Statistical inference
The term statistical inference describes statistical techniques that obtain information about a population parameter (or parameters) based on a single random sample from that population. There are two different but related types of question about the population parameter (or parameters) that we might ask:
What parameter values would be consistent with the sample data?
This branch of inference is called estimation and its main tool is a confidence interval. We described confidence intervals in the previous chapter.
A biologist is studying broad-tailed hummingbirds (Selasphorus platycerus), a common bird in the western USA. Incubation times were recorded for 18 eggs from nests in mountain country above 8000 feet. These values are be modelled as a random sample from the hypothetical population of incubation times for this type of bird in the same area.
The sample mean (17.83 days) is a point estimate of the population mean, and a 95% confidence interval can also be found.
Are the sample data consistent with some statement about the parameters?
This branch of inference is called hypothesis testing and is the focus of this chapter.
It is known from much larger studies that the mean incubation time of eggs from broad-tailed hummingbirds in lower elevations is 16.5 days.
The biologist must assess whether the data are consistent with the statement (hypothesis) that the underlying population mean for birds in mountain country above 8000 feet is also 16.5 days.
Uncertainty and strength of evidence
When we studied parameter estimation, we saw that a population parameter cannot be determined exactly from a single random sample — there is a 5% chance that a 95% confidence interval will not include the true population parameter.
In a similar way, a single random sample can rarely provide enough information about a population parameter to allow us to be sure whether or not any statement about that parameter will be true. The best we can hope for is an indication of the strength of the evidence against the statement.
The remainder of this chapter explains how this evidence is obtained and reported.