What type of variable can assume any value in an interval on the real line or in a collection of intervals?

A continuous random variable is defined as a type of variable that can take on any value within a given range or interval on the real number line. This means it can represent measurements or counts that can fall anywhere along a continuum, such as height, weight, temperature, or time. Unlike discrete random variables, which can only take specific, separate values (like integers), continuous random variables have an infinite number of possible values within their range.

This characteristic of continuous random variables allows for a more nuanced understanding of phenomena that can change gradually rather than in distinct steps. For example, if measuring the height of students, a continuous random variable could represent any height in centimeters, including values like 170.5 cm or 170.55 cm, rather than just whole centimeters.

In contrast, the other types of variables mentioned do not fit this definition. A discrete random variable takes on distinct, separate values (like the number of students in a class), a nominal random variable categorizes data without a structured order (such as types of fruits), and a binomial random variable specifically deals with the number of successes in a fixed number of binary trials, also yielding separate values. Thus, the correct choice reflects the nature of continuous random variables and their ability to encompass a