Sampling methods

Sampling Methods

Slide 1

Shakeel Nouman
M.Phil Statistics

Sampling Methods By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

16

Slide 2

Sampling Methods

• Using Statistics
• Nonprobability Sampling and Bias
• Stratified Random Sampling
• Cluster Sampling
• Systematic Sampling
• Nonresponse
• Summary and Review of Terms


16-2 Nonprobability Sampling
and Bias

Slide 3

• Sampling methods that do not use
samples with known probabilities of
selection are know as nonprobability
sampling methods.
• In nonprobability sampling methods,
there is no objective way of evaluating
how far away from the population
parameter the estimate may be.
• Frame - a list of people or things of
interest from which a random sample
can be chosen.

16-3 Stratified Random
Sampling

Slide 4

In stratified random sampling, we assume that the population of N units may
be divided into m groups with Ni units in each group i=1,2,...,m. The m strata
are nonoverlapping and together they make up the total population: N1 + N2
+...+ Nm =N.

Population
Stratum 1

N1

Stratum 2

N2

The m strata are
non-overlapping.

m

 Ni  N

i 1

Stratum m

Nm


16-3 Stratified Random Sampling
(Continued)

Slide 5

In stratified random sampling, we assume that the population of N
units may be divided into m groups with Ni units in each group
i=1,2,...,m. The m strata are nonoverlapping and together they make
up the total population: N1 + N2 +...+ Nm =N.
Ni

ni

1

2

3

4

5

6

7

Population Distribution

Group

1

2

3

4

5

6

7

Group

Sample Distribution

In proportional allocation, the relative frequencies in the sample (ni/n) are the
same as those in the population (Ni/N) .

Relationship Between the
Population and a Stratified
Random Sample

Slide 6

N

True weight of stratum i : W  i
i
N
n
Sampling fraction in stratum i : f  i
i
n
True mean of population : 
True mean in stratum i : 

i

True variance of the population :  2
True variance of stratum i :  2
i
Sample mean in stratum i : X
i
Sample variance in stratum i : s 2
i

The estimator of the population mean in stratified random sampling :
m
X  W X
st i1 i i

Properties of the Stratified
Estimator
of the Sample Mean

Slide 7

1. If the estimator of the mean in each stratum, Xi , is unbiased then the stratified
estimator of the mean, Xst , is an unbiased estimator of the population mean,  .
2. If the samples in the different strata are drawn independently of each other, then the
variance of the stratified estimator of the population mean, Xst , is given by:
m 2
V ( X st ) =  Wi V ( Xi )
i=1
3. If sampling in all strata is random, then the variance of Xst is further equal to:
m 2   2i 
 (1  f )
V ( X st ) =  Wi 
i=1  n 
i
i
When the sampling fractions, f , are small and may be ignored, we have:
i
m 2   2i 

V ( X st ) =  Wi 
i=1  n 
i

Properties of the Stratified
Estimator
of the Sample Mean (continued)
4.


If the sample allocation is proportional n
i

 1 - f  mW  2i

V ( X st ) = 
 n  i=1 i

 Ni 
 n 
 N


for all i  ,



Slide 8

then

which reduces to

 1 m W  2 i
V ( X st ) =    i
 n  i=1
when the sampling fraction is small.
In addition, if the population variances in all strata are equal, then
 2
V ( X st ) = 

n 

when the sampling fraction is small.

When the Population Variance is
Unknown

Slide 9

An unbiased estimator of the population variance of stratum i,  2 , is :
i
( X  X )2
i

S2 
i data in i n  1
i
If sampling in each stratum is random :

W S 2 
m i i 
2
S (X ) =  
(1  f )
st
i=1 n 


2

i

i


Confidence Interval for the
Population Mean in Stratified
Sampling

Slide 10

A (1 -  )100% confidence interval for the population mean,  , using stratified
sampling :
x

st

 z s( X


st

)

2

When the sample sizes are small, and the population variances are unknown,
use the t - value in the above formula.
The effective degrees of freedom :
2


s2 
 m N (N  n ) i 

i = 1 i i i n 
i



Effective df =
2
 N ( N  n )/n  s 4
m  i i


i i i

i 1

( n  1)
i


Example 16-2
Population
Number
Group
1. Diversified service companies
2. Commercial banking companies
3. Financial service companies
4. Retailing companies
5. Transportation companies
6. Utilities
N = 500

Slide 11

True
Weights
of Firms
100
100
150
50
50
50

1 f

Sampling
Sample
Fraction
(Wi)
Sizes
(fi)
0.20
20
0.20
0.20
20
0.20
0.30
30
0.30
0.10
10
0.10
0.10
10
0.10
0.10
10
0.10
n = 100

2

W s
Variance
ni
Wi
Wixi
n i i
97650
20
0.2
10.54
156.240
64300
20
0.2
22.52
102.880
76990
30
0.3
25.68
184.776
18320
10
0.1
1.26
14.656
9037
10
0.1
0.89
7.230
83500
10
0.1
5.23
66.800
Estimated Mean: 66.12 532.582
Estimated standard error of mean:
23.08

Stratum Mean
1
52.7
2
112.6
3
85.6
4
12.6
5
8.9
6
52.3

95% Confdence Interval:
x  z s( X )
st

st
2

66.12  (1.96)( 23.08)
66.12  45.24
[ 20.88,111.36]


Example 16-2 Using the template

Slide 12

Observe that the computer gives a slightly
more precise interval than the hand
computation on the previous slide.

Stratified Sampling for the
Population Proportion

Slide 13

Stratified estimator of the population proportion, p ,
m
  W P

Pst
i i
i 1

The approximate variance of Pst ,
m
 )  W2
V( Pst
i 1 i

 
P Qi
i
ni
When the finite - population correction factors, fi , must be considered:
 
P Qi
1 m 2
i

 N (N  n )
V( Pst ) 
i
i ( N  1) ni
N 2 i 1 i
i
When proportional allocation is used:
m
 )  1 f W PQ
 
V( Pst
i i i
n i 1


Population Proportion: Example 16-1
(Continued)
Group
Metropolitan
Nonmetropolitan

Number
Wi
ni fi
Interested
0.65 130 0.65
28
0.35
70 0.35
18
Estimated proportion:
Estimated standard error:


W p
i i
0.14
0.09
0.23

Slide 14

 
Wipi qi
n
0.0005756
0.0003099
0.0008855
0.0297574

90% confidence interval:[0.181,0.279]

90% Confdence Interval:


p  z s( P )
st

st
2

0.23  (1.645)( 0.297 )
0.23  0.049
[ 0.181,0.279 ]

Population Proportion:Example 16-1
(Continued) using the Template


Rules for Constructing Strata

Slide 16

1. Preferably no more than 6 strata.
2. Choose strata so that Cum f(x) is approximately
constant for all strata (Cum f(x) is the cumulative
square root of the frequency of X, the
variable of interest).
Age
20-25
26-30
31-35
36-40
41-45

Frequency (fi)
f(x)
1
16
4
25
5
4
9
3

Cum f(x)
1
5
5
2
5


Optimum Allocation

Slide 17

For optimum allocation of effort in stratified random sampling, minimize the
cost for a given variance, or minimize the variance for a given cost.
Total Cost = Fixed Cost + Variable Cost
C = C  C n
0
i i
(W )/ C
n
ii
i
i
Optimum Allocation : n
 (Wi i )/ Ci
If the cost per unit sampled is the same for all strata (C = c) :
i
Neyman Allocation :

n
(W )
i
ii
n  (W )
ii


Optimum Allocation: An
Example

i

Wi

s

1
2
3

0.4
0.5
0.1

1
2
3

i

C

4
9
16

i

Wi s i
0.4
1.0
0.3
1.7

Ws
i i
C
i

0.200
0.333
0.075
0.608

Slide 18

Optimum
Allocation

Neyman
Allocation

0.329
0.548
0.123

0.235
0.588
0.176


Optimum Allocation: An Example
using the Template


16-4 Cluster Sampling

1

2

3

4

5

6

7

Slide 20

Group

Population Distribution
Sample Distribution

In stratified sampling a
random sample (ni) is
chosen from each
segment of the
population (Ni).

In cluster sampling
observations are drawn from
m out of M areas or clusters
of the population.


Cluster Sampling: Estimating
the Population Mean

Slide 21

Cluster sampling estimator of  :
m

X cl 

n X
i 1
m

i

i

n
i 1

i

Estimator of the variance of the sample mean:
m

 M  m
s ( X cl )  
2
 Mmn 
2

ni2 ( X i  X cl ) 2

i 1

m1

where
m

n
n =

i 1

i

m


Cluster Sampling: Estimating
the Population Proportion

Slide 22

Cluster sampling estimator of p :
m

 ni Pi

Pcl  i 1m
 ni
i 1

Estimator of the variance of the sample proportion:
m
ni2 ( Pi  Pcl ) 2
  
2
cl )   M  m i 1
s (P

2
m1
 Mmn 


Cluster Sampling: Example 16-3

21
22
11
34
28
25
18
24
19
20
30
26
12
17
13
29
24
26
18
22

xi

8
8
9
10
7
8
10
12
11
6
8
9
9
8
10
8
8
10
10
11

ni

168
176
99
340
196
200
180
288
209
120
240
234
108
136
130
232
192
260
180
242

nixi

-0.8333
0.1667
-10.8333
12.1667
6.1667
3.1667
-3.8333
2.1667
-2.8333
-1.8333
8.1667
4.1667
-9.8333
-4.8333
-8.8333
7.1667
2.1667
4.1667
-3.8333
0.1667

3930
xcl =

2
2
 M  m ni ( X i  X cl )
 -x )22 
xi-xcl
(xiMmn 
m1
 cl
0.694
0.00118
0.028
0.00005
117.361
0.25269
148.028
0.39348
38.028
0.04953
10.028
0.01706
14.694
0.03906
4.694
0.01797
8.028
0.02582
3.361
0.00322
66.694
0.11346
17.361
0.03738
96.694
0.20819
23.361
0.03974
78.028
0.20741
51.361
0.08738
4.694
0.00799
17.361
0.04615
14.694
0.03906
0.028
0.00009

s2(Xcl)=
21.83

Slide 23

95% Confdence Interval :
x  z  s( X )
cl
cl
2

21.83  (1.96)( 1.587 )
21.83  2.47
[19.36,24.30]

1.58691


Cluster Sampling: Example 16-3
Using the Template

Slide 24


Cluster Sampling: Using the
Template to Estimate Population
Proportion

Slide 25


16-5 Systematic Sampling

Slide 26

Randomly select an element out of the first k elements in the population, and
then select every kth unit afterwards until we have a sample of n elements.
m
X
i 1 i
Systematic sampling estimator of  : X sy 
n
2
Estimator of the variance of the sample mean: s ( X sy ) 

 N  n S 2


 Nn 

When the mean is constant within each stratum of k elements but different between strata:
n
2
 ( Xi  X
)
ik
 N  n  i 1
2
s ( X sy )  

 Nn 
2 ( n  1)
When the population is linearly increasing or decreasing with respect to the variable of interest:
n
2
 ( Xi  2 X
 X i 2 k )
ik
 N  n  i 1
2
s ( X sy )  

 Nn 
6( n  2 )

Systematic Sampling: Example
16-4

Slide 27

m
 Xi
2
X sy  i 1
 0.5
s  0.36
n
 N  n  S 2   2100  100  0.36  0.0034
2


s ( X sy )  

 ( 2100)(100) 
 Nn 
A 95% confidence interval for the average price change for all stocks:
X sy  (1.96) s ( X sy )
0.5  (1.96)( 0.0034 )
0.5  0.114
[ 0.386, 0.614 ]


16-6 Nonresponse



Slide 28

Systematic nonresponse can bias estimates
 Callbacks of nonrespondents
 Offers of monetary rewards for nonrespondents
 Random-response mechanism


Name
Religion
Domicile
Contact #
E.Mail
M.Phil (Statistics)

Shakeel Nouman
Christian
Punjab (Lahore)
0332-4462527. 0321-9898767
sn_gcu@yahoo.com
sn_gcu@hotmail.com
GC University, .
(Degree awarded by GC University)

M.Sc (Statistics)
Statitical Officer
(BS-17)
(Economics & Marketing
Division)

GC University, .
(Degree awarded by GC University)

Livestock Production Research Institute
Bahadurnagar (Okara), Livestock & Dairy Development
Department, Govt. of Punjab


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