Final answer:
Random variables are categorized as discrete if their values are countable, such as the number of heads when flipping a coin or the number of correct answers on a test. They are continuous if their values are the result of measurement, like an average age or length in inches. In the given scenarios, flipping a coin, selecting a student, and choosing a mutual fund involve discrete random variables, while measuring average age and length of a newborn are examples of continuous random variables.
Step-by-step explanation:
When identifying if a random variable (RV) is discrete or continuous, it's important to consider if the values the RV can take are countable or measurable. Below is the analysis for each given scenario:
- Flip a coin three times: Let X be the total number of heads. This is a discrete RV because the total number of heads that can be obtained by flipping a coin three times is countable (0, 1, 2, or 3).
- Randomly select a student who took a true/false test with 100 questions. Let X be the number of questions answered correctly. As the number of correct answers can only be whole numbers between 0 and 100, X is a discrete RV.
- Randomly select a mutual fund. Let X be the number of companies in the portfolio. Again, since you count the number of companies, X is a discrete RV.
- Randomly select 50 community college students. Let X be the exact average age of the group. X is a continuous RV because the average age is a result of measurement and can take an infinite number of possible values within a range.
- Randomly select a newborn baby. Let X be the exact length in inches. Since the length is measured and not counted, X is a continuous RV.