered Systematic Sampling, when there is linear trend in the population.
18. a) List all possible Balanced Systematic Samples if N = 40,  n= 8.
19. b) List all possible Circular Systematic Samples if N = 7, n = 3.

 

 

 

 

SECTION-C

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

 

19. a) Describe the principal steps involved in the planning and execution of a survey. (14)
20. b) Let denote the sample mean of only distinct units under SRSWR. Find E

and V .                                                                                                     (6)

20. a) A population contains 5 units and it is known that Compare

with .  Find the values of a for which

is less efficient than .                                                 (12)

1. b) Show that Lahiri’ s method of selection is a PPS selection. (8)
2. a) Show that an unbiased estimator of V() is

(10)

1. b) Derive values of n_h such that C_o + is minimum for a given value of

.                                                                                                                                  (10)

22. a) Compare and assuming N_h is large for all h = 1,2, …., L.

(12)

1. b) A sampler has 2 strata. He believes that S₁ and S₂ can be taken as equal. For a given

cost c = c₁ n₁ + c₂ n₂,   show that   =   .                      (8)

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2004

ST-4500- BASIC SAMPLING THEORY

02.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION -A

 

Answer ALL questions. Each carries TWO marks.                                    (10 ´ 2 = 20 marks)

 

1. Define Sampling frame. Give two examples.
2. If there are two unbiased estimators for a parameter, then show that one can construct uncountable number of unbiased estimators for that parameter.
3. If T is an estimator for , then show that MSE (T) = V (T) +
4. Explain Lottery method of drawing simple random sample of size n.
5. Show that the probability of selecting a given subset consisting of ‘n’ units of the population of N units is .
6. Find the probability of selecting ith population unit in a given draw in PPS sampling.
7. Examine whether the estimator is unbiased for the population total under PPSWR.
8. Prove that the sample mean is a more efficient estimator of population mean under SRSWOR than under SRSWR.
9. Explain circular systematic Sampling Scheme.
10. Compute the number of units to be sampled for each stratum under proportional allocation scheme, when the total sample size is 40 and there are 4 strata of sizes 40, 30, 60, and 70.

 

SECTION – B

 

Answer any FIVE questions.  Each carries EIGHT marks.                           (5 ´ 8 = 40 marks)

 

11. Examine the validity of the following statement using a proper illustration: ‘An unbiased estimator under one method of sampling can become a biased estimator under another method of sampling’.

 

12. Show that, in SRSWOR,

cov (y_i , y_j)  = –

 

13. Prove that sample mean is unbiased for population mean in SRSWOR by using the probability of selecting a subset of the population as a sample.

 

14. What is PPS sampling? Describe cumulative total method.

 

15. Derive the variance of Hansen Hurwitz estimator for population total.

 

16. Derive E (and Var , where denotes the sample mean based on only distinct units under SRSWR.

 

17. Write a descriptive note on centered Systematic Sampling.

 

18. Derive the formula for under Neyman allocation.

 

SECTION – C

 

Answer any TWO questions.  Each carries TWENTY marks.                     (2 ´ 20 = 40 marks)

 

19. a) Derive the variance of sample mean, V(in SRSWOR by using probabilities of inclusion.                                                                                                                             (10)
20. b) Describe Lahiri’s method. Show that Lahiri’s method of selection is a PPS selection.

(10)

20. (a) A population contains 5 units and it is known that

Compare   Find the values of for which

is less efficient than .                                                           (10)

1. b) Derive the expressions for in SRSWR using the expressions for

available in PPSWR.                                                                                  (10)

 

21. a) Show that when irrespective of the random start ‘r’ .                                                                                                      (10)
22. b) In Linear systematic Sampling, when N is not a multiple of n, explain the undesirable situations encountered using suitable illustrations. (10)

 

22. a) Compare Vand assuming is large for all h = 1, 2, … , L.                                                                                                                            (10)
23. b) A sampler has two strata. He believes that stratum standard deviations are the same. For a give cost C = C₁n₁+C₂n₂, show that

.                                                            (10)

 

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 14

FOURTH SEMESTER – APRIL 2006

                                                   ST 4500 – BASIC SAMPLING THEORY

(Also equivalent to STA 504)

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

 

1. Define a population and give two examples.
2. State the difference between parameter and  statistic.
3. If there are two unbiased estimators for a parameter, then show that there are uncountable number of unbiased estimators for that parameter.
4. Explain Lottery method of selecting a  Simple Random Sample.
5. In SRSWOR , find the probability of selecting a given subset consisting of ‘n’ units of the population consisting of  ‘N’ units.
6. In PPS sampling , find the probability of selecting  i^th  population unit when a

simple random without replacement sample of size ‘n’ is drawn from a population

containing ‘N’ units.

1. Show that Hansen – Hurwitz estimator is unbiased for population total.
2. Show that the sample mean is more efficient under SRSWOR  scheme than under SRSWR scheme.
3. In SRSWR , obtain the variance of the sample mean based on only distinct units.
4. Describe Balanced Systematic Sampling Scheme.

 

SECTION – B

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

 

1. Illustrate that one can obtain more than one unbiased estimator for a parameter.
2. In SRSWOR , derive the variance of sample mean by using probabilities of inclusion.
3. Deduct the formulae for Ŷ,  V( Ŷ ) and  v(Ŷ ) under SRSWR using the formulae for  Ŷ , V(Ŷ) and v(Ŷ) available under PPSWR.
4. Explain the consequences of  using an LSS scheme  when  ‘N’ is not a  multiple of  ‘n’.
5. In  Stratified Random Sampling , derive the variance and  estimated variance of the sample mean.
6. Derive Neyman allocation formula.  Hence what do we conclude about the size of the sample taken from any stratum?
7. Explain Circular  Systematic Sampling Scheme.  List all possible samples under this scheme when N = 7 and n = 3.
8. In PPSWR , derive the variance of Hansen – Hurwitz  estimator for population total.

SECTION – C

 

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

 

19(a).  In SRSWOR , show that sample mean is unbiased  for population mean by

using  the probability of selecting a subset of the population as a sample.  (8)

19 (b). Describe Lahiri’s  method  and prove that this method  of selection is a

PPS  selection.  (12)

 

20(a). A population contains  5 units and it is known that

( (Y₁ /P₁) –  Y)² P₁ +  …  + ( (Y₅ / P₅) – Y)²P₅  =  100.

Compare  Ŷ₁  =  ((y₁ /P₁) + ( y₂ / P₂ )) / 2   and

Ŷ₂  = (2/3) (y₁ /P₁) + (1/3) ( y₂ / P₂).

Find the values of  α for which

Ŷ_α  =  α (y₁/ P₁)  +  (1 – α ) (y₂ / P₂)  is  less efficient than Ŷ₁ .(12)

20(b).  Verify whether or not the sample mean coincides with the population

mean in Centered Systematic Sampling scheme when the population is

linear. ie Y_i  =   α + βi  ,   i = 1,2,…,N.   (8)

 

21. In a population consisting of linear trend, show that a systematic sample is

less precise than a stratified random sample, with a strata of size 2 k and

2 units chosen per stratum, if n > (4k + 2)/ (k  + 1), when the 1^st    stratum

contains the 1^st   set of  2k units,  the 2^nd stratum contains the 2^nd set of 2k units

in   the population and so on.  (20)

 

22. With 2 strata , a sampler would like to have n₁  =   n₂  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 (V-Vopt )/Vopt  = ( (r-1) / ( r+1)) ² , where

r = n₁ / n₂  as given by Neyman allocation.   Assume that N₁  and N₂  are large.

(20)

 

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                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 y_i denote the y-value of the i^th drawn unit. Find the discrete probability distribution of y_{i .}
5. In PPS sampling, find the probability of selecting   i^th   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)

 

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

Y_i / P_i   –  Y)² P _i  =  100 .  Under PPSWR, compare

 

‘   =        and    ”   =   .

 

16. Deduct the formula for ,  V()   and  v() in SRSWR using the formula  for  ,  V()  and   v() available in PPSWR.
17. Describe circular systematic sampling with an example.
18. Derive values of n_h  such that c_o  +  is minimum for a given

value of  V (_st).

 

SECTION – C

——————-

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

 

19. ( 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 )

20. ( 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 )

21. ( 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 )

 

22. 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 V_opt 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 (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – STATISTICS

NO 13

THIRD SEMESTER – APRIL 2008

ST 3502/ 4500 – BASIC SAMPLING THEORY

 

 

 

Date : 29-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL questions:                                                       (10×2=20)

1. In what situations sampling is inevitable?
2. How would you distinguish between estimate and estimator?
3. Differentiate between SRSWR and SRSWOR.
4. Define standard error of an estimator.
5. Explain pps sampling.
6. Explain cumulative total method of selecting pps sampling.
7. What do you understand by stratified random sampling?
8. Distinguish between systematic sampling and stratified random sampling.
9. What is circular systematic sampling?
10. What are the advantages of systematic sampling?

 

SECTION-B

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

1. Explain the principles of sampling methods.
2. In SRSWOR, prove that sample mean is an unbiased estimator of the population mean and its sampling variance is given by .
3. Explain the various types of allocations used in stratified random sampling method.
4. Find the relative precision of systematic sample mean with simple random sample mean.
5. A village has 10 holdings consisting of 50, 30, 45, 25, 40, 26, 24, 35, 28 and 27 fields. Select a sample of size 4 by using Lahiri’s method.
6. Explain sampling and non sampling errors.
7. Derive the formulae for the mean and variance of systematic sample..
8. Distinguish between probability and non-probability sampling and write down their advantages and disadvantages.

 

SECTION – C

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

1. a) Describe Lahiri’s method of selection and it’s merits over cumulative method.

1. b) A random sample of size n=2 was drawn from a colony of 5 households having monthly income as follows:

Household:      1          2          3          4          5

Income(Rs):    156      149      166      164      155

• Calculate population mean and population variance .
• Enumerate all possible samples of size 2 by the replacement method and show that the sample mean gives an unbiased estimate of the population mean and find its sampling.

Variance: Also show that s² is an unbiased estimate of the population variance .

1. a) Show that under fixed cost.

1. b) Compare the variances of the sample mean under systematic sample with stratified and simple random sampling, assuming linear trend.

1. a) Explain different systematic sampling schemes.

1. b) Derive the variance of Hansen-Hurwitz estimator for population total.

1. a) Show that in SRSWOR the probability of selecting a sample of size ‘n’ from ‘N’ units is 1/Ncn.

1. b) In a bank there are 5000 deposit accounts. A random sample of 50 accounts was drawn WOR and the following information was obtained and . Where Y_i denotes the amount in a deposit account. Find an unbiased estimate of the average amount in a deposit account and find its estimated variance.

 

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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 (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 13

THIRD SEMESTER – April 2009

ST 3502 / ST 4500 – BASIC SAMPLING THEORY

 

 

 

Date & Time: 02/05/2009 / 9:00 – 12:00   Dept. No.                                                   Max. : 100 Marks

 

 

PART A (10 x 2 = 20 Marks)

Answer All Questions:

1. Distinguish between census method and sampling procedure.
2. Define Sampling design and Probability Sampling.
3. What is simple random sampling?
4. Prove that SRSWOR sample mean is an unbiased estimate of the population mean.
5. Write down the two steps to select one unit by Lahiri’s method of PPS selection.
6. What is PPSWR sample?
7. Explain the need of stratified sampling.
8. What is proportional allocation?
9. What is circular systematic sampling?
10. What is a balanced systematic sampling scheme?

PART B (5 x 8 = 40 Marks)

Answer any Five Questions:

1. What are the advantages of sampling over census method? What are its limitations?
2. Under simple random sampling is unbiased for the population total and also obtain its variance.
3. Marks secured by over one lakh students in a competitive examination were displayed jin 39 display boards. In each board marks of approximately 3000 students were given. Kiran, a student who scored 94.86 marks wanted to know how many candidates have scored more than him. In order to estimate the number of student who have scored more than him, he took a SRS of 10 boards and counted the number of students in each board who have scored more than him. The following is the data collected.

14, 29, 7, 13, 0, 36, 11, 43, 27 and 5.

Estimate the number of student who would have scored more than Kiran and also estimate the variance of its estimate.

 

 

1. Prove that the probability of selecting the i^th unit in the first effective draw is , in Lahiri’s method of  PPS selection.
2. Derive the variance for Hansen – Hurwitz estimator.
3. Explain the procedure of optimum allocation of sample size.
4. Derive an unbiased estimator for the population total and its variance under proportional allocation.
5. Prove that , where  and  are the conventional expansion estimators under linear systematic sampling and simple random sampling respectively.

PART C (2 x 20 = 40 Marks)

Answer any two Questions:

1. (a) Prove that under SRS, S_xy.

(b) Derive average mean squared errors under balanced systematic and modified systematic sampling schemes and compare them.

1. (a) Derive an unbiased estimator for population total in PPS sample and also obtain its variance.

(b) Prove that in Stratified random sampling with given cost function of the form , is minimum if _.

1. If the population consists of a linear trend, then prove that .
2. (a) Obtain the relative efficiency of systematic sample as compared to simple random sampling without replacement.

(b) Describe Lahiri’s method of selection and its merits over cumulative method.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2010

ST 3504/ST 3502/ST 4500 – BASIC SAMPLING THEORY

 

 

 

Date : 02-11-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

 

ANSWER ALL QUESTIONS.                                                                                       (10 X 2 =20 marks)

 

1. What is meant by “Probability Sampling Design”?
2. Give an example of a “Statistic”.
3. What are the values of second order inclusion probabilities in simple random sampling without replacement when N=10 and n=3?
4. Show that the variance of sample mean under simple random sampling with replacement is greater than its variance under simple random sampling without replacement.
5. Under what condition PPSWR reduces to SRSWR?
6. Suggest an unbiased estimator for population total under PPSWR.
7. When do you recommend stratified random sampling?
8. Suggest an unbiased estimator of population mean in stratified random sampling.
9. List all possible circular systematic samples of size 3 when the population size is 10.
10. What is the probability of selecting a sample consisting of 3^rd and 4^th population units in LSS when N=12 and n=4?

            SECTION – B

 

ANSWER ANY FIVE QUESTIONS                                                                                (5 X 8 =40 marks)

 

11. Compare sampling and census.
12. Explain the terms : (i) Unbiasedness (ii) Variance (iii) Mean square error and (iv) Bias.
13. Show that under simple random sampling when N=3 and n=2 the estimator

is unbiased for the population mean and obtain its variance.

 

14. Explain Cumulative total method of PPS selection with an example.
15. Compare the efficiencies of under optimum and proportional allocation.
16. Explain proportional allocation for a given sample. Derive the variance of under proportional allocation.
17. Derive in terms of .
18. Compare the variances of sample mean under LSS and SRS in the presence of linear trend.

 

SECTION  – C

 

 

ANSWER ANY TWO QUESTIONS                                                                     (2 X 20 = 40 marks)

 

19. a) Establish the relationship between Bias, Variance and Mean Square error.
20. b) Illustrate with an example that unbiasedness of an estimator is design dependent.
21. a) Explain Lahiri’s method of PPS selection with proof.
22. b) Derive the variance of Hansen Hurwitz estimator under PPSWR and develop its unbiased

estimator.

21. a) A sampler has two strata with relative sizes  and  .  He believes that

and  can be taken as equal. For a given cost , assuming the stratum

sizes are large, show that

 

1. Derive the formula for Optimum allocation.

 

22. Write short notes on the following:
23. Balanced systematic sampling
24. Yates corrected estimator
25. Neyman allocation
26. Formation of Strata.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2012

ST 3504/ST 3502/ST 4500 – BASIC SAMPLING THEORY

 

 

 

Date : 26-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

Answer ALL the questions:                                                                                     (10×2= 20 Marks)

 

1. Define Population and sample.
2. Explain sampling frame and give an example.
3. Define simple random sampling without replacement.
4. Distinguish between a questionnaire and schedule.
5. In SRSWOR, find the number of samples of size 3 that can be drawn from a population of size 15.
6. Distinguish between bias and error.
7. Explain stratified random sampling.
8. What are the merits of stratified random sampling?
9. Define linear systematic sampling.
10. What is meant by circular sampling?

 

PART – B

 

Answer any FIVE questions            :                                                                                   (5×8=40 Marks)

 

1. Discuss briefly the basic principles of a sample survey.
2. Explain Lottery Method and Random Number Table Method of unit selection.
3. In SRSWOR, prove that the sample mean is unbiased estimator of population mean. Also find its variance.
4. In SRSWR, prove that .
5. Explain modified systematic sampling and derive the expression for variance of the sample mean.
6. Explain the advantages and disadvantages of systematic sampling.
7. Explain cumulative total method of PPS selection.
8. In usual notations, prove that the systematic sample mean is more precise than the mean of SRSWOR if .

 

 

 

 

 

PART – C

Answer any TWO questions:                                                                                         (2×20=40 Marks)

 

1. (a) Discuss the advantages of sampling over completer enumeration.

(b) A population contains 12 units with y-values arranged according to their labels as

3, 5, 4, 7, 8, 2, 9, 12, 11, 15, 13, 14. List all possible linear systematic samples of

size 4.

1. (a) Derive the variance of Hansen-Hurwitz estimator for population total.

(b) Explain proportional allocation and optimum allocation.

1. Prove that when we compare stratified random sampling with SRS                              .
2.  Compare simple random sampling and linear systematic sampling in the presence of linear trend.

 

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3504/3502/4500 – BASIC SAMPLING THEORY

 

 

 

Date : 05/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

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

 

1. What is meant by sampling frame?
2. What is pilot survey?
3. Define simple random sampling with replacement.
4. Define unbiased estimator of a parameter.
5. Distinguish between SRSWR and SRSWOR.
6. Explain stratified random sampling.
7. Write any two advantages of stratified sampling.
8. Define Lahiri’s method.
9. Define linear systematic sampling.
10. Write down the merits of systematic sampling.

 

PART – B

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

 

1. What are the advantages of sampling over census method?
2. List out the dangers in using statistical packages.
3. Derive any two properties of sample mean in SRSWR.
4. Prove that in stratified sampling, sample mean is an unbiased estimator of population mean.

Also find its variance.

1. Write a descriptive note on cluster sampling.
2. Explain ‘Lottery Method’ of selecting a simple random sample.
3. Explain the advantages and disadvantages of systematic sampling.
4. Explain cumulative total method of PPS selection.

 

PART – C

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

 

1. (a) What are non-sampling errors? Explain its sources.

(b) Write a note on simple random sampling of attributes.

1. (a) If the population consists of linear trend, then prove that

.

(b) Compare

1. (a) Derive the variance of unbiased estimator for mean per element under cluster sampling in

terms of the intra cluster correlation.

(b) Prove that is unbiased for in SRSWOR.

22. Define systematic sampling. Obtain the sampling variance of the mean and

compare with that of SRSWOR and stratified sampling.

 

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