Abstract
In the last chapter, we discussed the first goal of statistics: summarization. Summarization of data is an incredibly important aspect of statistics, but our goal in summarizing sample data is not usually to simply to report the characteristics of a sample. Instead, we are usually interested in using our sample to draw some conclusions about the population from which the sample was drawn, and to place limits on the conclusions we can reach. This is the role of statistical inference. For example, suppose I had a sample of 50 persons drawn at random from the population and had measured the heights and weights of all of the sample members. Suppose that the mean height was 70 in., with a standard deviation of 5 in. As we discussed in the previous chapter, if the sample is random, then our sample mean (\(\bar{x}\)) and standard deviation (s) may be a good guess about the population mean (μ) and standard deviation (σ). However, it is unreasonable to expect that this sample mean would be a perfect reflection of the average height in the population, because we may have a few people in our sample who were unusually tall (or short). In other words, every sample we draw from a population is likely to have slightly different means and standard deviations. The goal of statistical inference in this example would be to attempt to quantify our uncertainty about the true mean and standard deviation in the population, given the sample data that we have. Thus, we might end-up with a statement like: we are 95 % confident that the mean height in the population is 70 in., give or take an inch. In common language, this “give or take an inch” is called the “margin of error,” as we will discuss in more detail in the next chapter.
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Lynch, S.M. (2013). Probability Theory. In: Using Statistics in Social Research. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8573-5_5
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