HW5.

Problem 6.3.

The structure described is completely hierarchical,
except that some individuals (those on the 30th line 
of selected pasges) are sampled together with ALL of
their fellow Household members. Thus, there are:

multiple census pages associated with each 
enumeration district, 1 in 11 selected randomly (SRS);

30 lines per page, systematic sample of every 5th line;
(with extra questions asked of the individual 
corresponding to the 30th line; also ALL individuals 
in the Household or "sample unit" for this individual 
on the 30th line are questioned as well.

The sample has SRS, systematic, and cluster aspects, 
and you are to discuss them in the problem.

The "sample" in #3 consists of the individuals on the 
30th line of the (randomly) selected census pages 
TOGETHER WITH ALL OF THE MEMBERS OF THE SELECTED 
INDIVIDUALS' HOUSEHOLDS.  So question (c) requires some
interpretation: what are the inclusion probabilities 
for individuals in households of size k, for each k ? 
(You can assume that the census pages are constructed 
in such a way that all individuals are equally likely 
to fall on the 30th line of a page.)


Problem 6.16.

In #16, take the first expression in (6.13) as the 
left-hand side of an equation and the second expression 
as the right-hand side. Then successively add or subtract 
the same terms  from both sides of the equation, using 
only algebra and (6.10)-(6.11), until you get 0=0. Try 
to separate the double-sum terms in which k=i from those 
with k>i and k<i, and observe that the summations over 
all i,k with  k<i and over all i,k with k>i give exactly 
the same result.

Problem 6.18.

In #18, you have to understand that there is only one 
Horvitz-Thompson estimator, and that its precise form is 
(6.12), but you have to figure out what the PSU's i and 
their inclusion probabilities are. We began discussing 
this in class today (11/16) and will go over it in 
further detail next Monday .