23  Statistical inference

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We have now learned something about random samples and about the sampling distribution of the sample mean. In the next several pages we will consider how to learn from the sample mean \(\bar X_n\) about the mean \(\mu\) of the population distribution when this is unknown. Beyond this, we will also consider what we can learn about the population variance \(\sigma^2\) from the sample variance \(S_n^2\).

Firstly, we will “learn” by constructing what are called confidence intervals. These are intervals constructed with observed sample data which are intended to capture the unknown values of quantities describing the population distribution, for example the mean \(\mu\) and the variance \(\sigma^2\). Secondly, we will learn how to formulate and test hypotheses concerning the values of population parameters \(\mu\) or \(\sigma^2\).

Once we have a handle on building confidence intervals and formulating and testing hypothesis concerning the mean or variance of a single population based on a single random sample, we will consider making comparisons between two populations when we observe a random sample from each one; these methods we will also extend to an arbitrary number of populations.