Part 5: Inference in Practice
Now you put inference to work.
In Part 4, you learned the theory — the Central Limit Theorem, confidence intervals, hypothesis testing. You understand the framework. But so far, the examples have been relatively tidy: testing one mean, constructing one interval, working through the logic step by step. Real statistical work is messier than that. You need to test proportions, not just means. You need to compare two groups, not just one group to a benchmark. You need to know whether a "statistically significant" result actually matters in practical terms. And you need backup plans for when the standard methods don't quite fit.
That's what this part of the book is about: applying the inference framework to the kinds of questions people actually ask.
Chapter 14 focuses on inference for proportions. What percentage of voters support a policy? What fraction of manufactured parts are defective? Has the graduation rate changed? Proportions come up everywhere — especially in polling, public health, and quality control — and they require their own version of confidence intervals and hypothesis tests. You'll learn the one-sample z-test for proportions and see how the margin of error you hear about in election coverage actually works.
Chapter 15 returns to means but gets more precise. You'll master the one-sample t-test, learn when to use z versus t procedures, and develop a sense for when your assumptions are solid and when they're shaky. The t-distribution is one of the workhorse tools of applied statistics, and by the end of this chapter you'll use it with confidence.
Chapter 16 is where things get really interesting: comparing two groups. Is there a difference between men and women in salary? Does the treatment group recover faster than the control group? Did the new website design perform better than the old one? You'll learn the two-sample t-test, the paired t-test, and the two-proportion z-test, and — just as importantly — you'll learn to choose the right test based on how the data was collected.
Chapter 17 tackles a question that too many statistics courses ignore: what does "significant" actually mean? You'll learn about statistical power, effect sizes, and the crucial difference between statistical significance and practical significance. This chapter will make you a much sharper reader of published research, because you'll understand why a result can be "statistically significant" and completely meaningless at the same time.
Chapter 18 offers a modern alternative. The bootstrap and permutation tests let you conduct inference without relying on mathematical formulas or distributional assumptions. Instead, you use the computer to simulate what would happen under different scenarios. These methods are increasingly popular in data science and machine learning, and they beautifully reinforce the logic of inference — even if you find the formulas intimidating.
By the end of Part 5, you'll have a complete toolkit for the most common inference problems. You'll know which test to use, when to use it, and how to interpret the results honestly.