Which of the following random variables is continuous?
A. the number of seniors in a college
B. the number of gold medals won at the 2012 Summer Olympics by athletes from Germany
C. the number of schools in a city
D. the number of registered physicians in the United States
E. the amount of gasoline used in the Unites States in 2012
The correct answer and explanation is:
The correct answer is E. the amount of gasoline used in the United States in 2012.
Explanation:
A random variable is considered continuous if it can take on any value within a given range, including fractional values. It contrasts with a discrete random variable, which takes on specific, distinct values. To determine whether a random variable is continuous or discrete, we need to analyze the nature of the variable.
- A. the number of seniors in a college: This is a discrete random variable because it represents a count of people, and people can only be counted in whole numbers (you can’t have a fraction of a person).
- B. the number of gold medals won at the 2012 Summer Olympics by athletes from Germany: This is also a discrete random variable. The number of gold medals is a count and, therefore, takes distinct values such as 0, 1, 2, and so on.
- C. the number of schools in a city: Similar to the previous examples, this is a discrete random variable. The number of schools is a whole number and cannot take fractional values.
- D. the number of registered physicians in the United States: This is a discrete random variable, as it represents a count of people (physicians), which can only be expressed as whole numbers.
- E. the amount of gasoline used in the United States in 2012: This is a continuous random variable. Gasoline usage can take any value within a range and can be measured in units such as gallons, liters, or cubic meters, allowing for fractional values (e.g., 10.5 million gallons or 10.75 million gallons).
In summary, a continuous random variable can take on any value within a range, including fractional values, while discrete variables can only take specific, whole-number values. Hence, the amount of gasoline used is continuous because it can be measured in infinitely precise units, whereas the other options are discrete, as they represent counts of things.