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.
The difference estimator for the population mean is used when there is an auxiliary variable $X$ correlated with the study variable $Y$. Let the population consist of $N$ units, and a sample of size $n$ is drawn. Denote the population means as $\bar{Y}$ and $\bar{X}$, and the sample means as $\bar{y}$ and $\bar{x}$. The difference estimator _______ ______ _______ ____ _________.
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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 |
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
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
| 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.
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.
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.