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"(ch:inf_pred_theory)=\n",
"# Theory for Inference and Prediction\n",
"\n",
"When you want to generalize your findings beyond descriptions for your collection of data to a larger setting, the data needs to be representative of that larger world. For example, you may want to predict air quality at a future time based on a sensor reading ({numref}`Chapter %s `); test whether an incentive improves the productivity of contributors based on experimental findings ({numref}`Chapter %s `); or construct an interval estimate for the amount of time you might spend waiting for a bus ({numref}`Chapter %s `). We have touched on all of these scenarios in earlier chapters, and now, in this chapter, we formalize the framework for making a prediction or an inference.\n",
"\n",
"At the core of this framework is the notion of a distribution, be it a population, empirical (aka sample), or probability distribution. Understanding the connections between these distributions is central to the basics of hypothesis testing, confidence intervals, prediction bands, and risk. We begin with a brief review of the urn model, first introduced in {numref}`Chapter %s `, then introduce formal definitions of hypothesis tests, confidence intervals, and prediction bands. We use simulation in our examples, including the bootstrap as a special case. We wrap up the chapter with formal definitions of expectation, variance, and standard error--essential concepts in the theory of testing, inference, and prediction."
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