1. Chapter 5 Stratified Random Samples

2. What is a stratified random sample and how to get one Population is broken down into strata (or groups) in such a way that each unit belongs to one AND ONLY ONE stratum. Select a SRS within each stratum

3. 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.

4. 5.3 Estimation of Population Mean and Total Recall that the population total estimate is  - hat = Ny-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 )

5. Example 5.10

6. 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

7. 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

8. 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

9. Calculating n 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

10. 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. 5.7,5.8, 5.9

11. 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

12. 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.

13. 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 )

14. Examples 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.