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Forest Survey
Forestry researchers used a two-stage cluster sampling design to
estimate the total volume of 9000 black cherry trees in a forest region.
The region was first divided into 100 transects. Then 3 of these
transects were selected by a probability sampling design. Finally within
each of these transects some of the trees were selected using simple
random sampling. The total number of trees within each of these three
transects as well as the number of sampled trees and the mean volume (in
cubic feet) of those sampled trees are shown in the table below.
|
|
Number of Trees
|
|
|
Transect
|
Total Trees
|
Sampled Trees
|
Mean Volume
|
|
1
|
100
|
20
|
10
|
|
2
|
110
|
30
|
12
|
|
3
|
90
|
25
|
11
|
Use the information given above to answer in the following concerning
the estimation of the total volume of all the black cherry trees in the
region.
Assume that the transects were selected using simple random
sampling. Confirm that the estimate of the total volume of all the
trees in the region using the unbiased estimator is 110333.3
cubic feet, and that the estimate using the ratio estimator is
99300 cubic feet.
Assume that the transects were selected using sampling with
replacement with probabilities proportional to size.
Confirm that the estimate of the total volume of all the trees in the
region using the Hansen-Hurwitz estimator is 99000 cubic
feet.
Educational Testing
Consider a cluster sampling design using a population of 2000 second
grade students in a school district. Researchers would like to estimate
the average score of these students on a particular test based on a
sample of students since the testing is expensive. They decided to use a
two-stage cluster sampling design where they selected a sample of 3
classrooms of students out of the 100 classrooms in the district, and
then selected some of the students from each of these selected
classrooms using simple random sampling. The table below shows the mean
test score for the sampled students from each of the sampled classrooms.
It also shows the number of students in the classroom as well as the
number of students that were sampled for testing.
|
|
Number of Students
|
|
|
Classroom
|
Total Students
|
Sampled Students
|
Mean Score
|
|
1
|
16
|
5
|
70
|
|
2
|
20
|
7
|
80
|
|
3
|
19
|
6
|
75
|
Use the information given above to answer in the following concerning
the estimation of the mean test score for all the second grade students
in the school district.
Assume that the classrooms were selected using simple random
sampling. Confirm that the estimate of the mean test score for all
second grade students in the district using the unbiased
estimator is approximately 69.1, and that the estimate using the
ratio estimator is approximately 75.4.
Assume that the classrooms were selected using sampling with
replacement with probabilities proportional to size.
Confirm that the estimate of the mean test score for all second grade
students in the district using the Hansen-Hurwitz estimator is
75.
Optimum Sample Sizes for Educational Testing
Suppose that based on either a previous survey or a pilot survey the
researchers in the previous problem had estimates of the
between-classroom and within-classroom mean squares. Assume that the
researchers also have a fixed budget of 500 units, and that the
estimated cost per classroom selected is 20 units, and the estimated
cost per student selected and tested is 5 units (here the cost units
represents both financial cost as well as time). The table below shows
the approximate optimal sample sizes for the number of students per
classroom to sample as well as the number of classrooms to sample
(assuming simple random sampling at both stages) for four hypothetical
populations.
|
|
Mean Square
|
|
|
|
Population
|
between
|
within
|
\(m_{\text{opt}}\)
|
\(n_{\text{opt}}\)
|
|
A
|
62.5
|
50.00
|
17.89
|
4.57
|
|
B
|
250.0
|
25.00
|
2.98
|
14.33
|
|
C
|
500.0
|
15.00
|
1.57
|
17.95
|
|
D
|
750.0
|
6.25
|
0.82
|
20.75
|
Not all classrooms are the same size, but they are usually around 20
students. So the calculations use \(\bar{M}\) = 20 in their calculations.
Confirm the optimal sample sizes shown above (within rounding
error).