# Is it better to estimate a population parameter directly or via a proportion?

Situation

I'll describe the situation through a made-up example. Suppose there is a village of 1,000 households where people collectively bought $$F$$ kg of food over some period of time, and ended up wasting some proportion of it ($$W$$) equal to $$F_w$$ kg (where $$F_w = F \times W$$). Assume we know there are no households that bought no food ($$F_i$$ is always positive).

Imagine I am the Secretary of State for the Environment who wants to charge the village mayor a fixed penalty for each kg of food wasted. The total amount, $$F$$, is known, but $$F_w$$ is unknown and needs to be estimated.

I can't survey every household so I need to take a sample. I select $$n$$ random households and for each household record the amount of food bought ($$F_i$$) and wasted ($$F_{wi}$$).

I now have two options for estimating the total $$F_w$$ from the sample.

Option 1: direct estimation

Here, the estimate would be based on the average per household extrapolated to all 1,000 households:

$$1000\times\left(\overline{F}_w\pm \frac{s}{\sqrt{n}}\right)$$

where $$s$$ is the sample standard deviation.

Option 2: estimation via the proportion

Here, the average proportion wasted would be worked out from the two measures: $$\overline{W} = \frac{\sum F_{wi}}{\sum F_i}$$ and extrapolated to the total population parameter $$F$$:

$$F\left( \overline{W} \pm \text{std. err.} \right)$$

The standard error above is not specified as there are several definitions of it for a proportion estimate. It could be based, for example, on the Wilson or the Agresti-Coull method.

Question

Which method would be most appropriate and recommended for this type of situation?

• Option 2 has problems. For instance, what estimate would it produce when one or more families buy no food $(F_i=0)$? – whuber Apr 23 '19 at 15:49
• @whuber this is only a made up example; but assume everyone buys at least some food. – Mihael Apr 23 '19 at 15:52
• That doesn't change the point: Option 2 is terrible. Just consider cases where some people buy little food and others buy a great deal. – whuber Apr 23 '19 at 15:53
• @whuber I don't disagree, can you show it so that it can be grasped intuitively. I honestly fail to see it. – Mihael Apr 23 '19 at 16:24