Problem 4 - Basic Statistics for Business & Economics

Question

Which of these variables are discrete and which are continuous random variables? a. The number of new accounts established by a salesperson in a year. b. The time between customer arrivals to a bank ATM. c. The number of customers in Big Nick's barber shop. d. The amount of fuel in your car's gas tank. e. The number of minorities on a jury. f. The outside temperature today.

Step-by-Step Solution

Answer

a-discrete, b-continuous, c-discrete, d-continuous, e-discrete, f-continuous

Step 1: Define Discrete and Continuous Variables

A **discrete random variable** is a variable that can take on a countable number of distinct values. Examples include the number of heads in coin tosses or the number of customers. A **continuous random variable** is a variable that can take on an infinite number of possible values within a given range. Examples include weight and temperature, which can vary continuously.

Step 2: Analyzing Variable a

The number of new accounts established by a salesperson in a year is discrete because you can count the number of accounts, and it doesn't take on fractional values. For example, they might establish 3, 4, or 5 new accounts, but not 3.5.

Step 3: Analyzing Variable b

The time between customer arrivals to a bank ATM is continuous because time can be measured to infinitely fine precision (e.g., 1.1 seconds, 1.12 seconds, 1.123 seconds, etc.).

Step 4: Analyzing Variable c

The number of customers in Big Nick's barber shop is discrete because you can count the exact number of customers, such as 1, 2, or 34, but not 3.5 customers.

Step 5: Analyzing Variable d

The amount of fuel in your car's gas tank is continuous because fuel can be measured in infinitely precise quantities, such as 9.75 liters, 9.751 liters, etc.

Step 6: Analyzing Variable e

The number of minorities on a jury is discrete because you count the number of individuals, which are whole numbers.

Step 7: Analyzing Variable f

The outside temperature today is continuous because temperature can vary continuously and can be measured with great precision, such as 72.3°F, 72.35°F, etc.

Key Concepts

Random Variables

Random variables are fundamental elements in statistics and probability. They represent outcomes of random phenomena, each capturing some aspect of unpredictability. Imagine you are tossing a coin. The result of heads or tails can be represented as a random variable. Depending on the situation, random variables can be classified into two main types: discrete and continuous. This classification is based on the type of values a variable can assume and how these values can be mathematically described.

Random variables play a vital role in modeling and analyzing random processes. They allow us to use mathematical models to predict or describe these processes. For example, a random variable can help us understand the variability in daily temperatures or in the count of people visiting a store. By identifying and analyzing random variables, statisticians can derive meaningful insights from data and make informed decisions.

Discrete Variables

A discrete random variable is characterized by a set of distinct, countable outcomes. This means the possible values the variable can take are isolated points on a number line. For example, the number of customers at a store, test scores, or the number of new accounts opened by a salesperson. These outcomes are often expressed as whole numbers.

Discrete variables are particularly useful when dealing with data that involves counting items or occurrences. They are easy to visualize using graphs like bar charts or pie charts, which display the frequency of each possible outcome.

• Example of Discrete Variables: The number of customers in a store, number of students in a class.
• Characteristics: Countable, Distinct values, Represent whole numbers.

Discrete variables are represented mathematically using probability mass functions. These functions provide the probability that a discrete random variable equals a particular value, helping in visualizing the likelihood of various outcomes.

Continuous Variables

Continuous random variables differ from discrete ones as they can take any value within a given range. This means their possible values are not limited to isolated points, but can include every number in a range, even fractions. Continuous variables are measured, rather than counted.

Imagine the amount of fuel in a car tank or the temperature at a specific moment. These are examples of continuous variables. They change fluidly and can be measured with varying degrees of precision. For instance, you might measure temperature to the nearest degree, half-degree, or even minute fraction of a degree.

• Example of Continuous Variables: Time intervals, Temperature.
• Characteristics: Infinite possibilities, Fractional values, Measured not counted.

In mathematical terms, continuous variables use probability density functions to define the probability of the variable falling within a certain interval. Understanding continuous variables is crucial in fields like physics and engineering, where precision measurements are central.