Suppose that I have the following sample
| $$\text{Farmer}_{\text{ID}}$$ | $$\text{Farmer}_{\text{Size}}$$ | $$\text{Range}$$ |
|---|---|---|
| $1$ | $27$ | $a) \leq 100$ |
| $2$ | $82$ | $a) \leq 100$ |
| $3$ | $91$ | $a) \leq 100$ |
| $4$ | $18$ | $a) \leq 100$ |
| $5$ | $320$ | $b) > 100$ |
| $6$ | $205$ | $b) > 100$ |
| $7$ | $92$ | $a) \leq 100$ |
| $8$ | $48$ | $a) \leq 100$ |
| $9$ | $180$ | $b) > 100$ |
| $10$ | $16$ | $a) \leq 100$ |
and I have some additional information about the two groups, where $87\%$ of the farmers are in the group $a) \leq 100$ and $13\%$ are in group $b) > 100$. How I could calculate a weighted average farm size?
Base in this sample the simple mean of each group is
Group $a \leq 100$: $$\frac{27 + 82 + 91 + 18 + 92 + 48 + 16}{7}\approx 53.4$$
Group $b > 100$: $$\frac{320 + 205 + 180}{3}\approx 235$$
Now if I use the additional information as weights $$\mu = 0.87 \times 53.4 + 0.13 \times 235 \approx 77$$
this makes any sense?