How Many Villages Make 500?

A nutrition survey uses cluster sampling: it randomly selects villages and interviews 15 households in every selected village.

Households in the same village tend to answer alike, because they share markets, water sources and local habits. So 15 households from one village carry less new information than 15 households picked independently from across the whole region. Survey planners put a number on this.

• The intraclass correlation $\rho$, between 0 and 1, measures how alike households in the same cluster are. $\rho = 0$ means no more alike than strangers; $\rho = 1$ means every household in a village gives the same answer. Past surveys in this region give $\rho = 0.06$.
• For clusters of $m$ households each, the design effect is

$$\text{DEFF} = 1 + (m - 1)\,\rho$$

• A cluster sample of $n$ households in total is worth an effective sample size of

$$n_{\text{eff}} = \frac{n}{\text{DEFF}}$$

which is the number of independently chosen households (a simple random sample) that would estimate the average just as precisely.

The funder requires precision at least as good as a simple random sample of 500 households. The original plan was 34 villages, chosen because $34 \times 15 = 510$ households is more than 500.

What is the smallest number of villages the survey must select to meet the funder's requirement?

Give a whole number of villages.