This chapter explores key concepts of inferential statistics, essential for drawing conclusions from data and making inferences. It explains the purpose and significance of inferential statistics in research, covering foundational concepts such as random sampling, probability distributions, and the central limit theorem, which are critical tools for statistical inference. The chapter also guides you through point and interval estimation, with a focus on calculating confidence intervals and understanding the differences between one-tailed and two-tailed intervals. Additionally, the chapter discusses hypothesis testing, explaining the difference between one-tailed and two-tailed tests, along with the concepts of Type I and Type II errors. Practical advice is provided on minimizing these errors to enhance the accuracy of statistical inferences. Examples throughout the chapter illustrate these concepts, making them more accessible and easier to apply.

The previous chapter considered the following problem: given a distribution, deduce the characteristics of samples drawn from that distribution. This chapter goes in the opposite direction: given a random sample, infer the distribution from which the sample was drawn. It is impossible to infer the distribution exactly from a finite sample. Our strategy is more limited: we propose a hypothesis about the distribution, then decide whether or not to accept the hypothesis based on the sample. Such procedures are called hypothesis tests. In each test, a decision rule for deciding whether to accept or reject the hypothesis is formulated. The probability that the rule gives the wrong decision when the hypothesis is true leads to the concept of a significance level. In climate studies, the most common questions addressed by hypothesis test are whether two random variables (1) have the same mean, (2) have the same variance, or (3) are independent. This chapter discusses the corresponding tests for normal distributions, called the (1) t-test (or difference-in-means test), (2) F-test (or difference-in-variance test), and (3) correlation test.