Exploration and description / Formulae

Let us now extend these calculations to estimate the total number of cattle in the sub-region.

Let T₁ ... T_n be the estimated totals of cattle in the n dip-tank areas sampled, and M₁ ... M_n be the numbers of homesteads in these dip-tank areas.

As we have already seen, the estimated total T_i for the i^th dip-tank area can be written: T_i = _i M_i (i = 1... n). Also, the variance of T_i is given by S_i² = M_i (M_i-m_i) s_i²/m_i, where, as before, m_i is the number of homesteads sampled in dip-tank area i, and s_i² is the variance of this sample.

An unbiased estimate of the average number of animals per homestead across the n sampled dip-tank areas (n=4 in Lobamba - see three pages back) can then be written as = (T₁+...+T_n) / (M₁+...+M_n).

Suppose that there are N dip-tank areas in the sub-region (N=13 in Lobamba) and that the number of homesteads with livestock M_i is also known in each of the dip-tank areas that were not sampled. Then an unbiased estimate of the total for the sub-region, assuming that the sampled dip-tank areas were selected at random, is: T = (M₁+...+M_N)_.