1. 1 ALLOCATION

2. 2 DETERMINING SAMPLE SIZE Problem: Want to estimate. How choose n to obtain a margin of error not larger than e? Solution: Solve the inequality for n! Ex. Want to estimate. Solve the following inequality for n:

3. 3 DETERMINING SAMPLE SIZE, CONT’D Let where a h is the proportion of the total sample allocated to stratum h. Ignore the fpc:s. Then, the solution is where

4. 4 ALLOCATION OF THE SAMPLE OVER STRATA Or: determining. 1.The stratum constitutes a large part of pop 2.The stratum is suspected to be less homogeneous with respect to what you are studying than other strata 3.The stratum is cheap to observe Makes sense to sample more from a stratum if How formalize this?

5. 5 ALLOCATION, CONT’D Consider a linear cost function where c 0 =overhead cost, c h =the cost of observing one unit in stratum h. Assume that you want to estimate Remember that the variance of is given by

6. 6 ALLOCATION, CONT’D Determine so that 1. (Fix variance) The total cost C is minimized for a given variance V or 2. (Fix cost) The variance is minimized for a given total cost C

7. 7 ALLOCATION, CONT’D Both problems are solved by - so called optimal allocation! In case 1 (fix variance): insert a h in the formula for n In case 2 (fix cost): insert a h in the cost function and solve for n Depending on the problem, however, n differs:

8. 8 SIMPLIFICATIONS Neyman allocation A special case of optimal allocation if Only case 1 (fix variance) relevant. Allocate according to and insert the result in the formula for n.

9. 9 SIMPLIFICATIONS, CONT’D Proportional allocation A special case of Neyman allocation if Only case 1 (fix variance) relevant. Allocate according to and insert in the formula for n.

10. 10 OTHER ALLOCATION PROBLEMS Optimal allocation if you have more than one study variable? Optimal allocation if you have precision demands both for separate strata and for the pop as a whole? 1.Allocate with regard to the most important variable. 2.Solve the full optimization problem. 1.Determine n separately for each stratum. 2.Solve the full optimization problem.