Sampling joke: “If you don’t believe in random sampling, the next time you have a blood test, tell the doctor to take it all.” At the beginning of Chapter 7 we introduced the ideas of population vs. sample and parameter vs. statistic. We build on this in the current chapter. The key concept in this chapter is that if we were to take different samples from a distribution and compute some statistic, such as the sample mean, then we would get different results.

The last two chapters have covered the basic concepts of estimation. In Chapter 9 we studied the problem of giving a single number to estimate a parameter. In Chapter 10 we looked at ways to give an interval that we believe will include the true parameter. In many applications, we want to ask some very specific questions about the parameter(s).

We begin this chapter with a review of hypothesis testing from Chapter 12. A hypothesis is a statement about one or more parameters of a model. The null hypothesis is usually a specific statement that encapsulates “no effect.” For example, if we apply one of the two treatments, A or B, to volunteers we may be interested in testing whether the population mean outcomes are equal.

Up to this point we have been talking about what are often called frequentist methods, because a statistical method is based on properties of its long-run relative frequency. With this approach, the probability of an event is defined as the proportion of times the event occurs in the long run. Parameters, that is values that characterize a distribution, such as the mean and variance of a normal distribution, are considered fixed but unknown.

The provincial coinage was transformed during the new regime of Augustus and the adoption of his portrait. Roman interventions, however, were rare and localised, except for Nero.

The diverse system of provincial city coinage saw the appearance of many personal names, including those of women, and the coinage was controlled mostly by the city elites.

We are often interested in how one or more predictor variables are associated with some outcome or response. We might postulate that the outcome $Y$ depends on the predictors $x_1,x_2,\dots ,x_p$ through some function.

In statistics, we are often interested in some characteristics of a population. Maybe we are interested in the mean of some measurable characteristic, or maybe we are interested in the proportion of the population that have some property. In all but the simplest cases, the population is so large that it is impossible, or at least impractical, to take the measurement on every item in the population. We therefore have to settle on taking a sample and measuring those units selected for this sample.

Silver coinage developed accompanied by locally produced silver. Gold was introduced in the late first century bce. Both were reformed by Nero, and the system eventually collapsed.

The ideas of the Second Sophistic were reflected in Asia. A new method of production was introduced. Small denominations were discontinued. The cities struggled to recognise Hadrian’s lover Antinous.