Statistical Inference Is Always Conditional

Under a wide range of common conditions, good data generally leads to good conclusions. Learning to reason statistically involves knowing what those conditions are, knowing what constitutes good data, and knowing how to move logically from data to conclusions. If we are working with sample data, we need to recognize that we simply cannot infer one characteristic (parameter) of a population without making some assumptions about the nature of the sampling process and at least some of the other attributes of the population.

As we introduce the techniques of inference for a population proportion, we start with some basic conditions. If these conditions accurately describe the study, the conclusions we draw from the techniques are most likely trustworthy. To the extent that the situation substantially departs from the conditions, we should doubt that the results can generalize to the population. Thus, our usual approach is to ask if we can reasonably assume that a given set of conditions applies to a particular situation.

At a minimum, we should be able to assume the following about our sample data:

• Each observation is independent of the others. Simple random sampling assures this, and it is possible that in a non-SRS we preserve independence.

• Our sample is no more than about 10% of the population. When we sample too large a segment of the population, in essence we no longer preserve the condition of independent observations.

We'll take note of other conditions along the way, but these form a starting point. In our first example, we will look again at the pipeline safety incident data. This is not a simple random sample, even though it may be reasonable to think that one major pipeline interruption is independent of another. Because we are sampling from an ongoing nationwide process of having pipelines carry natural gas, it is also reasonable to assume that our data table is less than 10% of all disruptions that will ever occur. We'll go ahead and use this data table to represent pipeline disruptions in general.