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STA 304 /1003 Stratified Random Sampling Design

创建日期：2024-04-07 16:41:02

所属分类：统计学statistics

标签： STA 1003

Stratified Random Sampling Design

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STA 304 /1003

Module 5 Outline
Stratified Random Sampling Design:
I Inference for the population mean §4.3
I Inference for the population total §4.4
I Inference for the population proportion §4.5
Shivon Sue-Chee Module 5- Stratified Random Sampling 1
Term Test #1 Info

Quercus Quiz
Coverage: tested on Chps 1-5; except 4.6, 5.11, slides (up to Wed.
Feb. 10), homework, past tests
Practice tests available in Quercus
Student Solutions Manual: worked-out answers to odd-numbered
questions
Support hours during Reading week
Shivon Sue-Chee Module 5- Stratified Random Sampling 2
Stratified random sampling- Ch. 5
Definition
[STRS] Divide the target population into strata; draw SRS within each
stratum
Shivon Sue-Chee Module 5- Stratified Random Sampling 3
Stratified random sampling- Ch. 5
Definition
[STRS] Divide the target population into strata; draw SRS within each
stratum
Why?
1 more efficient- smaller margin of error
2 more convenient-$$ cost less
3 get estimates for subgroups
4 protects against imbalance
Example: study of job status of undergraduate students
Possible strata:
Shivon Sue-Chee Module 5- Stratified Random Sampling 4
Stratified random sampling- Ch. 5
Text Examples:
USA Canada
Consumer Price Index
Current Population Survey
Establishment Survey
HW: What are the comparable Statistics Canada surveys?
Shivon Sue-Chee Module 5- Stratified Random Sampling 5
Notation and inference in Stratified RS
Strata and number of elements:
{1, . . . , L} =⇒ {N1, . . . ,NL} where
∑L
`=1N` = N
Samples: means and sizes
{n1, . . . , nL} =⇒ {y¯1, . . . , y¯L} where
∑L
`=1 n` = n
Population mean and total:
µ =
τ
N
=
τ1 + . . .+ τL
N
=
N1µ1 + . . .+ NLµL
N
Stratum means and estimate of µ
µ̂ = y¯st =
N1
N
y¯1 + . . .+
NL
N
y¯L
Shivon Sue-Chee Module 5- Stratified Random Sampling 6
Notation and inference in Stratified RS
Strata and number of elements:
{1, . . . , L} =⇒ {N1, . . . ,NL} where
∑L
`=1N` = N
Samples: means and sizes
{n1, . . . , nL} =⇒ {y¯1, . . . , y¯L} where
∑L
`=1 n` = n
Population mean and total:
µ =
τ
N
=
τ1 + . . .+ τL
N
=
N1µ1 + . . .+ NLµL
N
Stratum means and estimate of µ
µ̂ = y¯st =
N1
N
y¯1 + . . .+
NL
N
y¯L
Shivon Sue-Chee Module 5- Stratified Random Sampling 7
Inference in STratified RS: Estimator of µ (§5.3)
Toolbox (5.1) & (5.2)
Estimator of population mean:
µ̂st = y¯st =
L∑
i=1
Ni
N
y¯i
Estimated variance of estimator
Using {y¯i ,Var(y¯i ), V̂ar(y¯i )} :
V̂ar(µˆst) = V̂ar(y¯st) =
L∑
i=1
N2i
N2︸︷︷︸
weights
s2i
ni
(
1− ni
Ni
)
︸︷︷︸
from SRS
Shivon Sue-Chee Module 5- Stratified Random Sampling 8
Examples 5.1, 5.2
see data in Table 5.1
Summary: {i = 1, 2, 3}
Ni ni Mean (y¯i ) Median si
Town A 155 20 33.90 34.50 5.95
Town B 62 8 25.12 26.00 15.25
Rural 93 12 19.00 17.50 9.36
Shivon Sue-Chee Module 5- Stratified Random Sampling 9
...example 5.2, using R
> A=scan()
1: 35 43 36 39 28 28 29 25 38 27
11: 26 32 29 40 35 41 37 31 45 34
21:
Read 20 items
> B = scan()
1: 27 15 4 41 49 25 10 30
9:
Read 8 items
> Rural = scan()
1: 8 14 12 15 30 32 21 20 34 7 11 24
13:
Read 12 items
Shivon Sue-Chee Module 5- Stratified Random Sampling 10
...example 5.2: boxplots
To
w
n
A
To
w
n
B
R
ur
al
10 20 30 40 50
hours
S
tra
tu
m
Q: Sketch one boxplot of the hours using the data from all 3 areas.
Shivon Sue-Chee Module 5- Stratified Random Sampling 11
...example 5.2: an R function
> msn<- function(v){
+ nv=length(v);
+ meanv=mean(v);
+ sdv=sqrt(var(v));
+ return(c(meanv,sdv,nv))
+ }
> #using msn function
> msn(A)
[1] 33.90000 5.94625 20.00000
> msn(B)
[1] 25.12500 15.24502 8.00000
> msn(Rural)
[1] 19.00000 9.36143 12.00000
Shivon Sue-Chee Module 5- Stratified Random Sampling 12
...example 5.2: getting margin of error
> nA = msn(A)[3] # or length(A)
> nB = msn(B)[3]
> nR = msn(Rural)[3]
> NAA = 155
> NB = 62
> NR = 93
> c(nA/NAA,nB/NB,nR/NR)
[1] 0.1290323 0.1290323 0.1290323
> #notice that fpc’s are the same
> fpc = 1-.129
> NAA^2*fpc*var(A)/nA + NB^2*fpc*var(B)/nB
+ NR^2*fpc*var(Rural)/nR
[1] 189277.8
> .Last.value/((NAA+NB+NR)^2)
[1] 1.969592
Therefore, m.e.= ±2×
Shivon Sue-Chee Module 5- Stratified Random Sampling 13
Class data example
Use our class data to obtain a stratified random sample of n = 20
heights with job status as our stratification variable. Find a 95%
interval estimate for the mean population height.
Compare results with simple random sampling.

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