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.
In the field of design of experiments, factorial designs are widely used to study the effects of multiple factors simultaneously. A full factorial experiment in a $2^k$ design investigates all possible combinations of $k$ factors, each at two levels, usually denoted as low $(-1)$ and high $(+1)$. For instance, a $2^2$ factorial design involves two factors, say $A$ and $B$, each at two levels, leading to $2^2 = 4$ treatment combinations. These four combinations are $(--), (-+), (+-), (++))$. Such an arrangement allows the experimenter to estimate the main effects _______ ___ ________ _____ ___ ________ _______.
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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
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 |
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.
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.
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?
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 |
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 |