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Math 325: Sampling Methods I Question Paper
Math 325: Sampling Methods I
Course:Bachelor Of Science
Institution: Chuka University question papers
Exam Year:2012
1
CHUKA UNIVERSITY
And
COLLEGE
UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION (ARTS)
MATH 325: SAMPLING METHODS I
STREAM: B.ED (ARTS) Y3S1 TIME: 2 HOURS
DAY/DATE: TUESDAY 6/4/2010 8.30A.M.-10.30A.M.
INSTRUCTIONS:
ATTEMPT QUESTION ONE AND ANY OTHER TWO.
QUESTION ONE (30 MARKS)
(a) What reasons would motivate one to use sample information to say something
about the population instead of carrying out complete enumeration? [4 marks]
(b) What are the main sources of errors in sample surveys? [6 marks]
(c) In a population with N=15, the values of yi
are 2,3,1,5 and 4. Calculate the
sample mean for all possible simple random samples of size 2 without
replacement. Verify that
y
is an unbiased estimate of
Y
and that its variance is
given by
) 1 (
2
f
n
S
?
. [8 marks]
(d) Show that in stratified sampling, if in every stratum
h
y
is unbiased the
st
y
is an unbiased estimation of the population mean
. Y
[4 marks]
(e) Define systematic sampling and show that under certain condition its mean
is more precise than the mean of SRS
wor
. [6 marks]
2
(f) Under what condition(s) do we say that the sample parameter (t) is unbiased
estimate of the population parameter (T)? [2 marks]
QUESTION TWO
(a) In SRS with replacement, derive an expression for
) ( y V
and show that the
sample mean
) ( y
is an unbiased estimate of the population mean
) (Y
.
[10 marks]
(b) Two dentists Alex and Okwiri make a survey of the state of the teeth of 200
children in Chuka town. Dr. Alex selects a SRS of 20 children and counts the
number of decayed teeth for each child, with the following results.
1
10
1
9
0
8
0
7
0
6
1
5
1
4
2
3
2
2
4
1
8
0 /
children of Number
child teeth
decayed of Number
Dr. Okwiri, using the same dental techniques, examines 200 children, recording
only those who have no decayed teeth. He finds 60 children with no decayed
teeth.
Estimate the total number of decayed teeth in Chuka town’s children.
(i) Using Dr. Alex’s results
(ii) Using both Dr. Alex’s and Dr. Okwiri’s results.
(iii) Which estimate do you expect to be more precise? [10 marks]
QUESTION THREE (20 MARKS)
(a) If the variants yi
and x
i
are measured on each unit of a SRS of size n, assumed
large, show that the MSE and the variance
R
ˆ
=
x
y
are each approximated
by
2
1
)
ˆ
( )
ˆ
(
x n
f
R V R MSE
?
? ?
1
) (
1
2
?
? ?
?
N
Rx y
N
i
i i
[8 marks]
3
(b) The following data gives the number of persons (x
1
), the monthly family income
(x
2
) in hundred shillings and the monthly expenditure on food (y) in hundred
shillings in a SRS of 10 families.
29
60
8
10
21
59
5
9
40
80
6
8
30
79
9
7
20
40
7
6
29
55
4
5
21
65
5
4
33
87
3
3
20
62
3
2
14
62
2
1
) ( cos
) (
) (
2
1
y t Food
x Income
x size Family
number Family
Estimate the following:
(i) The mean monthly expenditure on food per family.
(ii) The mean monthly expenditure on food per person.
(iii) The % income spent on food
(iv) Compute the standard error of each of the above estimates. [12 marks]
QUESTION FOUR
(a) A researcher intends to take a stratified random sample. He expects his field cost
to be of the form
.
h
h
h
n C ?
His advance estimates of relevant quantities for the
two strata are as follows:
Stratum Wh Sh Ch
1 0.4 10 Sh.40
2 0.6 20 Sh.90
(i) Find the values of
n
n
1
and
n
n
2
that minimize the field cost for a given
Value of
). (
st
y V
[3 marks]
4
(ii) Find the sample size n, under this optimum allocation if
) (
st
y V
=1,
ignore fpc. [3 marks]
(iii) Find the total cost the researcher will incur in the field. [4 marks]
(b) Discuss by first stating your main points, clearly the principle steps involved
in the planning and execution of a sample survey. [10 marks]
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