Problem 32 - Basic Statistics for Business & Economics

Question

For each of the following, indicate whether the random variable is discrete or continuous. a. The length of time to get a haircut. b. The number of cars a jogger passes each morning while running. c. The number of hits for a team in a high school girls' softball game. d. The number of patients treated at the South Strand Medical Center between 6 and 10 p.m. each night. e. The distance your car traveled on the last fill-up. f. The number of customers at the Oak Street Wendy's who used the drive- through facility. g. The distance between Gainesville, Florida, and all Florida cities with a population of at least 50,000 .

Step-by-Step Solution

Answer

(a) Continuous, (b) Discrete, (c) Discrete, (d) Discrete, (e) Continuous, (f) Discrete, (g) Continuous.

Step 1: Discrete vs. Continuous Variables

A discrete random variable can only take a countable number of distinct values, such as whole numbers. On the other hand, a continuous random variable can take any value within a given range and is typically measurable.

Step 2: Analyze (a) Length of time to get a haircut

Time is typically measured and can take any value within a range, such as 15 minutes, 15.5 minutes, or even 15.75 minutes. Because it can take an infinite number of possible values within a range, it is a continuous random variable.

Step 3: Analyze (b) Number of cars a jogger passes

The number of cars is counted in whole numbers (such as 0, 1, 2, etc.), which means it is a discrete random variable.

Step 4: Analyze (c) Number of hits in a softball game

The number of hits in a game can only be whole numbers and is countable, making it a discrete random variable.

Step 5: Analyze (d) Number of patients treated

Again, as the number of patients treated is countable and can only take whole numbers, it is a discrete random variable.

Step 6: Analyze (e) Distance car traveled on last fill-up

Distance can be measured to fractions of a unit (like kilometers or miles), making it possible to take an infinite number of values within a range. Thus, it is a continuous random variable.

Step 7: Analyze (f) Number of customers using drive-through

The number of customers is counted in whole numbers, making it a discrete random variable.

Step 8: Analyze (g) Distance between cities

Distance, much like in (e), is a measure that can take infinite possible values within a range, so it is a continuous random variable.

Key Concepts

Discrete Variables

Discrete variables are those that can only take on a finite or countably infinite number of distinct values. These values are often whole numbers or integers. With discrete variables, you usually count the instances to determine the variable's value. For example:

• Number of cars passed during a morning jog.
• Number of hits in a softball game.
• Number of patients treated at a medical center during a specific time period.
• Number of customers using the drive-through.

All these examples highlight how discrete variables represent things that can be counted distinctly without taking fractional values. They are essential in cases where outcomes are categorical or quantifiable in separate units.

Continuous Variables

Continuous variables differ from discrete variables as they can take on any value within a given range. These variables are often measured rather than counted, which means they can include fractional and decimal values. For example:

• The length of time required to get a haircut.
• The distance a car travels after a fill-up.
• The distance between cities.

Continuous variables offer a wide range of potential values since they are not restricted to integers. Instead, they can have infinite possibilities, including rational and decimal numbers. This makes them crucial for measurements where precision and variability are important factors.

Probability Distributions

Probability distributions are mathematical functions that give the probabilities of occurrence of different possible outcomes for an event. There are distinct types of probability distributions for discrete and continuous variables. For discrete variables, you might find a probability mass function (PMF). This function assigns a probability to each possible value that the discrete variable can assume. An example would be the number of cars passed while running, where each countable number has its own set probability. In contrast, for continuous variables, we use a probability density function (PDF). This function describes the likelihood of a continuous random variable taking a particular range of values. For instance, variable time to get a haircut doesn’t have individual probabilities but rather probabilities over intervals of time. Understanding how these distributions work helps in analyzing outcomes, making predictions, and making informed decisions based on the likelihood of various events.