Ratio and Regression Estimators, Systematic Sampling & Cluster/Multi-Stage Sampling

For estimating population mean $\bar{Y}$ of a study variable $Y$ using double sampling, information on an auxiliary variable $X$ was collected from a large preliminary sample of size $n'$ units of a population in which $X$ alone was measured to furnish an estimate $\bar{x}'$ of population mean of $X$, as the information about $X$ of the population was not readily available. From this large preliminary sample, a smaller sample of size $n$ was selected and the information on the main variable $Y$ was collected and sample mean $\bar{y}$ and corresponding sample mean $\bar{x}$ were computed. In both phases, SRSWOR was done. The estimates were then used to obtain a ratio estimator $\bar{y}_d = \frac{\bar{y}}{\bar{x}} \bar{x}'$. While deriving the variance of this ratio type mean estimator, the following errors were considered: $e_1 = \frac{\bar{x} - \bar{x}'}{\bar{X}}$ and $e_2 = \frac{\bar{x}' - \bar{X}}{\bar{X}}$. Then, if $S_x^2$ is the population variance of $X$, $E(e_1 e_2)$ is:

Consider the problem of estimating the total number of palm trees on 100 islands in a tropical paradise. The area of each island is known and it is reasonable to think that the number $y_i$ of palm trees on each island is approximately proportional to the size of the island. The total size of these 100 islands is 100 square miles. Let the units selected with replacement in 4 independent draws be the islands numbered 1, 29, 36 and 29 with information details as follows:

Then the Horvitz-Thompson estimate for estimating the total number of trees is:

Consider the following statements to estimate the total marine fish landings of our country:

Consider the following statements:

Choose the correct answer from the options given below: