Remember that the sample you are using to make an inference about the population is only one of many possible samples from the population. Briefly, the general model of inference-making is to use statisticiansвЂ™ 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. 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. 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. 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. In between discussing the normal and t-distributions, we will discuss the central limit theorem. 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. It turns out that t-statistics can be computed a number of different ways on samples drawn in a number of different situations and still have the same relative frequency distribution. The relative frequency distribution of these t-statistics is the t-distribution. For each sample, the same statistic, called the t-statistic, which we will learn more about later, is calculated. The t-distribution can be formed by taking many samples (strictly, all possible samples) of the same size from a normal population. This chapter will discuss the normal distribution and then move on to a common sampling distribution, the t-distribution. If you ever took a class when you were вЂњgraded on a bell curveвЂќ, the instructor was fitting the classвЂ™s 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. The normal distribution is the bell-shaped distribution that describes how so many natural, machine-made, or human performance outcomes are distributed. It is normal because many things have this same shape. The normal distribution is simply a distribution with a certain shape.

Standard normal table left tail