Discrete Variables and Continuous Variables In Exercises 13–18, determine whether the random variable x is discrete or continuous. Explain.
Let x represent the fitted hat sizes of members of a softball team.

Verified step by step guidance

1. Understand the definitions of discrete and continuous variables: A discrete variable is one that can take on a countable number of distinct values (e.g., integers or specific categories), while a continuous variable can take on any value within a given range, often involving measurements (e.g., height, weight).
2. Identify the nature of the variable x in the problem: Here, x represents the fitted hat sizes of members of a softball team.
3. Consider how hat sizes are typically measured: Hat sizes are usually represented by specific numerical values or categories (e.g., 6, 6.5, 7, 7.5, etc.), which are countable and not infinitely divisible.
4. Determine whether the values of x are countable or continuous: Since hat sizes are chosen from a set of predefined, countable options, x is a discrete variable.
5. Conclude and explain: The random variable x is discrete because it represents a countable set of specific hat sizes rather than a continuous range of values.

Key Concepts

Discrete Variables

Discrete variables are those that can take on a countable number of distinct values. They often represent items that can be counted, such as the number of students in a class or the number of hats. In the context of the question, fitted hat sizes can be categorized into specific sizes (e.g., small, medium, large), making them discrete.

Continuous Variables

Continuous variables, in contrast, can take on an infinite number of values within a given range. They are typically measurements that can be subdivided into finer increments, such as height or weight. If the variable were to represent something like the exact measurement of a hat size in inches, it would be considered continuous.

Random Variables

A random variable is a variable whose values are determined by the outcomes of a random phenomenon. It can be classified as either discrete or continuous based on the nature of its possible values. In this case, the fitted hat sizes of team members represent a random variable that can be analyzed to determine its classification as discrete.