In the context of hypothesis testing, ?? (also denoted by alpha, α) is the significance level, which is a pre-determined threshold for rejecting the null hypothesis. The significance level is the maximum probability of rejecting the null hypothesis when it is actually true (i.e., the probability of a type I error).?

In other words, if we set the significance level to ??, then we reject the null hypothesis only if the p-value (the probability of obtaining a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true) is less than or equal to ??.?

The choice of ?? depends on the researcher's desired trade-off between the risk of a type I error (rejecting the null hypothesis when it is actually true) and the risk of a type II error (failing to reject the null hypothesis when it is actually false). A smaller ?? corresponds to a smaller risk of a type I error but a larger risk of a type II error, while a larger ?? corresponds to a larger risk of a type I error but a smaller risk of a type II error.?

For example, if we set ?? = 0.05, then we are willing to tolerate a 5% chance of rejecting the null hypothesis when it is actually true. This is a commonly used significance level in many fields.?

The books [BT] Introduction to Probability and [DS] Probability and Statistics provide more detailed discussions on the significance level and its usage in hypothesis testing, including how to choose an appropriate ?? level for a given problem.

continue to explain in detail based on the book

Sure, I'd be happy to provide more detail based on the books [BT] Introduction to Probability and [DS] Probability and Statistics.

In hypothesis testing, the null hypothesis H0 is typically the hypothesis that we want to test and may represent the status quo or some default assumption, while the alternative hypothesis H1 is the hypothesis that we want to accept if there is enough evidence to reject H0.?

The decision to reject or not reject H0 is made based on the observed data and the pre-determined significance level ??. If the p-value is less than or equal to ??, then we reject H0 in favor of H1, otherwise we do not reject H0.?

The significance level ?? can be thought of as the "maximum tolerable probability of making a type I error," or the probability of rejecting H0 when it is actually true. The choice of ?? depends on the researcher's desired trade-off between the risk of a type I error and the risk of a type II error.

For example, if we set ?? = 0.05, then we are willing to tolerate a 5% chance of making a type I error. This means that if H0 is true, we will still reject it 5% of the time based on the observed data, which is a tolerable rate of error for many applications.?

The book [BT] Introduction to Probability also introduces the concept of power, which is the probability of correctly rejecting H0 when it is actually false (i.e., the probability of making a correct decision when H1 is true). Power is a function of the sample size, the effect size (the difference between the true value of the parameter under H0 and H1), and the significance level ??.?

By increasing the sample size or the effect size, we can increase the power of the test (i.e., increase the probability of correctly rejecting H0 when it is actually false). However, this may also increase the risk of a type I error if ?? is not adjusted accordingly.?

The book [DS] Probability and Statistics provides more detailed discussions on the concepts of type I and type II errors, including how to calculate the probabilities of these errors, and how to choose an appropriate significance level and power for a given problem. It also covers various types of hypothesis tests, such as tests for means, variances, proportions, and goodness of fit, as well as nonparametric tests and tests for independence.