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on or before Thursday, December 19 at 1:30 pm
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of the School of Statistics
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You must work alone and may only discuss these questions with the TA or Professor Meeden.
You may use the class notes, the text and any other sources of material you have access to.
Start each answer on a new page and make sure that you name is on each page.
If I discover a misprint or error in a question I will post a correction on my class web page. In
case you think you have found an error you should check the class home page before contacting us.
1. Find a recent survey reported in a newspaper, magazine or on the web. Briefly describe
the survey. What are the target population and sampled population? What conclusions are drawn
from the survey in the article. Do you think these conclusions are justified? What are the possible
sources of bias in the survey? Please be specific but brief.
2. Consider a population where you have in hand the values of an auxiliary variable which you
know from past experience has a large positive correlation with the y characteristic of interest. For
this problem use the auxiliary variable xprob2 which you can load into your R work space by doing
i) Find a sensible stratification based on xprob2. Briefly explain why you think your choice is a
ii) For your stratification find the optimal allocation for a sample of size n.
3. To do this problem you need to load into your R working director the file f19clus.RData
using the commands.
This file contains two R objects. The first is clussmp which contains the results of a two stage
cluster sample where 17 clusters where selected out of a population of 100 clusters. At both stages
simple random sampling without replacement was used. The second object is clussz which contains
the sizes of the clusters in the sample. The average cluster size of the 83 clusters not in the sample
Give the value of your estimate for the population mean and an estimate of its variance. Briefly
justify the choice of your estimate.
4. Let the distribution of y1, . . . , yn given θ be independent and identically distributed Bernoulli(θ)
random variables and let the distribution for θ be Beta(α, β) where α > 0 and β > 0.
i) Find an expression for the probability distribution p(y1, . . . , yn) and write a function in R
which will calculate this value.
ii) Let f1 and f2 be two different beta distributions and let 0 < λ < 1 be given. Then f =
λf1 + (1 − λ)f2 is a new prior distribution for θ. Given Pyi find the posterior distribution of θ
when f is the prior.
5. In Minnesota there are 327 school districts. School districts are classified as urban, suburban
or other. In these districts there are 987 elementary schools. The total number of students in these
schools was 395,449. A state agency wants an estimate of the total number of student absences
there were in all the elementary schools last January. Note, an absence is a day when the school is
open but a student is not there. Assume the agency has a list of the schools and the total number
of students in each school.
Suggest a sampling design for the agency. What additional information, if any, could be provided
by the state agency to help you choose the design. Do you see any problems in implementing your
design? Specify your estimator and its estimated variance. Be brief but specific.
6. For this problem you need to get the data with the commands
The data set has three vectors. The vector y is the quantity of interest while x1 and x2 are auxiliary
variables. Each are of length 2,000 and x1 is an increasing function of its labels. You be asked
to do several simulation studies. In each one, the sample size n = 100 and you should take 300
samples. For both parts you should use three different designs: rep(1,2000), seq(3,1,length=2000)
i) In this part you will be estimating the population total of y. You need to compare three
estimators. The first is the usual Horvitz-Thompson estimator. The second is the calibrated HT
estimator where the calibrate weights must satisfy three conditions. The sum of the calibrated
weights should be 2000. In addition the sum of the calibrated weights times the sample values of
x1 and x2 should equal the population totals of x1 and x2. Sometimes the design weights are not
available when the data are analyzed. So for the third estimate we always assume that the design
was simple random sampling without replacement even when is not. The corresponding design
weights for this case are calibrated with the same three conditions that were used when creating
the second estimator.
For each estimator you should compute its average value, its average relative bias,(est-tru)/tru,
its average absolute error, the average length of its approximate 95% confidence interval and its
frequency of coverage.
ii) In this part we are interested in estimating
the ratio of the total amount of y that belongs to the quarter of the poluation consisting of the 500
units with the smallest values of x1 to the quarter of the population consisting of the 500 units with
the largest values of x1. In addition to the calibration constraints used in part i) you should add
the constraints that the weights for the units in the smallest group in the sample and the weights
for the units in the largest group in the sample should each sum to 500.
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