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 “graded on a bell curve”, the instructor was fitting the class’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.
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