In the Sign Test, ties between paired samples are generally discarded and not included in the final calculation.

The Sign Test is a non-parametric test used to test the median of a distribution, or to compare the medians of two distributions.

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.

JSON (JavaScript Object Notation) is commonly used for transmitting data in social media integrations and APIs due to its lightweight and human-readable format. It is easy to parse and generate, making it a popular choice for data exchange.

The 'application' directory in CodeIgniter is primarily used for configuration and customization. It houses files related to controllers, models, views, and other application-specific components. This directory is crucial for organizing and managing the core functionality of a CodeIgniter application.

Consent management is a crucial aspect of social media integrations using OAuth. It involves handling and storing user consent preferences and managing the lifecycle of access tokens issued during the authorization process. Proper consent management enhances user privacy and security.