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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?

θθ ˆ with

Solution: Solve the inequality for n!

eθVz ˆ

Ex. Want to estimate . Solve the following inequality for n:

str with yyU

enS

Nn

NNz

H

h h

h

h

hh

1

221

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DETERMINING SAMPLE SIZE, CONT’D• Let where ah is the proportion of

the total sample allocated to stratum h.• Ignore the fpc:s.

vezn 2

2

hh nan

Then, the solution is

where

H

h h

hhaS

NNv

1

22

4

ALLOCATION OF THE SAMPLE OVER STRATAOr: determining .

1. The stratum constitutes a large part of pop2. 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

Hh aaa ,...,,...,1

Makes sense to sample more from a stratum if

How formalize this?

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ALLOCATION, CONT’DConsider a linear cost function

H

hhhnccC

10

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

. with stryyUstry

H

h h

h

h

hhnS

Nn

NNyV

1

22

str 1ˆ

6

ALLOCATION, CONT’DDetermine 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

Hh aaa ,...,,...,1

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ALLOCATION, CONT’DBoth problems are solved by

H

hhhh

hhhh

cSN

cSNa

1

- so called optimal allocation!

In case 1 (fix variance): insert ah in the formula for nIn case 2 (fix cost): insert ah in the cost function and solve for n

Depending on the problem, however, n differs:

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SIMPLIFICATIONSNeyman allocationA special case of optimal allocation if

Only case 1 (fix variance) relevant.

Allocate according to

H

hhh

hhh

SN

SNa

1

and insert the result in the formula for n.

Hh ccc ......1

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SIMPLIFICATIONS, CONT’DProportional allocationA special case of Neyman allocation if

Only case 1 (fix variance) relevant.

Allocate according toNN

N

Na hH

hh

hh

1

and insert in the formula for n.

Hh SSS ......1

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OTHER ALLOCATION PROBLEMSOptimal 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.