STAT 3507 Exercise Questions Solutions
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Bernoulli Sampling Questions
Cluster Sampling Questions
Ratio, Regression and Difference Estimation
Stratified Random Sampling
Systematic Sampling
Two-stage Cluster Sampling Questions
1-A university professor who is correcting 600
written examinations decides to get a preliminary idea
of the passing rate on the test. He decides to use a simple randomized scheme
to single out a smaller number of exam copies for first-hand
correction. In passing through the pile of exams, he tosses
an ordinary six-sided die, once for each exam copy. If the die shows a 6, he
corrects the corresponding exam, otherwise not. Suppose the
sample selected in this way consists of 90 students and that 60 out of these are found to
have passed.
a.
Identify the sampling design implemented.
b.
Estimate the total number of students who
passed the exams
c.
Compute a 95% confidence interval, based on the normal
approximation, for the number of passing students (among the 600).
d.
Repeat (b) and (c) using the alternative (improved)
estimator.
2-
Table: Number of
Employees and Range
|
Division
|
Number of employees
|
Cumulative range
|
|
1
|
1200
|
1— 1200
|
|
2
|
450
|
1201 —1650
|
|
3
|
2100
|
1651— 3750
|
|
4
|
860
|
3751— 4610
|
|
5
|
2840
|
4611 — 7450
|
|
6
|
1910
|
7451— 9350
|
|
7
|
390
|
9361 — 9750
|
|
8
|
3200
|
9751 - 12950
|
|
|
12950
|
|
i.
Suppose 2011, 7972 and 10281 are the random
numbers generated between 1 and 12950. Using this information what are the clusters selected?
ii.
Suppose the total number of sick days used
by the three sampled divisions during the past quarter are respectively, y, =
4320 y2 = 4160 y3 = 5790 .
Estimate
the average number of sick clays used per person for the entire firm and place a bound on the
error of estimation.
Ratio
and Regression Estimators (Examples) under Simple Random Sampling Without
Replacement
Set
1: Estimation of Ratio
Let
us consider a community having eight community areas. Suppose that we wish to
estimate the ratio R of total pharmaceutical expenses Y to total medical
expenses X among all persons in the community. To do this, a simple random
sample of two community areas is to be taken and every household in each sample
community area is to be interviewed. The data for the community areas am as
given in Table 1 below:
Table
1: Pharmaceutical Expenses and Total Medical Expenses Among All
Residents
of Eight Community Areas
Community
Area Total Pharmaceutical Expenses, Y ($) Total Medical Expenses, X ($) 1
100,000 300.000 2 50,000 200.000 3 75,000 300.000 4 200,000 600.000 5 150,000
450,000 6 175,000 520.000 7 170,000 680.000 8 150,000 450.000 Total 1,070,000
3,500,000
Suppose
that community areas 2 and 5 were selected in the sample.
a.
State the target population for the study b. What are the elements in this
sampling design? c. What am the sampling units? d. Estimate the ratio of total
pharmaceutical expenses to total medical expenses. e. Estimate the variance of
the estimate in (d). f. Based on the sample data, how many community areas
would have to be sampled if it is desired to estimate the population ratio with
95% certainty to within 5% of its true value?
Set
2: Estimation of Total
Suppose
that a road having a length of 24 miles traverses areas that can be classified
as urban and rural and that the road is divided into eight segments, each
having a length equal to 3 miles. A sample of three segments is taken, and on
each segment sampled, special equipment is installed for purposes of counting
the number of total motor vehicle miles traveled by cars and trucks on the
segment during a particular year. In addition, a record of all accidents
occurring on each sample segment is kept.
The
number of truck miles and the number of accidents in which a truck was involved
during a certain period are given in Table 2 for each of the eight segments in
the population.
Suppose
that we take a simple random sample of three segments for purposes of
estimating the total number of truck miles traveled on the road.
Table
2: Truck Miles and Number of Accidents Involving Trucks by Type of Road Segment
Segment
Type Truck Miles x 1000 Number of Accidents Urban 6327 oo in Cs CT in ,-1 cr%
ct Rural 2555 Urban 8691 Urban 7834 V, Rural 1586 Rural 2034 Rural 2015 Rural
3012
Suppose
the segments 1, 3 and 4 were selected in the sample.
a.
Estimate the total number of truck miles traveled on the mad using the
customary and ratio estimators. b. Estimate the 95% confidence interval for the
total number of truck miles using the customary and the ratio estimators. c.
How do these estimators compare? d. Based on the sample data, how many road
segments would have to be sampled if it is desired to estimate the total number
of truck miles with 95% certainty to within 10% of its true value? Use the
customary estimator and the ratio estimator.
Example I:
Cavities and Post -stratification
Two dentists conduct a survey on the
condition of teeth of 200 children in a village. The
first dentist selects using a simple
random sampling 20 children among 200, and counts
the data in the sample according to
the number of teeth with cavities. The results are
presented
in
the Table
I.
The second dentist examines the 200
children but with the sole
goal of determining who has no
cavities. He notices that 50 children are in this category.
Table I: Teeth with cavities
Number of teeth with cavities
0
1
2
3
4
5
6
7
8
Number of children
8
4
2
2
1
2
0
0
1
a)
Estimate the mean number of teeth
with cavities per child in the village using only
the results of the first dentist.
What is the accuracy of the unbiased estimator
obtained? Estimate this accuracy and
the associated confidence interval.
b)
Propose another estimator for the
mean number of teeth with cavities per child
using the results of the two
dentists. Calculate the new estimate, and estimate the
gain in efficiency obtained.
Example
II: Foot Size
The
director of a business that makes shoes wants to estimate the average length of
right
feet of adult men in a city. Let _v be the characteristic length of right foot
(in
centimetres)
and x be the height of the individual (in centimetres). The director
knows
moreover from the results of a census that the average height of adult men in
this
city is 168 cm. To estimate the average foot length, the director draws a
simple
random
sample without replacement of 100 adult men. The results are the following:
A A
Y=169,
7:24, sn,=l5. st=lO, s‘_=2.
Knowing
that 400,000 adult men live in this city,
a)
Calculate the I-lorvitz-Thompson (customary) estimator, the ratio estimator,
the
difference estimator and the regression estimator.
b)
Estimate the variances of these four estimators.
c)
Which estimator would you recommend to the director?
d)
Express the literal difference between the estimated variance of the ratio
estimator
and the estimated variance of the regression estimator, as a function
A A
i j
of X
, Y and the slope bof the regression of _\~' on x in the sample. Comment
on
this.
The
lll'lIl'i - ~
<
t gt ment of a particular com ' '
-
panv is interested in cst' t‘ -
- *
- .tment ol , '_ _- _ _ . _ _ _
from
<~mplo\'e(-s loavin 1 flu. | -|d- P W) A 1 m 10 5Y$l9m&llC sample is
obtamed
in
“W M(m.],manying time ifliiiigt tile end of a Particular workday. Use the data
)h(_p
_ ‘ l - H - . ‘P. it proportion in favor of the new policy and
I *
‘I "HI!" OH the P1101‘ nf <~stunat1on. Assume N = 2000
Ii 1
13 Q
23 1
1993
1
2 .._
yk = 132
5-,
.'1,1.»w~~.-
11,, = 0.00, B_,, = 0.0037
edin
question, determine the sampie size required
For
the situation outlined in the pree g .
't.
What type of systematic sample should be run?
in
('SiiIlMi(' p to within 0.01 um
.'ln.su.'m".'
u = 10.30
' '
~ — ----\nrn\r\‘f\I A
Elements
of Sampling Theory and Methods - Zakkula Govindarajulu
Toyota
Manufacturing Company in Georgetown, Kentucky, wants to estimate the
ratio
of the number of man-hours lost due to sickness of its employees. It has
N = 7000
employees and it takes a random sample of n =10 employees and
obtains
the following data. Let x and y denote the number of man hours during
the
previous and current years respectively.
Employee
x .\’
l 15
l4
2 18
20
30
34
25
18
10
l5
20
25
16
20
12
l5
9 13
10
l0 2
5
0O\lO\LII-$0-J
Total
161 l 76
Estimate
the desired ratio and obtain 95% confidence interval for it.
Suppose
we wish to estimate the grade-point average of the graduating seniors at
Mars
University. Let N = 1000, the number of graduating seniors. We take a
random
sample of n = 9 students. Let x denote their SAT scores and y denote
the
grade-point average at the University. Assume that Y = 600.
Student
# x Y
OO\lO\£II-§UJl\J~—~
550
2.8
630
3.]
570
2.9
650
3.3
700
3.5
520
3.0
720
3.6
660
3.5
9
575 3.2
Total
5575 28.9
Obtain
a ratio estimate of the average grade-point average of a graduating senior
at
the University of Mars and also set up 95% confidence interval for the same.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29 3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44
4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
A
simple random sample of 10 customers was taken at a local grocery store and
the
following data were obtained:
Customer
Total Annual Income x Annual Amount Spent on
(in
thousands) Food y (in thousands)
25
3.7
32
4.1
40
4.5
35
3.9
29
3.8
42
4.6
50
4.8
38
4.0
41
4.5
44 4.4
‘5\ooo\1o\u14>wr\>--
Assuming
that N is effectively infinite, estimate R , the population ratio
(R =
Y/X ) , and obtain its standard error.
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