Assignment 5
TWO STAGE CLUSTER DESIGNS
due: November 25, 1997
 
     In stratification, we design the strata to be homogeneous within 
strata, but heterogeneous between strata.  We then sample in each stratum 
with the aim of improving on the precision of simple random sampling by 
ensuring a representative sample.
     In cluster sampling, we try to make the clusters homogeneous between 
clusters, but with each cluster roughly representative of the population 
with respect to the variables being measured.  When the cost of visiting 
each cluster is high (such as when travel costs are significant or when
preparing a frame of USU's within each cluster is costly), we try 
to reduce costs by limiting our sample to a small number of clusters.  
Although for a fixed sample size, cluster sampling is usually less 
precise than simple random sampling, we hope that by controlling our 
sampling costs we can increase the sample size to more than compensate 
for the lower precision.

     We will use the data set lc2stg.a.  lc2stg.a was created by Poisson
sampling from Lockhart City with the districts playing the role of PSU's.
PSU's were chosen with probability proportional to size and with an expected
first stage sample size of 10.  Thus
    pi1i = Prob of selecting cluster i = 10*Ni/N
The overall sampling scheme is supposed to be self-weighting with weights pik =
.01.  Thus
    .01 = pik = pi1i * pik|i = (10*Ni/N) * (ni/Ni), or
	ni = .01 * N/10 = 19.664.
Since ni must be integer, ni was chosen to be 20.

	The variable of interest is the average price a household is willing to
pay for cable TV service.  We consider two possible estimators:

(i) A ratio estimator with denominator variable the assessed valuation of the
house.  To use this estimator we must use that the average assessed value of
a house in Lockhart City is $71034.

(ii) A poststratified estimator with strata defined by the assessed valuation
of the house.  The Stephens County Tax Assessor has provided the following
information about assessed valuations in Lockhart City:

DISTRIBUTION OF RESIDENTIAL ASSESSED VALUATIONS LOCKHART CITY: 
House Value       Number
0 to 39999         1021
40000-49999        1786
50000-59999        2724
60000-69999        2603
70000-79999        4592
80000-89999        4608
90000-99999        1788
100000 and above    542

TOTAL             19664

Exercises:

1. Calculate both estimates together with their standard errors.

2. Notice that both (i) and (ii) use household assessed valuation as an
auxiliary variable.  What advantages and disadvantages can you think of
that would lead one to prefer one estimator over the other.

3. Use a regression estimate with predictor variables, intercept, family
size, and household valuation to estimate the average price a household
is willing to pay for cable TV service.  Remember that the population of
Lockhart City is 57007.

4. Estimate with standard errors (review exercise):
1. The proportion of households in Lockhart City willing to pay $10 for
cable TV service.
2. The total number of TV's in households willing to pay $10 for cable
TV service.
3. The average number of hours per week watching movies (on TV) in
households willing to pay $10 for cable TV service.
4. The average proportion of children (number of children/family size) in
households willing to pay $10 for cable TV service.