1. STT 350: SURVEY SAMPLING Dr. Cuixian Chen Chapter 5: Stratified Random Samples Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 1 Chapter 5

2. What is a stratified random sample and how to get one  Example: Suppose a public opinion poll designed to estimate the proportion of voters who favor spending more tax revenue on an improved ambulance service is to be conducted in a certain county.  The county contains two cities and a rural area.  First: our goal in designing surveys is to maximize the information obtained (or to minimize the bound on the error of estimation) for a fixed expenditure.  Second: the cost of obtaining observations varies with the design of the survey.  How to conduct sampling design? Chapter 5Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 2

3. What is a stratified random sample and how to get one  Population is broken down into strata (blocks, or groups) in such a way that each unit belongs to one AND ONLY ONE stratum.  Select a SRS within each stratum  Real life examples?

4. What is a stratified random sample and how to get one  Principal reasons for stratified random sampling rather than SRS are as follows:  1. Stratification may produce a smaller bound on the error of estimation than would be produced by a simple random sample of the same size. This result is particularly true if measurements within strata are homogeneous.  2. The cost per observation in the survey may be reduced by stratification of the population elements into convenient groupings.  3. Estimates of population parameters may be desired for subgroups of the population. These subgroups should then be identifiable strata.

5. Why stratified random sampling over simple random sampling?  Stratification may produce a smaller bound on the error of estimation than would be produced by a SRS of the same size (especially if homogeneous within strata).  The cost per observation in the survey may be reduced.  Estimates of population parameters may be desired for subgroups of the population.

6. 5.3 Estimation of Population Mean and Total  Recall that the population total estimate is  -hat = N*y-bar  So, to calculate the total from all the information from the individual strata, we use the total from within each group  -hat ST =  i (N i * y-bar i )  The Estimated variance of  -hat is  i N i 2 (1-n i /N i )(s i 2 /n i )

7. 5.3 Estimation of Population Mean

8. Estimation of a population mean in a stratified random sample  The population mean can be estimated by using the population total (remember,  -hat ST =  i (N i *y-bar i )), so y- bar ST = (1/N)  -hat ST The bound on this estimator is 2*sqrt( (1/N 2 )  i N i 2 (1-n i /N i )(s i 2 /n i ) ) Use this on 5.10 Do Exercise 5.3

9. Eg5.1, page 118 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 9

10. Eg5.2, page 120 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 10

11. 5.3 Estimation of Population Total

12. Eg5.3, page 120 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 12

13. 5.4 Selecting sample sizes  To estimate  and  : n = (  i N i 2  i 2 /a i )/(N 2 D +  i N i  i 2 ) Where a i is the fraction of observations allocated to stratum i. For estimating , D=B 2 /4 For estimating , D=B 2 /4N 2 Then to find n i, take a i *n

14. 5.4 Selecting sample sizes n Then to find n i, take a i *n.

15. Eg5.5, page 124 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 15

16. Eg5.6, page 124 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 16

17. 5.5 Allocation of the Sample  The allocation of sample among strata is affected by  The total number of elements in each stratum  The variability of observations within each stratum  The cost of obtaining observations from each stratum

18. Calculating n i  n = (  i N i 2  i 2 /a i )/(N 2 D +  i N i  i 2 ), where D=(B 2 /4) for  and D=B 2 /(4N 2 ) for   Three different ways to calculate allocations….  If allocation is to minimize cost then a i = (N i  i /sqrt(c i ))/  k (N k  k /sqrt(c k ))  If allocation is to minimize variation (and cost is the same within each stratum…or cost is not an issue) then (Neyman’s allocation) a i = (N i  i )/  k (N k  k )  If allocation is to proportional to size of strata, then (proportional allocation) a i = (N i )/  k (N k )  Then to calculate n i = a i *n

19. Calculating n i w/ given price for each stratum D=(B 2 /4) for  and D=B 2 /(4N 2 ) for  Then to calculate n i = a i *n

20. Eg5.7, page 127 (different Cost/Variance) Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 20

21. 5.4 Finding n i, by Neyman Allocation

22. Eg5.8, page 128 (same Cost/ different Variance): Neyman Allocation Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 22

23. Eg5.10, page 131 (same Cost/Variance): Proportional Allocation Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 23

24. Eg5.11, page 131 (same Cost/Variance): Proportional Allocation with cost constraint Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 24

25. EX5.3, page 152 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 25

26. Extra Examples  As class, work on 5.2 (using all 3 allocation methods….assume c1=$4 per unit, c2=$2 per unit, c3=$1 per unit). Want a bound of 100 for the total. Find n and n1, n2, n3 for all three methods.  Eg. 5.7,5.8, 5.9

27. 5.6 Population proportion  p-hat ST = (1/N)  i N i p-hat  Estimated variance of p-hat ST : (1/N 2 )  i (N i 2 (1-n i /N i )(p-hat*q-hat/(n i -1))) NOTE: if the fpc is not used, then n i -1 is simply n i

28. 5.6 Population proportion

29. Eg5.12, page 153 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 29

30. Sample Size  n = (  i N i 2 p i q i /a i )/(N 2 D+  N i p i q i ) Where D = B 2 /4 NOTE: If p can be estimated, use the estimated value of p. Otherwise, if an estimate is not given, use p=0.5.

31. Sample Size

32. Eg5.13, page 135 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 32

33. Allocation  Proportional allocation: a i = N i /  k N k  Neyman’s allocation: a i = N i  i /  k N k  k or in case of proportions: a i = N i sqrt(p i q i )/  k N k sqrt(p k q k )  Allocation including cost: a i = N i sqrt(p i q i /c i )/  k N k sqrt(p k q k /c k )

34. EX5.31, page 152 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 34

35. Extra Example  In a school that has 200 Freshmen, 250 Sophomores, 175 Juniors and 125 Seniors, it was desired to estimate the percent of students in favor of the new basic studies program being proposed. They randomly selected 23 freshmen, 33 sophomores, 28 juniors and 30 seniors. Of these random samples, 8 of the 23 freshmen supported the new basic studies program; 15 of the 33 sophomores supported it; 10 of the 28 juniors supported it and 7 of the 30 seniors supported it.  a. Find the overall percent of students at this school that support the new basic studies program and find an appropriate bound for it.  b. Estimate the difference in percent of freshmen and seniors that support the new basic studies program and find an appropriate bound for it.  ********************************************  If an administrator wanted to estimate this percent with a bound of 0.07, what sample size would be needed? Use Neyman's allocation.  ******************************************  Do problem 5.31.