6.2 The Stratification Principle

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 Unit Summary stratification principle optimal allocation of sample size examples

The Stratification Principle

If your only objective of stratification is to produce estimators with small variances, then we want to stratify such that within each stratum, the units are as similar as possible. In a survey of human population, stratification may be based on socioeconomic factors or geographic regions.

For example, to estimate the average starting income for recent Penn State graduates, it would make sense to stratify by department since the starting income for graduates of the same department would be similar.

Allocation in Stratified Random Sampling

The question is, given a total sample size of n, how do we allocate these among L strata?

If our objective is to use an allocation that gives us a specified amount of information at minimum cost, then the best allocation scheme is affected by what three factors?

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If we don't have all this information, but we know the total number, we can use a simplistic allocation. This is a proportional allocation that will maintain a steady sampling fraction throughout the population.

$$n_h=\dfrac{n\cdot N_h}{N}$$

This does not take into consideration the variability within each stratum and is not the optimal choice.

If the cost of sampling from each stratum is the same, then the optimal allocation (the allocation with the lowest variances) is:

$$n_h=\dfrac{n \cdot N_h \sigma_h}{\sum\limits_{k=1}^L N_k \sigma_k}$$ (read text section 11.8 for proof)

However, if the cost of sampling differs from stratum to stratum and the total cost is:

c = c0 + c1n1 + c2n2 + ... + cLnL

where c0 is the overhead cost, ch is the cost per unit for stratum h. The optimal allocation is:

$$n_h=\dfrac{(c-c_0)N_h \sigma_h/\sqrt{c_h}}{\sum\limits_{k=1}^L N_k \sigma_k \sqrt{c_k}}$$

Note:

1. the sample size is directly proportional to Nh and σh, i.e., allocate a larger sample size to the larger and more variable stratum.
2. the sample size is inversely proportional to $$\sqrt{c_h}$$, i.e., this allocates smaller sample sizes to the more expensive stratum.

In order to use the optimal allocation, one must be able to estimate σh

Let's take a look at this in the context of the TV Example...

Application Exercise

For the TV Example, if before the advertising the firm conducts the survey they have already estimated that σ1 ≈ 5, σ2 ≈ 15, σ3 ≈ 10. Now, if the cost of obtaining an observation is about the same for the three areas , (e.g., telephone interview), then what is the optimal allocation if they want to sample 40 households?

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