From observed data, statistical inference infers the properties of the underlying probability distribution. For hypothesis testing, the t-test and some non-parametric alternatives are covered. Ways to infer confidence intervals and estimate goodness of fit are followed by the F-test (for test of variances) and the Mann-Kendall trend test. Bootstrap sampling and field significance are also covered.

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.