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IGNOU MST-020 Solved Assignment | June 2024 to June 2025 | Survey Sampling and Design of Experiments-II | MSCAST Sample Solution

MST-020 Solved Assignment 2025

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 ({\overset{―}{Y}}_{Reg})

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

2^{4}

full factorial design will be denoted by a

2^{2}

Factorial design.

e) The Cluster estimator

{\overset{―}{\bar{y}}}_{n}

. becomes an unbiased estimator of population mean only if

M_{i}

and

{\bar{Y}}_{i}

. are uncorrelated

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	28	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.

Define the difference estimator for the population mean. Show that it is an unbiased estimator and its variance is given by $V ({\overset{―}{Y}}_{D}) = \frac{N - n}{Nn} S_{Y}^{2} (1 - ρ^{2})$ , where symbols have their usual meanings.
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}, 4^{th}

and

6^{th}

clusters be selected randomly in the first-stage sample. Further, let the

y

-values

2, 7, 3

of the

2^{nd}

cluster;

6, 3, 1

of the

4^{th}

cluster and

7, 4, 3

of the

6^{th}

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.

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?
Below is given the plan and yields of $2^{2}$ -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?

Explain what is meant by a one-quarter fractional factorial experiment of a $2^{k}$ factorial experiment. Give a notation which denotes the one-quarter fractional factorial design.
Let us consider the following Balanced Incomplete Block Design
(B.I.B.D.) with parameters $ϑ = b = 7, r = k = 4, λ = 2$ :

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		45

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

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.
Let $X = {1, 2, 3, 4, 5}$ and $A = {{1, 2, 3},, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}$ , ${2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}$ ; then $(X, A)$ is a $(5, 3, 3)$ – B.I.B.D. Find the incidence matrix of this B.I.B.D.

Expert Answer

Question:-1

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

Question:-1(a)

If X and Y are uncorrelated, the $V ({\overset{―}{Y}}_{Reg})$ reduces to that of Variance of Ratio Estimator of Population mean.

Answer:

The statement is false.

Justification:

The variance of the regression estimator of the population mean, denoted as

V ({\overset{―}{Y}}_{Reg})

, is generally lower than the variance of the ratio estimator, particularly when there is a strong linear relationship between

X

and

Y

. The regression estimator exploits this relationship by reducing variability due to the correlation between

X

and

Y

X

and

Y

are uncorrelated, the covariance term that helps reduce variance in the regression estimator disappears. However, this does not imply that the variance of the regression estimator will become equal to the variance of the ratio estimator. The regression estimator and the ratio estimator are derived differently and rely on different assumptions about the relationship between

X

and

Y

. Consequently, even if

X

and

Y

are uncorrelated,

V ({\overset{―}{Y}}_{Reg})

does not reduce to the variance of the ratio estimator.

Question:-1(b)

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

Answer:

The statement is false.

Justification:

In the General Class of estimators, which includes estimators such as the mean per unit estimator, ratio estimator, and regression estimator, it is not true that only the regression estimator is unbiased. In fact:

The mean per unit estimator (i.e., the sample mean) is an unbiased estimator of the population mean, as it is simply the average of the sample observations and does not depend on auxiliary variables.
The regression estimator can be unbiased under certain conditions, such as when there is a linear relationship between the study variable and the auxiliary variable with known parameters.
The ratio estimator is generally biased but can be nearly unbiased for large sample sizes if the correlation between the auxiliary variable and the variable of interest is high and the sample is large.

Thus, both the mean per unit estimator and the regression estimator can be unbiased under appropriate conditions, not just the regression estimator.

Question:-1(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.

Answer:

The statement is false.

Justification:

To check if an Incomplete Block Design is a Balanced Incomplete Block Design (BIBD), it must satisfy the following conditions:

$b$ : Number of blocks.
$v$ : Number of treatments.
$k$ : Number of treatments per block.
$r$ : Number of times each treatment appears across all blocks.
$λ$ : Number of times each pair of treatments appears together in the blocks.

For a design to be a BIBD, the parameters must satisfy the equation:

λ = \frac{r (k - 1)}{v - 1} .

In this case, the given parameters are:

$b = 8$ (number of blocks),
$k = 3$ (number of treatments per block),
$v = 8$ (number of treatments),
$r = 3$ (number of replications of each treatment).

Step 1: Check if $λ$ is an integer

Plugging these values into the BIBD condition:

λ = \frac{r (k - 1)}{v - 1} = \frac{3 \cdot (3 - 1)}{8 - 1} = \frac{3 \cdot 2}{7} = \frac{6}{7} .

Since

λ = \frac{6}{7}

is not an integer, the design does not satisfy the BIBD conditions.

Conclusion

The given Incomplete Block Design is not a Balanced Incomplete Block Design because

λ

is not an integer.

Question:-1(d)

A One-half fractional factorial design of a $2^{4}$ full factorial design will be denoted by a $2^{2}$ Factorial design.

Answer:

The statement is false.

Justification:

A one-half fractional factorial design of a

2^{4}

full factorial design means we are taking half of the total number of experimental runs that would be used in a

2^{4}

design.

In a $2^{4}$ full factorial design, there are $2^{4} = 16$ experimental runs, as each of the 4 factors has 2 levels.
A one-half fractional factorial design means that only half of these runs are used, resulting in $16 / 2 = 8$ runs.

In fractional factorial notation, this one-half fraction of a

2^{4}

design is written as a

2^{4 - 1}

2^{3}

design, not a

2^{2}

design. The notation

2^{4 - 1} = 2^{3}

indicates that this is a fraction of the

2^{4}

design with 3 "effective" factors, even though we are still considering 4 factors.

Thus, a one-half fractional factorial design of a

2^{4}

design is denoted by

2^{4 - 1} = 2^{3}

, not

2^{2}

Question:-1(e)

The Cluster estimator ${\overset{―}{\bar{y}}}_{n}$ . becomes an unbiased estimator of population mean only if $M_{i}$ and ${\bar{Y}}_{i}$ . are uncorrelated.

Answer:

The statement is false.

Justification:

In cluster sampling, the cluster estimator

{\overset{―}{\bar{y}}}_{n}

(the mean of cluster means) is an unbiased estimator of the population mean regardless of whether the cluster sizes

M_{i}

and the cluster means

{\bar{Y}}_{i}

are correlated or uncorrelated. The unbiasedness of

{\overset{―}{\bar{y}}}_{n}

does not depend on the correlation between

M_{i}

and

{\bar{Y}}_{i}

The estimator

{\overset{―}{\bar{y}}}_{n}

is simply the average of the means of randomly selected clusters, and as long as the clusters are selected randomly, it provides an unbiased estimate of the population mean. However, if

M_{i}

and

{\bar{Y}}_{i}

are correlated, this can affect the variance of the estimator but not its unbiasedness.

Therefore, the correlation between

M_{i}

and

{\bar{Y}}_{i}

affects the precision of the estimator, not its unbiasedness.

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IGNOU MST-020 Solved Assignment | June 2024 to June 2025 | Survey Sampling and Design of Experiments-II | MSCAST

Sample Solution

MST-020 Solved Assignment 2025

Expert Answer

Question:-1

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

Question:-1(a)

If X and Y are uncorrelated, the V ( Y ― Reg ) V Y ¯ Reg V( bar(Y)_(“Reg “))\mathrm{V}\left(\overline{\mathrm{Y}}_{\text {Reg }}\right)V(Y―Reg ) reduces to that of Variance of Ratio Estimator of Population mean.

Answer:

Justification:

Question:-1(b)

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

Answer:

Justification:

Question:-1(c)

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

Answer:

Justification:

Step 1: Check if λ λ lambda\lambdaλ is an integer

Conclusion

Question:-1(d)

A One-half fractional factorial design of a 2 4 2 4 2^(4)2^424 full factorial design will be denoted by a 2 2 2 2 2^(2)2^222 Factorial design.

Answer:

Justification:

Question:-1(e)

The Cluster estimator y ¯ ― n y ¯ ¯ n bar(bar(y))_(n)\overline{\bar{y}}_ny¯―n. becomes an unbiased estimator of population mean only if M i M i M_(i)\mathrm{M}_{\mathrm{i}}Mi and Y ¯ i Y ¯ i bar(Y)_(i)\bar{Y}_{\mathrm{i}}Y¯i. are uncorrelated.

Answer:

Justification:

If X and Y are uncorrelated, the $V ({\overset{―}{Y}}_{Reg})$ reduces to that of Variance of Ratio Estimator of Population mean.

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.

Step 1: Check if $λ$ is an integer

A One-half fractional factorial design of a $2^{4}$ full factorial design will be denoted by a $2^{2}$ Factorial design.

The Cluster estimator ${\overset{―}{\bar{y}}}_{n}$ . becomes an unbiased estimator of population mean only if $M_{i}$ and ${\bar{Y}}_{i}$ . are uncorrelated.