IGNOU MSCAST MST 20 Survey Sampling and Design of Experiments Solved Assignment ENGLISH

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MST 020 (January 2026 - July 2026) - ENGLISH

**Course Code: MST-020**
**Assignment Code: MST-020/TMA/2026**
**Maximum Marks: 100**

**Note: All questions are compulsory. Answer in your own words.**

1. State whether the following statements are true or false and also give the reason in support of your answer:

a) In Two-Stage Sampling if , it becomes the Cluster Sampling.

b) Among all the members of the General Class of the estimators only Product estimator is an unbiased estimator of the Population mean.

c) With the following parameters , an Incomplete Block Design is found to be a Balanced Incomplete Block Design.

d) a 2³ Factorial design is One-half fractional factorial design of a 2⁶ full factorial design.

e) If M_i and are correlated, the Cluster estimator becomes an unbiased estimator of the Population Mean.

2. In a population of size 500, the population mean of the auxiliary variable X was observed to be 950.00. On the basis of a sample of size 30, the following sample values were obtained:



Obtain the value of the Ratio estimate of population mean as obtained on the basis of the sample drawn. Also, find the estimated value of the Sample Mean Squared Error of the estimate. (15)

3. In a population of size 500, the population mean of the auxiliary variable X was observed to be 834.00. On the basis of a sample of size 40, the following sample values were obtained:



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$y = 740.00; ar{x} = 865.00; s_y^2 = 9560.00; s_x^2 = 15300.00; s_{yx} = 30540.2.$

Obtain the values of the regression estimate and the ratio estimate of the population mean as obtained on the basis of the sample drawn. $quad quad quad quad quad quad quad quad (08)$

4. Following data presents a population of 10 clusters of size 3 each. Each cluster represents a bunch of students of standard IV in a school. The marks obtained by students in a competitive test (out of 75 marks) have been shown in the data set. Find the Relative Efficiency (R.E.) of the Cluster Sampling with respect to the Simple Random Sampling without Replacement scheme:

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1	56, 34, 28	2	70, 56, 30	3	12, 57, 30	4	24, 70, 35	5	9, 44, 59
6	25, 69, 70	7	14, 38, 50	8	74, 47, 65	9	14, 48, 62	10	56, 59, 39

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5. A crime reporter wishes to publish a report in a national crime magazine highlighting crime of tax evasion happened in the hotel industry in a specific district in the preceding year. For this purpose, in order to estimate the proportion of owners of 3-star hotels, he selected a sample of size 350 hotel owners using simple random sampling with replacement scheme out of a population of 2000 hotels. Every owner in the sample was asked to randomly pick a card from a deck of cards, in which 86% of the cards were marked "I cheated on last year's income tax return" and the remaining 14% were marked "I did not cheat on last year's income tax return". The survey gave 100 "True" responses and 250 "False" responses.

Find the value of the unbiased estimator of the unknown population proportion of tax evaders. Also, find the variance of the estimator.

6. In order to obtain the idea of interactions effects of two factors, A and B, each available at two levels; a 22-factorial experiment was conducted in six randomized blocks. The two levels of the factors A and B are denoted respectively by (a1, a2) and (b,, b₂). Below is given the plan and yields of the experiment. Analyze the design for finding out if there are any significant treatment effects- either main or interaction:

Based on the image provided, here is the transcribed text formatted to match the layout of the blocks:

Block I

$a_1b_1$	$a_1b_2$	$a_2b_2$	$a_2b_1$
50	46	52	40

Block II

$a_2b_2$	$a_1b_1$	$a_1b_2$	$a_2b_1$
39	42	47	49

Block III

$a_1b_1$	$a_2b_1$	$a_1b_2$	$a_2b_2$
47	51	42	45

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Block IV

$a_2b_2$	$a_2b_1$	$a_1b_2$	$a_1b_1$
51	50	47	40

Block V

$a_2b_2$	$a_1b_2$	$a_1b_1$	$a_2b_1$
55	48	53	57

Block VI

$a_2b_1$	$a_1b_1$	$a_2b_2$	$a_1b_2$
55	52	47	54

7. Explain what is meant by the one-half fraction of a 2k

(k ≤ 5)

factorial experiment. Considering a 2ª factorial design, show how would you get a one-half fractional factorial design.

8. What is a Complimentary B.I.B.D. of a given symmetric B.I.B.D.? Explain the method of obtaining a complimentary B.I.B.D. given the following B.I.B.D.:

9.Let us consider a B.I.B. design as given below with parameters

9=b=7,r=k=3,λ = 1:

Given the above Balanced Incomplete Block Design (B.I.B.D.), construct a residual design and find the parameters of this design.

 

MST 020 (January 2025 - July 2025) - ENGLISH

TUTOR MARKED ASSIGNMENT

MST-020: Survey Sampling and Design of Experiments-II

Course Code: MST-020

Assignment Code: MST-020/TMA/2025

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are true or false and also give the reason in support of your answer:

a) If X and Y are uncorrelated, the V(Rag) reduces to that of Variance of Ratio Estimator of Population mean

b) Among all the members of the General Class of the estimators only Regression estimator is an unbiased estimator of the Population mean.

c) An Incomplete Block Design with the following parameters b=8,k=3,9=8,r=3 is found to be a Balanced Incomplete Block Design.

d) A One-half fractional factorial design of a 2ª full factorial design will be denoted by a 22 Factorial design.

e) The Cluster estimator ỹ, becomes an unbiased estimator of population mean only if M and , are uncorrelated

2. Suppose we wish to estimate the total number of Cows in 2026 in certain state. The total number of cows for 2024 was X = 5000. The Sampling unit was the farm, and it is assumed that there has been no change in the number of farms which we shall assume to be N=500. A sample of n=20 farms is selected, and the data is as follows:

Farm	2024	2026	Farm	2024	2026
1	12	14	11	11	14
2	22	25	12	17	19
3	38	37	13	12	12
4	15	18	14	22	23
5	18	20	15	14	16
6	31	30	16	26	28
7	15	15	17	08	09
8	20	21	18	16	15
9	10	12	19	13	15
10	25	12	20	19	20

Estimate the average number of cows for 2026 by Ratio Method of estimation and obtain the estimate of the MSE of Ratio estimator of Population Mean

3. Define the difference estimator for the population mean. Show that it is an unbiased estimator and its variance is given by where symbols have their usual meanings. 

 4. A population consisting of 6 clusters, each of size 6 is given below. The values of the study variable Y, noted on each of the units within each cluster are also provided. A random sample of size 3 clusters was selected from the population and 3 elementary units from the selected clusters were randomly chosen

Cluster	Y - values	Cluster	Y - values
1	2, 4, 6, 1, 3, 5	4	3, 2, 5, 1, 6, 4
2	2, 5, 3, 4, 7, 4	5	2, 4, 6, 8, 3, 5
3	4, 3, 6, 2, 1, 5	6	4, 1, 2, 7, 5, 3

Let the 2 nd, 4th and 6thclusters be selected randomly in the first-stage sample. Further, let the y-values 2, 7, 3 of the 2nd cluster; 6, 3, 1 of the 4th cluster and 7, 4, 3 of the 6th cluster be selected randomly for the second-stage sample

Estimate the population mean on the basis of the suggested estimator and compare it with the actual value of the population mean

5. Suggest an estimator of population mean in cluster sampling with unequal size clusters, which is based upon the means of selected clusters. Determine whether it is an unbiased estimator?

6. Below is given the plan and yields of 2² -Factorial Experiment involving 2 factors A and B each at two levels 0 and 1. Analyse the design

Block I
117 (1)	106 b	129 ab	124 a
Block II
124 ab	120 (1)	117 b	124 a
Block III
111 (1)	127 a	114 b	126 ab
Block IV
125 ab	131 a	112 b	108 (1)
Block V
95 ab	97 b	73 (1)	128 a
Block VI
158 a	81 (1)	125 ab	117 b

Does treatment effect A differs from treatment effect B?

7. Explain what is meant by a one-quarter fractional factorial experiment of a 2 ² factorial experiment. Give a notation which denotes the one-quarter fractional factorial design.

 

8. Let us consider the following Balanced Incomplete Block Design (B.I.B.D.) with parameters

Block Label	Design
I	1    3    4    5
II	1    4   6   7
III	1   2     5    7
IV	3    5   6   7
V	2    3    4    7
VI	1   2   3    6
VII	2   4   5   6

Obtain a derived design from the above Balanced Incomplete Block Design (B.I.B.D.) and find the parameters of the obtained design.

9. Define the Residual B.I.B.D. and Derived B.I.B.D. of a given symmetric B.I.B.D. Mention the rule of constructing a residual B.I.B.D. corresponding to a specific B.I.B.D.

then (X, ) is a (5, 3, 3)- B.I.B.D. Find the incidence matrix of this B.I.B.D.

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