Interest in a single population parameter
Although we sometimes want to fully discover a population distribution, in most applications we are mainly interested in a single parameter of this distribution. Estimating a population parameter is more focused and there is a well-established body of statistical theory to support it.
To estimate a population parameter from a sample, we usually use the corresponding sample statistic.
Some scenarios are presented below where estimating a population proportion or mean (using a corresponding sample statistic) might be justified.
Proportion of farmers growing sorghum
Because of possible climate change, some farmers in a mainly maize-growing district of Africa have started growing sorghum in some of their fields.
The Ministry of Agriculture is interested in estimating the proportion of farmers who are currently growing sorghum. It is known that there are 3535 farms in the district but collecting information from all farms would be very expensive, so data are collected from a random sample of 100 farms.
The sample proportion, p, of farmers growing sorghum is an estimate of the unknown population proportion, π.
It was found that 37 out of the 100 sampled farms used sorghum, so our estimate of the proportion using the crop in the whole district is
= p = 37/100 = 0.37
Note: A 'hat' is often drawn over a parameter to denote an estimate of it.
Average area growing sorghum
Studies often have multiple objectives. In the survey described above, the Ministry is also interested in the current average area under sorghum for those farmers who are currently growing that crop. Data collected from the 37 farms in which sorghum is used can provide an estimate.
The sample mean area devoted to sorghum in farms where sorghum is grown, ,
is an estimate of the unknown population mean, µ.
In the 37 farms growing sorghum, the average area devoted to sorghum was 0.68 hectares, so we estimate the average area in sorghum-growing farms in the whole district to be
=
= 0.68 hectares
Total area growing sorghum in district
The sample data above can also be used to estimate the total area growing sorghum in all 3535 farms in the district. The total area in the 100 sampled farms was
37 × 0.68 = 26.16 hectares
We therefore estimate the total area in the whole 3535 farms in the district to be
26.16 × 3535/100 = 889 hectares
Comparing beers
A brewery wishes to assess the effect of an additive on the alcohol content of its premium beer. Several batches are brewed, both with and without the additive, and the alcohol content of each batch is recorded. We therefore have two random samples; one from a hypothetical population of alcohol contents of batches brewed with the additive, and the other from the corresponding hypothetical population without the additive.
We could estimate the two population distributions and attempt to compare them. However the brewer's interest is mostly about whether the alcohol content increases or decreases with the additive, and this can be phrased in terms of the two population means — are they different?
The sample means provide estimates of the corresponding population means and are the basis for answering the question.
The difference in sample means is an estimate of the difference between the unknown population means.