Question:
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 brief.
2. A doctor at a university with 3,100 students is interested in knowing how many of the
students get a flu shot.
i) Last year in a random sample of 100 students she found that 65 had gotten a shot. Give
the usual 95% confidence interval for the true proportion of students that got a shot.
i) This year she would like to know how large her sample size should be for the resulting
95% confidence interval for p, the proportion of students who had a shot, is no longer
than 0.08.
3. On the clas web page under Data sets you wil se the link, house sales. This is a link
to the R data file glen.rda which is a 597 by 3 matrix caled glen. You can load this
matrix directly into you working R directory using the command
load(url("http:/users.stat.umn.edu/~gmeden/clases/5201/datasets/glen.rda"))
For a recent year this contains information about 597 house sales in two zip codes in St
Paul. A row gives y, the sale price of a home in thousands of dollars, x, the amount of
taxes paid for the house in thousands of dolars and a zip code identifier.
To answer the following questions asume you know both the sales price and tax amount
for every house. In each case asume that we are estimating the population mean.
i) Give the true variance of the sample mean under simple random sampling without
replacement for a sample of size n = 60.
i) Suppose we form. three strata, the first 300, the next 200 and the final 97. Find the
optimal alocation for a sample of size 60 and the true variance of this estimator.
ii) Find approximately the true variance of the Ratio Estimator for a simple random of
size 60.
4) For this problem you wil be using the house sales population of problem 3. In your R
working directory run the following three commands
> set.sed(878787)
> smp1 smp2<-sort(sample(1:597,30,prob=tax))
i) For smp2 give the value of the Horvitz-Thompson estimator for the population total of
the house sales and its estimate of variance.
denote the sale price of the ith house and xdenote the amount of taxes for the
house. Consider the model . For each of the two samples find the ratio estimate of the population total
and their estimates of variance. 5) For a population with a y of interest and an auxiliary x which is correlated with y and a
design you need to write a program which alows you to compare the behavior. of three
estimators under repeated sampling from the design. The three estimators are the Horvitz-
Thompson (HT) estimator, the HT estimator that simultaneously constrains the weights
so that the add to the population size and are calibrated on x, (one way to do this is
explained in the R handout on the clas web page under the link calibration using
quadprog) and the estimator which asumes that the design was srs with replacement and
is again adjusted so that its new weights sum to the population size and are calibrated on
x. For each estimator you need to compute its average value and average absolute error
for 500 samples taken using the design.
Apply your function to the population of house sales in problem 3) for three different
designs. The designs are pps using x, using x in reverse order, i.e. in R use rev(x) and
simple random sampling without replacement. Take the sample size to be n = 30.
6) Again the population of interest is the population of house sales used in problem three.
In this case we are interested in estimating the median of the price of the houses sold. For
500 samples of size 30, where the sampling design is simple random sampling without
replacement, find the point estimate and 95% confidence interval for the median based on
the polya posterior. Also find the average absolute error of your point estimate and the
frequency of coverage of your interval estimate.
Next do the problem but now use pps sampling proportional to tax as your design. In this
case you wil need to use the function wtpolyap that is described in the R handout
usingpolyapost on the clas web page.