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:
Ratio and Regression Estimators, Systematic Sampling & Cluster/Multi-Stage Sampling - Agricultural Statistics | Agrigyan
1
1. $\left(\frac{1}{n} - \frac{1}{n'}\right) S_x^2$
2. $\left(\frac{1}{n'} - \frac{1}{N}\right) S_x^2$
3. $0$
4. $\left(\frac{1}{n} - \frac{1}{N}\right) S_x^2$
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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:
Island $i$ | Size of island $i$ sq miles | Palm trees $y_i$ | |||
|---|---|---|---|---|---|
| 1 | 1 | 10 | |||
| 29 | 5 | 50 | |||
| 36 | 2 | 25 | |||
| 29 | 5 | 50 |
3
Consider the following statements to estimate the total marine fish landings of our country:
(A) . Even though the sampling design adopted for the purpose is ‘stratified multistage random sampling’ and stratification
being done over space and time, cluster sampling and double sampling are also involved for selecting first stage units.
(B) . A month is divided into 3 groups of 10 consecutive days. Suppose that the 4th date (day) was selected from the group of
first 10 days of a month. Then 6 consecutive days from the selected day will be 4, 5, 6, 7, 8, 9 and these days are grouped
into 3 clusters of 2 consecutive days, i.e., dates (4 and 5) will form one cluster, while (6 and 7) and (8 and 9) will form the
other two clusters. From the 2nd group of 10 days, 6 days are systematically selected with a sampling interval of 10 days
from the first date selected from the group of first ten days. Thus 6 days In the 2nd group will be 14, 15, 16, 17, 18 and 19
forming 3 clusters of dates (14 and 15), (16 and 17) and (18 and 19). From the 3rd group. 6 days are selected with the
sampling interval of 10 days from the first day selected in the 2nd group, ie., the dates will be 24, 25, 26, 27, 28 and 29
whose clusters are (24 and 25), (26 and 27), (28 and 29). Thus there are 9 clusters of two days each in a month
(C) . Each maritime state is divided into several zones on the basis of geographical consideration and fishing practices. Nine
landing centres are selected at random from each zone for recording fish landings. Then 9 clusters, formed by a certain
procedure involving systematic sampling also within a month, are allotted randomly to the 9 selected landing centres
(D) . On the first day of observation data are collected from 6 AM to 12 Noon and in the next day, 12 Noon to 6 PM. The data
on night landings are collected by enquiry covering the period from 6 PM of the first day to 6 AM of the next day. Thus a 24
hour period is covered for a landing centre. This forms the ‘landing centre day’ and is the first stage sampling unit.
Choose the correct statements from the options given below:
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Consider the following statements:
(A) . Suppose the N units of a population are numbered 4 to N in some order. Suppose further that N is expressible as a
product of two integers n and k such that N = nk. Select a number between 1 to k. Suppose it is i (this i can be any of
numbers 1, 2, ..., k). Select the first unit, whose serial number is i. Select every k-th unit thereafter till the required sample
size of nis taken. Then the last unit will be i + (k-1) n fori=1, 2, ..., 0
(B) . Suppose the N units of a population are numbered 1 to N in some order. Suppose further that N is expressible as a
product of two integers n and k such that N = nk. Select a number between 1 to k. Suppose it is i (this i can be any of
numbers 1, 2, ..., k). Select the first unit, whose serial number is i. Select every k-th unit thereafter till the required sample
size of nis taken. Thus there are k possible systematic samples. In other words, a systematic sample is a simple random
sampling of one cluster unit from a population of k cluster units each of size n.
(C) . Higher heterogeneity within the units of a systematic sample makes the mean estimator more efficient.
(D) . Systematic sampling is more efficient than Simple Random Sampling (SRS) if the variation within the units in the
systematic sample is less than the variation within the units of the SRS sample
Choose the correct answer from the options given below:
Design of Experiments
4 Topics
Linear Estimation, Gauss-Markov Theorem & Basic Experimental Designs (CRD, RCBD, Latin Square Design)
Factorial Experiments (2^n, 3^n Series), Complete & Partial Confounding Mechanics
Incomplete Block Designs: Balanced Incomplete Block Designs (BIBD), PBIBD & Lattice Designs
Split-Plot, Strip-Plot, Crossover Designs & Analysis of Covariance (ANCOVA)
Sample Surveys
3 TopicsGeneral Agriculture
5 TopicsStatistical Genetics
3 Topics
Population Genetics: Hardy-Weinberg Equilibrium, Selection, Mutation, Migration & Inbreeding Coefficients
Quantitative Genetics: Variance Partitioning, Heritability, Repeatability, Path Analysis & Selection Indices
Mendelian Genetics, Linkage Estimation, Recombination Frequencies & Three-Point Test Cross Mapping
Statistical Methods I
3 TopicsStatistical Methods II
3 Topics
Multivariate Analysis (Wishart, Hotelling T2, MANOVA, PCA, Factor & Cluster Analysis) & Linear Programming Optimization
Hypothesis Testing: Neyman-Pearson Fundamental Lemma, UMP Tests, Likelihood Ratio Tests (LRT) & Non-Parametric Tests
Correlation, Multiple/Partial Regression, Collinearity Diagnostics & Econometric Residual Analysis
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29-06-2024