The Chi-square test for goodness of fit is used with categorical data. It compares the observed frequencies in each category with the frequencies we would expect to see if the data followed the theoretical distribution.

In the Mann-Whitney U test, the null hypothesis is that there is no significant difference between the groups. More specifically, it states that the probability that a randomly selected value from the first group is greater than a randomly selected value from the second group is equal to 0.5.

Increasing the sample size decreases the width of the confidence interval. The larger the sample size, the more information you have, and thus the less uncertainty (which translates into a smaller standard error and narrower confidence interval).

If A and B are independent events, the probability of both occurring is P(A ∩ B) which is equal to P(A) * P(B). This is the fundamental characteristic of independent events in probability.

The power of a test is the ability of the test to detect an effect if it truly exists. It's the probability that the test correctly rejects a false null hypothesis. High power is desirable because it means the test is less likely to make a Type II error (false negative). When designing a study, it's important to choose a sample size and significance level that will provide enough power to detect an effect if one exists.

In the context of statistics, a 'population' refers to the entire set of individuals or items that we are interested in studying. This could range from all the people living in a country, to all the particles in a chemical solution, to all the sales transactions in a store, and so forth. When we collect data from every individual or item in the population, it's called a census. The information gained from a population is reliable and accurate because it accounts for everyone or everything we want to study.

An event in the context of probability refers to a possible outcome or combination of outcomes from a random experiment. For instance, if we toss a coin, the possible outcomes are 'Heads' and 'Tails'. Each of these outcomes is an event.

The probability of the union of two mutually exclusive events A and B is given by P(A ∪ B) = P(A) + P(B). This is because mutually exclusive events cannot occur at the same time, so the probability of either event A occurring or event B occurring is the sum of their individual probabilities.

The alternative hypothesis proposes that there is a difference or an effect in the population and the sample observations are influenced by some non-random cause.

To create a user-defined exception, one must usually extend the Exception class. By extending this base class, developers can define their own custom exceptions with specific behaviors, making the code more modular and maintainable.