Graphical Analysis In Exercises 9–12, determine whether the graph on the number line represents a discrete random variable or a continuous random variable. Explain your reasoning.


The attendance at concerts for a rock group

Verified step by step guidance

1. Step 1: Understand the difference between discrete and continuous random variables. A discrete random variable takes on countable values, such as integers or specific points, while a continuous random variable can take on any value within a range, including fractions and decimals.
2. Step 2: Analyze the graph provided. The number line shows specific attendance values marked as distinct points (e.g., 40,000, 45,000, 50,000). These points are countable and do not include intermediate values between them.
3. Step 3: Consider the context of the problem. Attendance at concerts is typically recorded as whole numbers because it represents the count of people, which cannot be fractional (e.g., you cannot have 45,000.5 people attending).
4. Step 4: Based on the graph and the nature of attendance data, conclude that the variable is discrete. The graph represents specific, countable values rather than a continuous range.
5. Step 5: Explain the reasoning clearly: The attendance at concerts is a discrete random variable because it consists of distinct, countable values (whole numbers) and does not include fractions or decimals.

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. In the context of the attendance at concerts, this means that the number of attendees can only be whole numbers (e.g., 40,000, 45,000) and cannot be fractional. The graph shows specific points on a number line, indicating that only certain attendance figures are possible.

Continuous Random Variable

A continuous random variable, in contrast, can take on an infinite number of values within a given range. This means it can include fractions and decimals, representing measurements that can vary smoothly. For example, if attendance could be measured in fractions of a person (which is not realistic), it would be considered continuous.

Graphical Representation

Graphical representation is a visual way to display data, which can help in understanding the nature of the variable being analyzed. In this case, the number line with marked points indicates that the attendance figures are specific and separate, reinforcing the idea that the variable is discrete. The presence of distinct points rather than a continuous line further clarifies this distinction.