Stat 3203 -multphase sampling

Multiphase sampling

1. Stat-3203: Sampling Technique-II (Chapter-3: Double sampling and multiphage Sampling) (Section-B) Md. Menhazul Abedin Lecturer Statistics Discipline Khulna University, Khulna-9208 Email: [email protected]

2. Problem-1 S.N. of cut Harvest Yield (in kg) (x) Dry yield (in kg) (y) S.N. of cut Harvest Yield (in kg) (x) 1 16.8 15.2 21 8.7 2 12.7 11.8 22 11.6 3 18.8 17.5 23 11.5 4 13.8 12.5 24 14.4 … … … … … 20 14.0 12.8 40 10.9

3. Problem-1 Auxiliary variable is Harvest Yield (in kg) (x ) easy to collect, less time need. Study variable is Dry yield (in kg) (y ) size is less than auxiliary variable.

4. Problem-10.2 Strata No.of eye estimation No. of crop-cutting experiments Yield rate in kg/ha based on cuting 101-200 Unknown Poplation Size Is N So that Preliminary Sample of Size 154 40 200, 208, 152, 224, …, 440, 400, 400. 201-300 189 46 104, 152, 148, 256, …, 206, 496, 535. 301-400 91 22 280, 192, 192, 280, …, 496, 304, 243. 401-500 40 15 288, 156, 280, 136, …, 672, 568, 520. 501-600 13 6 428, 368, 506, 824, 624, 768. Above 600 12 2 344, 712 Total Strata Yield rate in kg/ha based on cuting 101-200 154 40 200, 208, 152, 224, …, 440, 400, 400. 201-300 189 46 104, 152, 148, 256, …, 206, 496, 535. 301-400 91 22 280, 192, 192, 280, …, 496, 304, 243. 401-500 40 15 288, 156, 280, 136, …, 672, 568, 520. 501-600 13 6 428, 368, 506, 824, 624, 768. Above 600 12 2 344, 712 Total

5. Multiphase Sampling Multiphase sampling: A sampling method in which certain items of information are drawn from the whole units (x) of a sample and certain other items of information are taken from the subsample (y).

6. Difference multi-phase sampling ……necessary to have a complete sampling frame of the units multi-stage sampling……frame of the next stage units is necessary only for the sample units selected at the stage Precision is substantial as the increase in cost due to collection of information on the auxiliary variate for large samples.

7. Problem-10.2 Strata No.of eye estimation No. of crop-cutting experiments Yield rate in kg/ha based on cuting 101-200 Unknown Poplation Size Is N So that Preliminary Sample of Size 154 40 200, 208, 152, 224, …, 440, 400, 400. 201-300 189 46 104, 152, 148, 256, …, 206, 496, 535. 301-400 91 22 280, 192, 192, 280, …, 496, 304, 243. 401-500 40 15 288, 156, 280, 136, …, 672, 568, 520. 501-600 13 6 428, 368, 506, 824, 624, 768. Above 600 12 2 344, 712 Total Strata Yield rate in kg/ha based on cuting 101-200 154 40 200, 208, 152, 224, …, 440, 400, 400. 201-300 189 46 104, 152, 148, 256, …, 206, 496, 535. 301-400 91 22 280, 192, 192, 280, …, 496, 304, 243. 401-500 40 15 288, 156, 280, 136, …, 672, 568, 520. 501-600 13 6 428, 368, 506, 824, 624, 768. Above 600 12 2 344, 712 Total

8. Problem-2 be the proportion of units falling in the i th statum, (unknown) be the proportion of first sample units falling in the i th statum (preliminary sample) An estimator of the population mean is is the sample mean for the study variate in the i th stratum  

9. Double sampling for stratification See Problem-2 Theorem 10.2.1: If the values of do not depend on show that the estimator is an unbiased estimator of the population mean and its sampling variance is given by Where and  

10. Double sampling for stratification Whenever a new sample is drawn, it implies a fresh dwawing of first and second samples. Thus and the sample mean are both random variables. Since the first is a simple random sampling , if we take expectation first, over samples in which are fixed is the mean of a simple random sample from the stratum Hence, the expectation over different selections of the sample is given by  

11. Double sampling for stratification Which shows that the estimator is unbiased To calculate the the sampling variance See book With corollary 1:  

12. Double sampling for stratification Corollary 1 : For large population, prove that tends to 0 and tends to 1 For proportional allocation the variance appxoxomately  

13. Double sampling for stratification Optimal Allocation : The cost functio n of double sampling can be written as is the over head cost, and and are the costs per u nit measuring the auxiliary variate and study variate respectively.  

14. Double sampling for stratification The problem is to obtain the values of so as to minimize the variance of the estimator for a given cost Lagrange function Derivative with respect to , and and set equal to zero then find the solution of and .  

15. Double sampling for stratification The solution will be For this values of and , the mimimum variance of is   Proof: Board

16. The difference estimator of may be defined by Where is taken as known in the population; and are sub sample means for and , respectively, and is the preliminary sample mean for .   Double Sampling for Difference estimator

17. Double Sampling for Difference estimator Theorem 10.4.1: Show that is an uniased estimator of the populatin mean, its sampling varianceis given by  

18. Double Sampling for Difference estimator Proof: Given the first sample. Let be the mean value. This shows that the estimator is unbiased. [Like SRS]  

19. Double Sampling for Difference estimator Combining them will get the proof  

20. Double Sampling for Ratio estimator The ratio estimator of may be defined by Where and are sub sample means for and , respectively, and is the preliminary sample mean for .  

21. Double Sampling for Regression estimator The regression estimator of may be defined by Where is the the least square estimate of the regression coefficient; and are sub sample means for and , respectively, and is the preliminary sample mean for .  

22. Further study Double Sampling for PPS estimator Optimum allocations are like proportional allocation