ST 432 Reiland
HW 4
page 1
Homework 4 (20 points) due Mon. 3/25/2015
General Notes: Homework will be due at the beginning of class, unless I notify you otherwise. Please
hand in a typed, stapled assignment. Your homework should be not only legible, but well-organized and
of a semi-professional appearance. It is in your best interest to keep an electronic copy of all assignments
you submit.
The course text is on reserve in D.H. Hill Library.
1.
2.
3.
4.
Chapter 5, p. 154, #5.7
Chapter 5, p. 154, #5.8
Chapter 5, p. 154, #5.9
See problem below
(p. 154, #5.6 was assigned in homework 3; it is included to serve as a reference for problem 1 below)
p. 154, #5.6
A school desires to estimate the average score on a reading comprehension exam for students in the
sixth grade. The school’s students are grouped into three tracks, with the fast learners in track I and
the slow learners in track III. The school decides to stratify on tracks because this method should
reduce the variability of test scores. The sixth grade contains 55 students in track I, 80 in track II and
65 in track III. A stratified random sample of 50 students is proportionally allocated and yields
simple random samples of n1 = 14, n2 = 20, n3 = 16 from tracks I, II, and III. The test is administered
to the sample of students; the results are available in the Excel file linked to under HW #3 on our
class webpage.
a. Estimate the average score for the sixth grade, and place a bound on the error of estimation.
b. Construct parallel boxplots for these data and comment on the pattern you see. Do you think
there could be a problem in placing students in tracks?
c. Estimate the difference in average scores between track I and track II students. Are track I
students significantly better, on average, than track II students?
1. (p. 154, #5.7)
Suppose the average test score for the class in Exercise 5.6 is to be estimated again at the end of
the school year. The costs of sampling are equal in all strata, but the variances differ. Find the
optimum (Neyman) allocation of a sample size 50, using the data in Exercise 5.6 to approximate
the variances.
2. (p. 154, #5.8)
Using the data in Exercise 5.6, find the sample size required to estimate the average score, with a
bound of 4 points on the error of estimation. Use proportional allocation.
ST 432 Reiland
HW 4
page 2
3. (p. 154, #5.9)
Repeat Exercise 5.8 using Neyman allocation. Compare the results with the answer to Exercise
5.8.
4. For a national comparison, a real estate magazine is interested in estimating the average price of a
house in an area of Raleigh. The area is composed of three neighborhoods consisting of 200, 250
and 400 homes, respectively. In each neighborhood, a SRS of homes is taken. The data are
summarized in the following table.
Neighborhood
1
2
3
Sample size
12
15
18
Sample Mean
150,000
223,000
318,000
Sample Variance
400,000
210,000
450,000
a. Using an unbiased estimator, give an estimate of the mean price of a house in this area of
Raleigh and determine the bound on the error of estimation.
b. A Raleigh real estate company would like to obtain its own sample to estimate the average
selling price of a house in this area of Raleigh with a bound of $150 on the error of
estimation. To access the county database to get the data, the real estate company will have
to pay the county c1 = $5, c2 =$3, and c3 = $8 for each observation from neighborhood 1, 2,
and 3, respectively. Determine the total sample size n and the sample sizes n1, n2 , and n3
from each stratum. Use the data in the table above to approximate the variances.