Final-Assignment-Q.1 (12 marks)
In a list of N individuals, we are interested in a variable y. The individuals are identified by their order on the list, so their order goes from 1 (for the first) to N (for the last). We use systematic sampling with interval k to select n individuals from the list. We assume that: k = N ∈ N.
Let S2 = 1 Σn (yij − y¯i)2 be the sample variance for the ith cluster, where y¯i = 1 Σn denote the ith cluster mean.
(a) (4 marks) Show that everything happens as if we selected a unique cluster of individuals from a population pre-divided into clusters. We will specify what the clusters are, what their size is, and how many there are in the population.
(b) (4 marks) Let yij be the value of y for jth record counted in cluster i, and µ denote the population mean for y.
(i) What is the unbiased estimator µ of µ? Show that µ is effectively unbiased.
(ii) What is the expression of the true variance of µ, as a function of µ, the ith cluster mean y¯i and k.
(c) (2 marks) Considering the natural splitting of the population into k clusters, show that the general expression of the population variance σ2 can be decomposed
(d) (1 mark) Show that if N is large, and if we denote:
(1 Mark) Show that systematic sampling is more precise than simple random sampling if and only if: σ2 < SSW by considering N as very large with respect to n.
Question 2. (12 marks)
Assume that the following are data from cluster sampling with simple random sampling of clusters.There are 10 clusters (primary units) and a total of 100 secondary units in the population. For each of the n = 3 selected clusters, yi is the cluster total for the variable of interest and Mi is cluster size: y1 = 4, M1 = 5; y2 = 12, M2 = 20; y3 = 7, M3 = 10.
(a) (4 marks) Give an unbiased estimate of the population total. Estimate the variance of that estimator.
(b) (4 marks) Give the ratio-type estimate of the population total and estimate the variance of that estimator.
(c) (4 marks) Assuming that the sample was obtained with selection probabilities proportional to cluster size (PPS), with replacement, give an unbiased estimate of the population total and estimate the variance of that estimator.
(d) (2 marks) Which of the sampling strategies in (a), (b) and (c) would you prefer?
Final-Assignment-Q.3 ( 26 marks)
We consider a population of individuals of size M =62 000. This population is made up of N = 15 000 households. We denote:
Mi = Size of the household i (Number of individual), yi = Number of men of the household i
The data from the sample required for the calculations are shown in Table below.
|
Household identifier |
Mi |
yi |
|
1 2 3 4 5 · · · · · · ·2·5· 26 27 28 29 30 |
5 6 3 3 2 · · · · · · · 2· · 4 3 4 2 4 |
1 3 1 1 1 · · · · · · · 1· · 3 1 2 1 2 |
In this question we have three parts: Part 1, Part 2 and Part 3. All these parts use the same sample date above.
Q3-Part 1. ( 9 marks):
First, we conduct a simple random sampling of n = 30 households among N (sample S), and we survey all the individuals from each of the n households selected.
(a) ( 1 mark) What do we call this type of sampling?
(b) ( 2 marks) What are the selection probabilities of the households, and what are the selection probabilities of the individuals?
(c) ( 3 marks) Let τ be the total number of men in the population.
(i) Give an unbiased estimate τb of τ .
(ii) Estimate the variance of that estimator.
(d) (3 marks) For this question, we are trying to estimate the total using a ‘ratio by size’ expression.
(i) Give ratio-type estimate τbr of τ .
(ii) Estimate the variance of that estimator.
Q3-Part 2. ( 9 marks):
Second, we decide to select n=30 households proportionately to their size Mi. We consider that this sampling, performed in reality without replacement, can be likened to be a sampling design with replacement.
(a) ( 1 mark) Give, as a function of Mi, the selection probability πi of household i at the time ofeach primary drawing.
(b) ( 4 Marks) Give an unbiased estimate τbpps, and give a 95% confidence interval estimated for τ.
(c) ( 4 marks) What do you conclude in comparison to the results of Part 1.(c) and Part 1.(d)?
Q3-Part 3. ( 8 marks) The goal of this part is to compare the accuracy of the estimate from SRS with one obtained in Part 1. (c) and (d). We assume that the individuals in the selected households had been selected by simple random sampling directly from the population of size M.
(a) 2 marks) How would we estimate the total τ . let τsrs be the obtained estimator.
(b) ( 2 marks) What would be the estimated variance of τ
(c) ( 2 marks) Find the relative efficiency of the estimator in Part 1 (c) τ relative to τ. What do you conclude?
(c) ( 2 marks) Find the relative efficiency of the estimator in Part 1 (d) τ relative to τ What do you conclude?
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