Double Sampling For Stratification

 

Author:  Vibhashree G M (2048142)

 

Double Sampling

Double Sampling basically designs in which initially a sample of units is selected for obtaining auxiliary information only, and then a second sample is selected in which the variable of interest is observed in addition to the auxiliary information. It is  useful for finding information for stratified sampling and also to obtain auxiliary variables for ratio and regression estimation.

 

Double Sampling for Stratification

In some sampling situations, units can be assigned to strata only after the sample is selected.

The method of post-stratification is useful only if the relative proportion of each stratum in the population Wh=Nh/N is known for each stratum h. If these proportions are not known, double sampling may be used, with an initial (large) sample used to classify the units into strata and then a stratified sample selected from the initial sample.

 

The two steps of double sampling for stratification:

Step 1: n' initial simple random samples are selected from a population of N units. These units are classified into strata, with nh′ observed to be in stratum h. The population proportion of Wh=Nh/N is estimated by the sample proportion: wh=nh′n′, h = 1, ..., L.

Step 2: A second sample is then selected by stratified random sampling from the first sample. These units are classified into strata, with nh units selected from the ng′ sample units in stratum h. Measurement of yhi is recorded for each unit in the second sample.

We denote the sample mean in stratum h in the second sample as:




An estimate for the population mean is thus: 

Hereis unbaised

The decomposition of variances of two-phase sampling is:

Thus, the variance of the estimate for the population mean is:


where σ2 is the overall population variance and σh(s1)2 is the population variance within stratum h for the particular first–phase sample s1.

An unbiased estimate for the variance of the estimate is:



 Where sh2  is the stratum sample variance from the second sample.

Applications:

A sampling procedure involving repeated application of double sampling for stratification on several successive occasions is considered. By using this design few researchers illustrated by its application to a recently conducted survey for estimation of coconut production in the State of Assam.

 

Double sampling for stratification is a sampling design that is widely used for forest and other resource inventories in forest ecosystems. It is shown that this sampling design can be adapted to repeated inventories including estimators of net change, even for non-proportional allocation of second-phase units and periodically updated stratification. The method accounts for the transition of sampling units among strata.

 

Example:

A shoe store wants to estimate the average number of pairs of shoes owned by the students who live in a certain college town neighborhood. They think that a stratified sample based on gender is a good approach to take but do not know the makeup of the gender in that neighborhood. They also do not know the gender of the respondent until after contacting them. So, they use double sampling by first contacting 160 randomly selected students in that neighborhood and ask them about their gender. It turns out that 64 are males and 96 are females. They then randomly sample 8 males and 12 females, provide them a $10.00 the incentive for going home to count the number of pairs of shoes and report them.

The data are given in the table below:

 

Male

5

6

9

5

9

7

5

8

 

 

 

 

Female

17

19

13

16

8

11

15

19

12

13

33

2

 

Also given that

Variable

N

Mean

StDev

male

8

6.750

1.753

female

12

16.33

6.37

 

Solution:

 

To estimate the average pairs of shoes, here we have to use a double sampling:

Step 1:  160 students are randomly sampled to find out their gender.

Result is 64 male, 96 female.

Step 2: stratify by gender and randomly sample 8 males and 12 females.

male: n1′ = 64 , female: n2′ = 96




 

Conclusion:

Double sampling is basically a common alternative to simple random sampling when there are expected to be gained from using stratified sampling, but the units cannot be assigned to strata prior to sampling. It is assumed throughout that the survey objective is an estimation of the finite population means.

 We can see that by using doubling sampling here we have obtained the least standard error and the variance So here the average number of pairs of shoes owned by the students is 12.498 with a variance of 1.411624 and a standard error of 1.188114.

 

 References:

 

https://www.epa.gov/radiation/what-double-sampling

https://www.statisticshowto.com/double-sampling-2/

https://economictimes.indiatimes.com/definition/stratified-sampling

https://www.fs.fed.us/rm/pubs/rmrs_rp007.pdf


 






 



 







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