Sampling from a population
Sampling from an underlying population (whether finite or infinite) gives us a mechanism to explain the randomness of data. The underlying population also gives us a focus for generalising from our sample data — the distribution of values in the population is fixed and does not depend on the specific sample data.
Unknown population
The practical problem is that the population underlying most data sets is unknown. Indeed, if we fully knew the characteristics of the population, there would have been little point in collecting the sample data!
Even though our model implies that we could take many different samples from the population,
In practice we only have a single sample.
However this single sample does throw light on the population distribution. In later chapters, we will go into much more detail about how to estimate population characteristics from a sample.
Detection of concealed weapons at LAX
For many years, authorities have been concerned about the risk of bombs and other weapons being taken onto scheduled flights by terrorists. Tests of airline and airport security are regularly conducted by the American Federal Aviation Administration (FAA).
Although no recent information is available, the Gainesville Sun on 11 December, 1987 reported the results from one series of tests, in which FAA inspectors tried to carry mock weapons on board planes at Los Angeles International Airport (LAX) 100 times. Only 72 of these were detected by security guards.
We are interested in the chances of a concealed weapon being discovered — the symbol π denotes the probability of detection.
The detection of the 100 mock weapons can be treated as a sample from a hypothetical infinite population in which a proportion π would be detected, but π is an unknown value. We are more interested in this longer-term proportion than in the proportion in our specific sample.
The sample proportion detected, p = 0.72, however throws some light on the likely value of π.