Chapter 5
Stratified Random Sampling
Advantages of stratified random
sampling
 How to select stratified random sample
 Estimating population mean and total
 Determining sample size, allocation
 Estimating population proportion; sample
size and allocation
 Optimal rule for choosing strata

Stratified Random Sampling

The ultimate function of stratification is to
organize the population into
homogeneous subsets and to select a
SRS of the appropriate size from each
stratum.
Stratified Random Sampling

Often-used option
– May produce smaller BOE than SRS of
same size
– Cost per observation may be reduced
– Obtain estimates of population parameters
for subgroups

Useful when the population is
heterogeneous and it is possible to
establish strata which are reasonably
homogeneous within each stratum
Chapter 5
Stratified Random Sampling
Improved Sampling
Designs with Auxiliary
Information
Stratified Random
Sampling
Chapter 6 Ratio and
Regression
Estimators
Stratified Random Sampling:
Notation
y i :sam ple m ean of data from stratum i,
i  1,
,L
n i :sam ple size for stratum i
 i : population m ean of stratum i
 i : population total of stratum i
population total    1   2 
L
Stratified Random Sampling
S R S w ithin each stratum , so:
E ( yi )   i
ˆi  N i y i ; E (ˆi )  E ( N i y i )  N i E ( y i )  N i  i   i
E stim ate population total  by sum m ing
estim ates of  i
ˆ 
ˆ1  ˆ2 
N
 ˆ L
 y st
Stratified Random Sampling: Estimate
of Mean
y st 

1
N
1
N
 N 1 y1  N 2 y 2

 N L yL 
L
N
i
yi
i 1
1
ˆ
 N 12Vˆ ( y1 )  N 22Vˆ ( y 2 ) 
V ( y st ) 
2
N 
1  2

N1
2 
N 
2
 N L Vˆ ( y L ) 

2
2



n 1  s1
n
s
2
2
2
1


N
1





2 
N 1  n1
N 2  n2


2



n
s
2
L
L
 N L 1 


N L  n L 

Stratified Random Sampling:
Estimate of Mean , BOE
Vˆ ( y st ) 
1
N
 N 1 Vˆ ( y 1 )  N 2 Vˆ ( y 2 ) 
2
2
1 
2
2
 N L Vˆ ( y L ) 
n1  s1
n2  s2


2
2

N1 1 
 N 2 1 



2 
N 1  n1
N 2  n2
N 


2
2
nL  sL 

2
 N L 1 


N L  nL 

2
BOE
 1.96
1 
n1  s1
n2  s2


2
2
N1 1 
 N 2 1 



2 
N 1  n1
N 2  n2
N 


2
2
nL  sL 

2
 N L 1 


N L  nL 

2
Stratified Random
Sampling: Estimate of
Population Total
N y st   N 1 y1  N 2 y 2 
 N L yL 
L

N
i
yi
i 1
2
Vˆ ( N y st )  N Vˆ ( y st )
2
2
  N 1 Vˆ ( y1 )  N 2 Vˆ ( y 2 ) 

 2
  N1

2
 N L Vˆ ( y L ) 

2
2



n 1  s1
n
s
2
2
2
1


N
1





2 
N 1  n1
N 2  n2


2



n
s
2
L
L
 N L 1 


N L  n L 

Stratified Random Sampling: BOE for Mean
and Total , t distribution

When stratum sample sizes are small, can use t dist.
 
L
2
2
ak sk
k 1
S atterw aith e d f 
a
L

2
sk
k

N k ( N k  nk )
w h ere a k 
2
nk
nk  1
k 1
B O E fo r  :
 
N 1 
 
1
t df
2
n



1
1
2
N
N
1
2
s
1
n
 N
1
2
2

1 

n
N



2
2
2
s
n
2

 N
2
2
L

1 

n
N



L
L
B O E fo r  :
 
N 1 
 
2
t df
n
1
1
N
1



2
s
1
n
1
 N
2
2

1 

n
N
2
2



2
s
n
2
2

 N
2
L

1 

n
N
L
L



2
s
n
L
L



2
s
n
L
L



Degrees of Freedom(worksheet cont.)
S tratified R andom S am ple S um m ary:
a

k
N (N
k
1
k
 155 , N
2
 8 , n  12
3
k
2
 62 , N
a  1046.25 , a
1
df 
, n  20 , n
1
n
N
 n )
k
2
3
 93,
 418.5 , a
 1046.25  5.95
 1046.25  5.95 
2
19
 21.09; t 21.09

2
2
3
 627.75
2
 418.5  15.25  627.75  9.36
 418.5  15.25 
2

2
2

2
 627.75  9.36 
2

7
2.08 (see E xcel w orksheet)
11
2
Compare BOE in Stratified Random
Sample and SRS (worksheet cont.)
S tratified R andom S am ple S um m ary:
n  40, y  27.7; Vˆ ( y st )  1.97.
Strat. random
sample has
more precision
If observations w ere from S R S :
2
40
11.31


s  11.31, Vˆ ( y )   1 
 2.79

310  40

Approx. Sample Size to Estimate
V ( y st )  B  V ( y st ) 
2
B
2
4
L et n i  a i n , a i  prop. of sam ple from stratum i
1
N
a n  s
B

N 1 



N  a n
4

2
L
2

2
i
2
i
i
i 1
i
i
L

n 
2
2
N i si
ai
i 1
w here D 
L
N D 
2

i 1
2
N i si
B
2
4
Approx. Sample Size to Estimate
B
V ( N y st )  B  V ( y st ) 
2
2
4N
2
L et n i  a i n , a i  prop. of sam ple from stratum i
1
N
a n  s
B

N 1 



2
N  a n
4N

2
L
2

2
i
2
i
i
i 1
i
i
L

n 
2
2
N i si
ai
i 1
w here D 
L
N D 
2

i 1
2
N i si
B
2
4N
2
Summary: Approx. Sample Size to
Estimate ,
L
N
2
i
s
2
i
ai
i 1
n 
L
N D
2
N
i
s
2
i
i 1
D 
B
2
w hen estim ating 
4
D 
B
2
4N
2
w hen estim ating 
Example: Sample Size to
Estimate (worksheet
cont.)
L
N
n 
2
i
2
si a i
i 1
N D
2
N
i 1
P rio r su rvey:  1  5,  2  1 5,  3  1 0 .
E stim ate  to w ith in 2 h rs w ith 9 5 % co n f.
allo catio n p ro p o rtio n s are a1  a 2  a 3  1 3 .
B  2 D 
B
4
3
N
2
i
s
2
i
ai 
 1; N D  3 1 0  9 6,1 0 0
2
2
2
155 ( 25 )
1 3

2
62 ( 225 )
1 3
2

2
93 (100 )
1 3
 6, 9 9 1, 2 7 5
i 1
3

N i s i  1 5 5(2 5)  6 2 (2 2 5)  9 3(1 0 0 )  2 7 ,1 2 5
2
i 1
n 
6, 9 9 1, 2 7 5
9 6,1 0 0  2 7 ,1 2 5
so n1  n 2  n 3 
1
3
 5 6 .7  5 7
(5 7 )  1 9
D 
L
i
s
2
i
B
2
4
Example: Sample Size to
Estimate (worksheet
cont.)
L
N
2
i
2
si a i
i 1
n 
N D
2
N
i 1
P rio r su rvey:  1  5,  2  1 5,  3  1 0 .
E stim ate  to w ith in 4 0 0 h rs w ith 9 5 % co n f.
allo catio n p ro p o rtio n s are a1  a 2  a 3  1 3 .
D 
B
2
4N
2

400
2
4N
2

160 , 00
2

3
N
2
i
s
2
i
ai 
4N
155 ( 25 )
1 3
2

40 , 000
N
2
2
62 ( 225 )
1 3
; N D  4 0, 0 0 0

2
2
93 (10 0 )
1 3
 6, 9 9 1, 2 7 5
i 1
3

N i s i  1 5 5(2 5)  6 2 (2 2 5)  9 3(1 0 0 )  2 7 ,1 2 5
2
i 1
n 
6, 9 9 1, 2 7 5
4 0, 0 0 0  2 7 ,1 2 5
so n1  n 2  n 3 
1
3
 1 0 4 .2  1 0 5
(1 0 5)  3 5
D 
L
i
s
2
i
B
2
4N
2
5.5 Allocation of the Sample
Objective: obtain estimators with small
variance at lowest cost.
 Allocation affected by 3 factors:

1. Total number of elements in each stratum
2. Variability in each stratum
3. Cost per observation in each stratum
5.5 Allocation of the Sample:
Proportional Allocation

If don’t have variability and cost
information for the strata, can use
proportional allocation.
S am ple size for stratum h :
nh  n 
Nh
N
In general this is not the optimum choice
for the stratum sample sizes.
5.5 O ptim al (m in V ( y st ) allocation
Vˆ ( y st ) 
of the sam ple: sam e cost/obs
L
1
N
2

i 1
2


n
s
2
N i 1  i  i
N i  ni

in each stratum
m in Vˆ ( y st ), subject to g ( n1 , n 2 ,
, n L )  0,
, n L )  n1  n 2 
 nL  n
n1 , n 2 ,
, nL
w here g ( n1 , n 2 ,
Directly proportional to
stratum size and stratum variability
U se L agrange m ultipliers:
 Vˆ ( y st )
 ni

g
 ni
 0, i  1,
, L  ni  n
N i si
N
k 1
T his m ethod of choosing n1 , n 2 ,
called N eym an allocation
, i  1,
L
, nL
k
sk
,L
5.5 O ptim al (m in V ( y st ) allocation
L

of the sam ple: sam e cost/obs
n 
in each stratum
ni  n
, i  1,
L
N D 
k
,L
sk
k 1
substitute
ni
n
for a i above gives


  N i si 
 i 1

L
n 
2
L
N D
2

i 1
2
N i si
ai
L

i 1
N i si
N
2
i 1
2
F rom previous slide
2
N i si
2
N i si
5 .5 O p tim al (m in V ( y st ) allo catio n o f th e sa m p le:
sam e co st/o b s in each stratu m

Worksheet 11
5.5 O ptim al (m in V ( y st ) allocation
Vˆ ( y st ) 
of the sam ple for fixed cost C : c i = cost/ob s
L
1
2
N
i 1
in stratum i.
m in Vˆ ( y st ), subject to g ( n1 , n 2 ,
n1 , n 2 ,
, nL
w here g ( n1 , n 2 ,
 Vˆ ( y st )
 ni

g
 ni
, n L )  0,
, n L )  c1 n1  c 2 n 2 
U se L agrange m ultipliers:
 0, i  1,

2


n
s
2
N i 1  i  i
N i  ni

 cL nL  C
Directly proportional to
stratum size and stratum variability
, L  ni  n
N i si
ci
, i  1,
L
N
k
sk
,L
ck
k 1
Inversely proportional
to stratum cost/obs
5.5 O ptim al (m in V ( y st ) allocation
L

of the sam ple: sam e cost/obs
n 
in each stratum
2
L
N D 

i 1
ni  n
N i si
ci
, i  1,
L
N
k
sk
,L
ck
k 1
substitute
ni
n
n 
for a i above gives
 L
  N k sk
 k 1
 L
c k    N i si
  i 1
L
N D
2

i 1
2
N i si

ci 

ai
i 1
2
From previous slide
2
N i si
2
N i si
5 .5 O p tim al (m in V ( y st ) allo catio n o f th e sa m p le:
c i = co st/o b s in each stratu m

Worksheet 12