Handout4

# Handout4 - Handout 4 Defining the Population Mean and...

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Handout 4 Defining the Population Mean and Population Variance In the last handout the subject of statistics was defined to be the art and science of making inferences about a population on the basis of a sample. But what sort of inferences about a population does one seek to make ? One often seeks to estimate the population mean and the population variance . So the point of taking a sample and computing a sample mean and a sample variance is to estimate the population mean and population variance, respectively. The foregoing discussion is getting ahead of itself, in the sense that it presumes that the terms population mean and population variance have already been defined – which they haven’t. There are two cases to consider : the case in which the population is finite, and the case in which the population is infinite. Finite versus Infinite Populations In the case of a finite population, say of size N, the definition of a population mean is the obvious analogy to the definition of a sample mean given in Handout 1 : the population mean is . For an infinite population this clearly makes no sense at all! Infinite populations are not just the province of pure mathematicians spinning abstract theories for the sake of abstraction itself ; the case of an infinite population is quite common in scientific settings. Suppose one is investigating the wing span ( in millimeters ) of the progeny resulting from the cross breeding of two strains of fruit fly. What is the population under investigation? The fruit flies bred by the investigator ? The fruit flies bred by all researchers in the US that year ? The fruit flies bred by anyone, anyplace to date ? If it’s a truly general scientific investigation, the population of interest is the set of all wing spans of all potential fruit flies that have been or could be bred, anytime or anyplace ; — which is, by the nature of the concept, an infinite set. And any scientific investigation of sufficient generality to be of significant enduring interest, is likely to be about an infinite population. Developing a Definition for the Case of Discrete Random Variables We will consider the problem of defining the population mean in the special case of a discrete random variable : that is, a random variable that takes on a finite number of different values. ( For example, consider the random variable X that takes on the value 1 if a coin toss comes up heads, and the value 0 if the coin toss comes up tails. This is a discrete random variable since there are only two different values the random variable takes on ; but one can, at least conceptually, keep tossing the coin from here to eternity, and hence the population of interest is infinite. )

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The goal is to develop a definition which makes sense whether the population is finite or infinite. We’re not going to just plop down an unmotivated definition !! The motivation will be provided by a consideration of how to efficiently compute a sample mean when one has so-called ‘grouped’ data. Keep in mind that the computational aspect is not what is of
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Handout4 - Handout 4 Defining the Population Mean and...

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