7.1 Introduction: Estimation Theory

So far we have developed a firm foundation in descriptive statistics. As we move forward, we will turn our attention to statistical inference. We study statistics, of course, because we have curiosity about a population we cannot study in its entirety. We must take a sample and use that information to make a generalization about a population or make a decision. There are two main focuses when we try to generalize or infer something about a population: estimation and hypothesis testing.

Estimation theory has to do with quantifying a certain aspect of a population. For example, consider the 2024 U.S. Presidential Race and the intense curiosity about the outcome. Opinion polling creates statements like this one from the Pew Research Center on April 24, 2024, “About half of voters say that, if given the chance, they would replace both candidates on the ballot.” Researchers polled U.S. adults and are reporting a point estimate in the hopes of quantifying the true proportion of U.S. adults who would replace both candidates if given the chance. How accurate is this? That is an excellent question. What does accuracy mean in this setting?

The Pew Research Center largely conducts polls via telephone or face-to-face interviews. In order to quantify the precision of their survey results, they will present the margin of error. For example, in a 2024 global attitudes survey in Canada, 1163 adults were surveyed and the percent the margin of error was calculated as plus or minus 3.4 percent. Where does this margin of error come from and what does it mean for the survey results?

There are two ideas in need of clarification here, accuracy and precision. In the discussion of these ideas, we will be lead to arguably the most important theorem in all of statistics, the Central Limit Theorem. Its use will form the foundation for our work with estimation and hypothesis testing.

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