The normal distribution is simply a distribution with a certain shape. It is "normal" because many things have this same shape. The normal distribution is the “bell-shaped distribution” that describes how so many natural, machine-made, or human performance outcomes are distributed. If you ever took a class when you were "graded on a bell curve", the instructor was fitting the class' grades into a normal distribution— not a bad practice if the class is large and the tests are objective, since human performance in such situations is normally distributed. This chapter will discuss the normal distribution and then move onto a common sampling distribution, the t-distribution. The t-distribution can be formed by taking many samples (strictly, all possible samples) of the same size from a normal population. For each sample, the same statistic, called the t-statistic, which we will learn more about later, is calculated. The relative frequency distribution of these t-statistics is the t-distribution. It turns out that t-statistics can be computed in a number of different ways on samples drawn in a number of different situations and still have the same relative frequency distribution. This makes the t-distribution useful for making many different inferences, so it is one of the most important links between samples and populations used by statisticians. In between discussing the normal and t-distributions, we will discuss the central limit theorem. The t-distribution and the central limit theorem give us knowledge about the relationship between sample means and population means that allows us to make inferences about the population mean.
The way the t-distribution is used to make inferences about populations from samples is the model for many of the inferences that statisticians make. Since you will be learning to make inferences like a statistician, try to understand the general model of inference making as well as the specific cases presented. Briefly, the general model of inference-making is to use statisticians' knowledge of a sampling distribution like the t-distribution as a guide to the probable limits of where the sample lies relative to the population. Remember that the sample you are using to make an inference about the population is only one of many possible samples from the population. The samples will vary, some being highly representative of the population, most being fairly representative, and a few not being very representative at all. By assuming that the sample is at least fairly representative of the population, the sampling distribution can be used as a link between the sample and the population so you can make an inference about some characteristic of the population.
These ideas will be developed more later on. The immediate goal of this chapter is to introduce you to the normal distribution, the central limit theorem, and the t-distribution.
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