# ALLOCATION - PowerPoint PPT Presentation

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ALLOCATION. 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 :.

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ALLOCATION

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### 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:

### DETERMINING SAMPLE SIZE, CONT’D

• Let where ah is the proportion of the total sample allocated to stratum h.

• Ignore the fpc:s.

Then, the solution is

where

### ALLOCATION OF THE SAMPLE OVER STRATA

Or: determining .

Makes sense to sample more from a stratum if

• The stratum constitutes a large part of pop

• The stratum is suspected to be less homogeneous with respect to what you are studying than other strata

• The stratum is cheap to observe

How formalize this?

### ALLOCATION, CONT’D

Consider a linear cost function

where c0=overhead cost, ch=the cost of observing one unit in stratum h.

Assume that you want to estimate Remember that the variance of is given by

### 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

### ALLOCATION, CONT’D

Both problems are solved by

- so called optimal allocation!

Depending on the problem, however, n differs:

In case 1 (fix variance): insert ah in the formula for n

In case 2 (fix cost): insert ah in the cost function and solve for n

### 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.

### 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.

### 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?

• Allocate with regard to the most important variable.

• Solve the full optimization problem.

• Determine n separately for each stratum.

• Solve the full optimization problem.