Conditions for Inference

Chapter 8 dealt with inference for categorical data and this chapter deals exclusively with inference for continuous quantitative data. As in Chapter 8, we'll learn to estimate a population parameter using a confidence interval, and we'll learn to conduct hypothesis tests. When we first learned about the distribution of continuous data, we learned that every variable or column of data can be summarized in terms of its center, shape, and spread. There are formal methods that one can use to infer specific population characteristics of center, shape, and spread based on a single sample of data. In this chapter we'll deal exclusively with the mean.

Just as with categorical data, we rely on the reasonableness of specific assumptions about the conditions of inference. Because the conditions are so fundamental to the process of generalizing from sample data, we begin each of the next several chapters by asking, "What are we assuming?"

Our first two assumptions are the same as before: observations in the sample are independent of one another and the sample is small in comparison to the entire population (usually 10% or less). We should also be satisfied that the sample is probabilistic and therefore can reasonably represent the population. If we know that our sample data violates one or more of these conditions, then we should also understand that we'll have little ability to make reliable inferences about the population mean.

The first set of procedures that we'll encounter in this chapter relies on the sampling distribution of the mean, which we introduced in Chapter 7. We can apply the t-distribution reliably if the variable of interest is normally distributed in the parent population, or if we can invoke the Central Limit Theorem because we have ample data. If the parent population is highly skewed, we might need quite a large n to approach the symmetry of a t-distribution; the more severe the skewness of the population, the larger the sample required.