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

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

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.

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

Does treatment effect A differs from treatment effect B?

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?

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

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

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. 

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:

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

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

Explain the terms PATH, CIRCUIT and CYCLES in context of Graphs

Explain and prove the Handshaking Theorem, with suitable example

Determine whether the above graph has a Hamiltonian circuit. If it has, find such a circuit. If it does not have, justify it.

What is a chromatic number of a graph? What is a chromatic number of the following graph?

Prove that the complement of  is G

Find the solution of the recurrences relation 

Find an explicit recurrence relation for minimum number of moves in which the n-disks in tower of Hanoi puzzle can be solved! Also solve the obtained recurrence relation through an iterative method.

Explain inclusion-exclusion principle and Pigeon Hole Principle with example.

What is the probability that a number between 1 and 10,000 is divisible by neither 2, 3, 5 nor 7?

How many words can be formed using letter of DEPARTMENT using each letter at most once?

i) If each letter must be used,

ii) If some or all the letters may be omitted.

There are three Companies, C1, C2 and C3. The party C1 has 4 members, C2 has 5 members and C3 has 6 members in an assembly. Suppose we want to select two persons, both from the same Company, to become president and vice president. In how many ways can this be done?

What is equivalence relation? Explain use of equivalence relation with the help of an example.