Hypothesis testing is an important concept in probability and statistics that helps researchers make decisions based on data. It is used to determine whether a particular hypothesis about a population is true or not, based on the evidence provided by a sample.

The process of hypothesis testing involves several steps. First, the researcher must formulate a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis is the hypothesis that is assumed to be true unless there is evidence to the contrary. The alternative hypothesis is the hypothesis that the researcher wants to test, and it represents a departure from the null hypothesis.

Next, the researcher collects data and calculates a test statistic based on the sample. The test statistic is a numerical measure that summarizes the evidence provided by the data in favor of one hypothesis or the other. The choice of test statistic depends on the type of data being analyzed and the hypothesis being tested.

Once the test statistic has been calculated, the researcher must determine the p-value, which is the probability of obtaining a test statistic as extreme or more extreme than the one calculated, assuming that the null hypothesis is true. The p-value is used to make a decision about whether to reject or fail to reject the null hypothesis. If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis. If the p-value is greater than the significance level, then the null hypothesis is not rejected.

It is important to note that hypothesis testing does not prove that the alternative hypothesis is true. Instead, it provides evidence for or against the alternative hypothesis based on the sample data. Additionally, hypothesis testing is subject to certain assumptions, such as the assumption that the sample is representative of the population and that the data are independent and identically distributed.

The books [BT] Introduction to Probability and [DS] Probability and Statistics provide a comprehensive overview of hypothesis testing, including different types of tests, such as t-tests and ANOVA, and how to interpret the results.