Wiki Page, Math 13 - Perez Sandbox : Module 11 - Introduction

Wiki Page
2026-01-17T21:15:10-08:00

Recall the Big Picture—the four-step process that describes a statistical investigation. It is easy to get lost in the details of each unit, so the Big Picture helps us see how the pieces fit together.

We are discussing parts of the Big Picture out of order, but soon we will have the whole picture.

Producing data—how we collect data and what considerations affect the data production process. In Unit 6 we will discuss surveys and experiments–and the important role of random selection and random assignment.

Exploratory data analysis—tools for representing data with graphs and numerical summaries and tools for exploring relationships between two variables. We discussed this in Units 2-4.

Our eventual goal is inference—drawing reliable conclusions about the population based on a sample.

Let’s suppose we do a good job of selecting an unbiased sample. Common sense tells us that samples will vary. Unfortunately, when looking at a particular sample, we never know how much it differs from the population. This can affect our confidence in our conclusions, so we need a way to talk about (and hopefully quantify) our uncertainty. Are we 95% confident? 80% confident? How likely are our sample results?

This uncertainty is where probability comes into the picture. If we randomly select a sample from the population, we can use probability to describe the uncertainty. We will learn more about the importance of random selection and random assignment in Unit 6. For now let's use an example to illustrate the Big Picture at this point in the course.

EXAMPLE Death penalty

Suppose we are interested in estimating the percentage of U.S. adults who favor the death penalty. We choose a random sample of 1,200 U.S. adults and ask their opinion: either in favor of or against the death penalty. We find that 744 of the 1,200, or 62%, are in favor.

Our goal is to make an inference about the entire population of U.S. adults regarding the death penalty on the basis of the opinions of only 1,200 of them.

Can we conclude that 62% of the population favors the death penalty? Another random sample could give a very different result, so we are uncertain. To describe our uncertainty (to quantify it), we use probability to describe the likelihood that our sample is within a desired level of accuracy. For example, probability can answer the question, How likely is it that our sample estimate is no more than 3% from the true percentage of all U.S. adults who are in favor of the death penalty?

In this module, we discuss probability more generally. Then we will develop the probability machinery that underlies statistical inference.

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